# Stochastic failure of cell infection post viral entry: Implications for
infection outcomes and antiviral therapy

Christian Quirouette1, Daniel Cresta1, Jizhou Li2, Kathleen P. Wilkie3,
Haozhao Liang${}^{4,\text{\textcurrency}}$, Catherine A.A. Beauchemin1,2,∗
1 Department of Physics, Ryerson University, Toronto, Canada
2 Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN,
Wako, Japan
3 Department of Mathematics, Ryerson University, Toronto, Canada
4 Nishina Center for Accelerator-Based Science (RNC), RIKEN, Wako, Japan
¤Current address: Department of Physics, University of Tokyo, Tokyo, Japan
∗ Corresponding author<EMAIL_ADDRESS>
###### Abstract

A virus infection can be initiated with very few or even a single infectious
virion, and as such can become extinct, i.e. stochastically fail to take hold
or spread significantly. There are many ways that a fully competent infectious
virion, having successfully entered a cell, can fail to cause a productive
infection, i.e. one that yields infectious virus progeny. Though many
discrete, stochastic mathematical models (DSMs) have been developed and used
to estimate a virus infection’s extinction probability, these typically
neglect infection failure post viral entry. The DSM presented herein
introduces parameter $\gamma\in(0,1]$ which corresponds to the probability
that a virion’s entry into a cell will result in a productive cell infection.
We derive an expression for the likelihood of infection extinction in this new
DSM, and find that prophylactic therapy with an antiviral acting to reduce
$\gamma$ is best at increasing an infection’s extinction probability, compared
to antivirals acting on the rates of virus production or virus entry into
cells. Using the DSM, we investigate the difference in the fraction of cells
consumed by so-called extinct versus established virus infections, and find
that this distinction becomes biologically meaningless as the probability of
extinction approaches 100%. We show that infections wherein virus is release
by an infected cell as a single burst, rather than at a constant rate over the
cell’s infectious lifespan, has the same probability of infection extinction,
despite previous claims to this effect [28]. Instead, extending previous work
by others [41], we show how the assumed distribution for the stochastic virus
burst size, affects the extinction probability and associated critical
antiviral efficacy.

##  1 Introduction

Typically, mathematical models describing the course of a virus infection
express the number of cells and infectious virions (virus particles) as real
positive numbers (continuous), and infection events as deterministic, e.g. one
virion infects 0.2 cell. By nature, however, the number of cells and virions
are whole numbers (discrete) and infection events are stochastic, e.g. one
virion will infect one cell 20% of the time. When dealing with large numbers
of infectious virions and cells, stochastic fluctuations can become
negligible, and the continuous, deterministic, mean-field approach can provide
an accurate representation of the kinetics of interest. Yet the small number
regime arises commonly. For example, antiviral therapy can reduce the
effective number of infection-capable virions to near or below unity. In such
cases, random fluctuations could have an important effect on the time course
and outcome of an infection.

Several discrete, stochastic mathematical models (DSMs) of viral infection
kinetics have been proposed to date, e.g. [7, 28, 8, 18, 41, 9, 33]. Notably,
Heldt et al. [18] predicted with an intracellular DSM that most cells
inoculated with a single infectious virion would lead to a non-productive cell
infection. But extracellular DSMs typically do not account for cell infection
failure post viral entry. When it has been included only some lethal
replication errors were represented, e.g. lethal reverse transcription errors
only [9]. In fact, there are many ways that infectious virions post cell entry
can fail to cause a cell infection that will produce fully infectious progeny.

For human immunodeficiency virus (HIV), entry begins with successful binding
to the cell surface, followed by fusion of the virus membrane with the cell
surface. Yet sometimes virions can be taken in via the endocytic route [29].
In this case, virions can fail to fuse with late endosomes and eventually be
degraded [15]. Or, if there is successful entry, virions can still be degraded
in the cytosol [38]. The virus can also fail to be imported into the nucleus
[5, 4]. In addition, fatal mutations can be acquired during reverse
transcription [19], when viral RNA is transcribed into complementary DNA
(cDNA) which is used to make typically a single copy of viral DNA for
integration. Finally, failure may also be the result of host cell mechanisms
that interfere with early replication steps [6, 21].

For influenza A virus (IAV), entry begins with successful binding to the cell
surface, followed by internalization by endocytosis. Failure to undergo fusion
with late endosomes can then occur [22, 31]. If uncoating happens, released
viral ribonucleoproteins (vRNPs) containing the viral RNA segments could be
subject to RNA degradation in the cytosol [34] or fail to enter the nucleus
[14, 31]. Degradation of one or more genome segments, following nuclear import
and prior to transcription could also occur [18]. There is also a multitude of
host cell mechanisms that can impair early steps of viral replication [39, 11,
37] and these may be yet more potential sources for failure.

These are just some of the ways in which otherwise fully physically infectious
virions could, through random chance, fail to complete a key step following
cell entry. In addition to such stochastic occurrences in fully functional
virions, a number of entry capable virions could have physical defects that
prevents them from completing one or more key replication steps, leaving them
physically unable to cause a productive infection. As such, failure of
productive cell infection post viral entry is a combination of both the
stochastic post-entry failure of otherwise fully infectious virions, and the
inevitable failure of replication defective, entry capable virions.

A common, important application of DSMs is to estimate the extinction
probability of an infection: the likelihood that the infection will fail to
take hold or spread significantly. It typically depends on both the
replication capabilities of the virus (i.e. infection parameters) and the
initial number of infectious virions or cells. The extinction probability is
an important quantity to derive as it can, for example, be used to evaluate
the probability of success of antiviral therapy [8, 10]. In particular,
Czuppon et al. [10] compared the ability of prophylactic antivirals acting
either on the rate of virus entry into cell or the rate of virus production,
to reduce the establishment probability, or $1-$(extinction probability), of
severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection in
vivo. However, the effect of certain classes of antivirals, such as endosomal
fusion inhibitors, would be better represented as reducing the probability
that, having successfully entered a cell, a virion will go on to cause the
productive infection of that cell. For influenza A virus, it would be a more
appropriate way to represent the mode of action of adamantanes, such as
amantadine and rimantadine which block the M2 ion channel necessary for
successful fusion of influenza A virus with the endosome [13, 1]. For SARS-
CoV-2, it has been suggested that cathepsin L inhibitors can reduce the
probability of successful endosomal fusion [25]. Failure to undergo endosomal
fusion would lead to loss of the infectious virion, but would not result in a
productive cell infection.

Although the extinction probability is often an important consideration in
comparing prophylactic antivirals, what is never discussed is the number or
fraction of cells that are actually consumed by infections that are said to
have gone “extinct” or to have “established”. It is possible that antivirals
with different modes of action, even with the same extinction probability,
could result in a very different fraction of cells consumed by so-called
established or extinct infections. This would have important implications for
how one should interpret the probability of success of a particular antiviral
therapy.

In addition, nearly all DSMs, including past works that used the extinction
probability to evaluate the probability of success of antiviral therapy [8,
10], assume the duration of the infectious phase, the period during which
infected cells are producing and releasing virus progeny, to be exponentially-
distributed. Careful pairing of mathematical models and experimental
measurements has established that the duration of the infectious phase in
vitro for cells infected with IAV [20], simian HIV [2], or Ebola virus [23],
follows a log-normal or normal-like distribution, and has clearly rejected the
probability of an exponentially distributed infectious phase duration. Yan et
al. [41] estimated the extinction probability for a DSM of IAV infection in
vivo that allowed the lifespan of infectious cells to follow an Erlang
distribution which, via its shape parameter ($k$), can capture exponential
($k=1$), log-normal-like ($k\sim[1,6]$), normal-like ($k>10$), and even Dirac
delta-like ($k\to\infty$) distributions. Yan et al. [41] have shown that
increasing the shape parameter ($k$) from exponential to log-normal leads to a
decrease in the extinction probability given an infection initiated with a
number of infectious virions. Hence, the distribution of the infectious phase
duration is also expected to affect the likelihood that an infection will
become established under antiviral therapy.

In this work, we construct a DSM for virus infection through the physical
consideration of key infection steps, rather than through a systematic, direct
mathematical conversion of our mean-field model. Our DSM explicitly represents
the probability that a virion, after having successfully entered a cell, will
fail to result in the productive infection of that cell. The DSM is first used
to estimate the extinction probability of an infection. Extending work by
Czuppon et al. [10], we show that prophylactic therapy with an antiviral that
blocks productive cell infection _after_ viral entry, is better at reducing
the establishment probability, than one acting to reduce either the virus
production rate or the rate of virus entry into cells. In addition, we
investigate the difference in the fraction of cells consumed by so-called
extinct versus established infections, and re-visit the comparison of
antivirals though this new lens. Finally, we demonstrate how the antiviral
efficacy required to achieve a desired infection extinction probability
critically depends on the assumed distribution of the infectious phase
duration.

##  2 Results

###  2.1 The mathematical models

The mean-field mathematical model (MFM) and its DSM counterpart used herein
are given by

$\displaystyle\frac{\mathrm{d}T}{\mathrm{d}t}$
$\displaystyle=-\gamma\beta_{\text{}}TV_{\text{}}/S$ $\displaystyle T^{t+1}$
$\displaystyle=T^{t}-N^{\text{inf}}$
$\displaystyle\frac{\mathrm{d}E_{1}}{\mathrm{d}t}$
$\displaystyle=\gamma\beta_{\text{}}TV_{\text{}}/S-\frac{n_{E}}{\tau_{E}}E_{1}$
$\displaystyle E_{1}^{t+1}$
$\displaystyle=E_{1}^{t}+N^{\text{inf}}-E_{1}^{\text{out}}$
$\displaystyle\frac{\mathrm{d}E_{i}}{\mathrm{d}t}$
$\displaystyle=\frac{n_{E}}{\tau_{E}}E_{i-1}-\frac{n_{E}}{\tau_{E}}E_{i}$
$\displaystyle E_{i}^{t+1}$
$\displaystyle=E_{i}^{t}+E_{i-1}^{\text{out}}-E_{i}^{\text{out}}\qquad
i=2,3,...,n_{E}$ (1) $\displaystyle\frac{\mathrm{d}I_{1}}{\mathrm{d}t}$
$\displaystyle=\frac{n_{E}}{\tau_{E}}E_{n_{E}}-\frac{n_{I}}{\tau_{I}}I_{1}$
$\displaystyle I_{1}^{t+1}$
$\displaystyle=I_{1}^{t}+E_{n_{E}}^{\text{out}}-I_{1}^{\text{out}}$
$\displaystyle\frac{\mathrm{d}I_{j}}{\mathrm{d}t}$
$\displaystyle=\frac{n_{I}}{\tau_{I}}I_{j-1}-\frac{n_{I}}{\tau_{I}}I_{j}$
$\displaystyle I_{j}^{t+1}$
$\displaystyle=I_{j}^{t}+I_{j-1}^{\text{out}}-I_{j}^{\text{out}}\qquad
j=2,3,...,n_{I}$ $\displaystyle\frac{\mathrm{d}V}{\mathrm{d}t}$
$\displaystyle=p_{\text{}}\sum_{j=1}^{n_{I}}I_{j}-cV_{\text{}}-\beta
TV_{\text{}}/S$ $\displaystyle V^{t+1}$
$\displaystyle=V^{\text{prod}}+V^{\text{remain}}$

Initially, all cells are uninfected, susceptible target cells, $T$, i.e.
$T(t=0)=N_{\text{cells}}$. Target cells, $T$, are then infected by infectious
virions, $V$, resulting in a successful cell infection ($T\rightarrow E_{1}$)
at rate $\gamma\beta V/S$, where $S$ is the volume of supernatant. Newly
infected cells enter ($T\rightarrow E_{1}$) and traverse ($E_{1}\rightarrow
E_{2}\rightarrow\ldots\rightarrow E_{n_{E}}$) the $n_{E}$ compartments of the
eclipse phase, during which cells are infected but are not yet producing
infectious virions. Infected cells then enter ($E_{n_{E}}\rightarrow I_{1}$)
and traverse ($I_{1}\rightarrow I_{2}\rightarrow\ldots\rightarrow I_{n_{I}}$)
the infectious phase, during which they produce infectious virions at constant
rate $p$. As infected cells leave the last compartment ($I_{n_{I}}$), they
cease virus production and thus cease to contribute to the infection kinetics,
and possibly undergo apoptosis. The exponentially-distributed durations of the
$n_{E}$ eclipse (or $n_{I}$ infectious) phase compartments together yield an
Erlang-distributed total duration for the eclipse (or infectious) phase of
mean duration $\tau_{E}$ (or $\tau_{I}$), and standard deviation
$\tau_{E}/\sqrt{n_{E}}$ (or $\tau_{I}/\sqrt{n_{I}}$), where $n_{E}$ (or
$n_{I}$) corresponds to the shape parameter of the Erlang distribution.
Infectious virions ($V$) are produced at a constant rate of $p$ per infectious
cell per hour, and are lost either through loss of infectivity at rate $c$, or
entry into susceptible cells at rate $\beta T/S$.

This MFM is similar to that widely validated and applied to analyze and
predict the course of in vitro infections with IAV [27, 30, 42], simian HIV
[2], Ebola virus [23], RSV [17] and rotavirus [16]. It differs from the latter
by explicitly accounting for the loss of infectious virions due to cell entry
at rate $\beta TV/S$, which causes a corresponding loss of uninfected target
cells becoming infected ($T\rightarrow E_{1}$) at rate $\gamma\beta TV/S$.
Parameter $\gamma$ therefore has units of cell per infectious virion (IV),
where
$\gamma\in$(0,1]\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{/}\mathrm{I}\mathrm{V}$$,
and corresponds to the average fraction of infectious virion entries into a
cell that results in a successful infection of the cell ($T\rightarrow
E_{1}$). This parameter is not typically included in ordinary differential
equation (ODE) models of viral infection kinetics.

Biologically, $\gamma$ accounts for several different causes of infection
failure post viral entry into a cell. It can represent semi-infectious virions
that are entry-competent, but are defective in their ability to complete
another downstream step, e.g. are missing one or more viral genome segment or
have deleterious genetic mutations. It can also represent fully infectious
virions that are not defective but rather, through random chance, fail to
achieve a key step they could have functionally achieved, e.g. 50% of
influenza A virions failing to fuse with endosome membrane following cell
entry [22, 36]. Herein it is assumed that when an infectious virion enters a
cell, the cell will either be successfully infected ($T\rightarrow E_{1}$)
with probability $\gamma$, or remain uninfected ($T$) with probability
$1-\gamma$. Cells being left in a semi-infected state by partially failed
infections are not considered [24, 3, 12].

The DSM is largely analogous to the MFM, where DSM variables, e.g. $T^{t}$ the
number of target cells at time $t$, each denoted with a superscript $t$, are
whole numbers. The remaining terms, corresponding to changes in these
variables, are random whole numbers drawn at each time step from distributions
thought to best represent the corresponding underlying biological process
where

  * •

$E_{i}^{\text{out}}=\text{Binomial}(n=E_{i}^{t},\,p_{E}=\Delta t\cdot
n_{E}/\tau_{E})$ corresponds to the number of cells in the $i^{\text{th}}$
eclipse compartment ($E_{i}$) that will transition to the $(i+1)^{\text{th}}$
compartment over a time interval $\Delta t$, given that there are $E_{i}^{t}$
cells in the $i^{\text{th}}$ compartment at time step $t$. Each of the
$E_{i}^{t}$ cells in eclipse compartment $i$ undergo an independent Bernoulli
trial with two possible outcomes: either the cell transitions to the
$(i+1)^{\text{th}}$ compartment with success probability $p_{E}=\Delta t\cdot
n_{E}/\tau_{E}$, or otherwise remains in the $i^{\text{th}}$ compartment. The
time step, $\Delta t$ is chosen to be sufficiently small to ensure $p_{E}<1$,
as discussed below.

  * •

$I_{j}^{\text{out}}=\text{Binomial}(n=I_{j}^{t},\,p_{I}=\Delta t\cdot
n_{I}/\tau_{I})$ is the number of cells in the $j^{\text{th}}$ infectious
compartment ($I_{j}$) that transition to the $(j+1)^{\text{th}}$ compartment
over a time interval $\Delta t$, given that there are $I_{j}^{t}$ cells in the
$j^{\text{th}}$ compartment at time step $t$, and the probability of
transition $p_{I}=\Delta t\cdot n_{I}/\tau_{I}$.

  * •

$V^{\text{prod}}=\text{Poisson}(\lambda=\Delta t\cdot
p\,\sum_{j=1}^{n_{I}}I_{j}^{t})$ is the number of infectious virions newly
produced into the supernatant over a time $\Delta t$, given that there are
$\sum_{j=1}^{n_{I}}I_{j}^{t}$ infectious cells at time step $t$, each
producing infectious virions at a rate of $p$ infectious virions per hour. The
Poisson-distributed random variable, $V^{\text{prod}}$ is the number of events
that occurred, given the expected number of occurrences over a time $\Delta
t$, namely $\lambda=\Delta t\cdot p\sum_{j=1}^{n_{I}}I_{j}^{t}$.

  * •

$V^{\text{decay}},V^{\text{enter}},V^{\text{remain}}=\text{Trinomial}(n=V^{t},\,p_{1}=\Delta
t\cdot c,\,p_{2}=\Delta t\cdot\beta T^{t}/S,\,p_{3}=1-p_{1}-p_{2})$
corresponds to the number of infectious virions in the supernatant at time
$t$, $V^{t}$, that end up in each of 3 possible fates, namely
$V^{t}=V^{\text{decay}}+V^{\text{enter}}+V^{\text{remain}}$.
$V^{\text{decay}}$ is the number of infectious virions that lose infectivity
with probability $p_{1}=\Delta t\cdot c$, $V^{\text{enter}}$ are lost from the
supernatant as they enter a target cell with probability $p_{2}=\Delta
t\cdot\beta T^{t}/S$, and $V^{\text{remain}}$ infectious virions do neither
and remain in the supernatant with probability $p_{3}=1-p_{1}-p_{2}$. Each
probability ($p_{1},p_{2},p_{3}$) can be considered constant over time
interval $\Delta t$, provided a sufficiently small time step is chosen, as
discussed below.

  * •

$N^{\text{inf}}=\texttt{len(numpy.unique(numpy.random.choice(a=}T^{t}\texttt{,size=}V^{\text{suc}}\texttt{,replace=True)))}$
is the number of target cells that become infected ($T\rightarrow E_{1}$) over
time $\Delta t$, given that $V^{\text{enter}}$ infectious virions enter into
target cells, out of which
$V^{\text{suc}}=\text{Binomial}(n=V^{\text{enter}},\,p_{V}=\gamma)$ infectious
virions ultimately lead to successful cell infection, given probability
$p_{V}=\gamma$. Random variable $N^{\text{inf}}$ does not correspond to any
named probability mass function. The Python expression simulates randomly
placing $V^{\text{suc}}$ infectious virions into $T^{t}$ cells chosen at
random with replacement, numpy.random.choice, and counts the number of
distinct cells that received one or more infectious virions,
len(numpy.unique(...)).

Table 1: Random variables of the DSM Random variable | Random number generator
---|---
$E_{i}^{\text{out}}$ |  $\text{Binomial}(n=E_{i}^{t},\,p_{E}=\Delta t\cdot n_{E}/\tau_{E})$ where $i=1,2,...,n_{E}$
$I_{j}^{\text{out}}$ |  $\text{Binomial}(n=I_{j}^{t},\,p_{I}=\Delta t\cdot n_{I}/\tau_{I})$ where $j=1,2,...,n_{I}$
$V^{\text{prod}}$ | $\text{Poisson}(\lambda=\Delta t\cdot p\sum_{j=1}^{n_{I}}I_{j}^{t})$
$V^{\text{decay}},V^{\text{enter}},V^{\text{remain}}$ | $\text{Trinomial}(n=V^{t},\,p_{1}=\Delta t\cdot c,\,p_{2}=\Delta t\cdot\beta T^{t}/S,\,p_{3}=1-p_{1}-p_{2})$
$N^{\text{inf}}$ | $\texttt{len(numpy.unique(numpy.random.choice(a=}T^{t}\texttt{,size=}V^{\text{suc}}\texttt{,replace=True)))}$
|  where $V^{\text{suc}}=\text{Binomial}(n=V^{\text{enter}},\,p_{V}=\gamma)$

Table 1 summarizes how each random variable is generated. The duration of the
DSM’s discrete time steps, $\Delta t$, which sets the probability of event
occurrences or the number of such events, is computed at each iteration step
$t$ as

$\displaystyle\Delta
t=\frac{\mathcal{P}_{\text{events}}}{\max\bigg{\\{}\frac{\beta
T^{t}}{S},c,\frac{n_{E}}{\tau_{E}},\frac{n_{I}}{\tau_{I}}\bigg{\\}}}$ (2)

where $\mathcal{P}_{\text{events}}$ is the probability of occurrence of the
most likely event, namely of virion loss due to cell entry ($\beta T^{t}/S$)
or loss of infectivity ($c$), or transition of cells from one infected state
to another ($n_{E}/\tau_{E}$ and $n_{I}/\tau_{I}$). A value of
$\mathcal{P}_{\text{events}}=0.05$ (or 5%) is used as it was found to be small
enough that choosing a smaller value did not affect the results presented
herein. The numerical solution of the DSM was further validated by replacing
the DSM’s random variables, e.g. $E_{i}^{\text{out}}$, by their expected
value, e.g. $E_{i}^{\text{out}}=E_{i}^{t}\cdot\Delta t\cdot n_{E}/\tau_{E}$,
(see Methods, Section 4.2) and comparing the solution against that obtained
for the MFM in Eqn. (2.1) with a standard numerical ODE solver. The solutions
were found to be in agreement over the wide range of parameter values explored
(not shown).

Some readers might prefer to see the DSM expressed as transitions [28, 41],
namely

$\displaystyle V$ $\displaystyle\xrightharpoonup{\mathmakebox[1cm]{\beta
T/S}}V^{\text{enter}}$ $\displaystyle V^{\text{enter}}+T$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{\gamma}}E_{1}$ $\displaystyle
V^{\text{enter}}+T$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{1-\gamma}}T$ $\displaystyle
E_{i}$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{n_{E}/\tau_{E}}}E_{i+1}\qquad
i=1,2,3,...,n_{E}-1$ $\displaystyle E_{n_{E}}$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{n_{E}/\tau_{E}}}I_{1}$ (3)
$\displaystyle I_{j}$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{n_{I}/\tau_{I}}}I_{j+1}\qquad
j=1,2,3,...,n_{I}-1$ $\displaystyle I_{n_{I}}$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{n_{I}/\tau_{I}}}\varnothing$
$\displaystyle I_{j}$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{p}}I_{j}+V\qquad
j=1,2,3,...,n_{I}$ $\displaystyle V$
$\displaystyle\xrightharpoonup{\mathmakebox[1cm]{c}}\varnothing$

One limitation of this representation is that it indicates that one virion
that enters a cell and that is successful at causing an infection with rate
$\gamma$ will cause the infection of one cell
($V^{\text{enter}}+T\xrightharpoonup{}E_{1}$). In reality, in our DSM, one
successful virion does not necessarily cause the infection of one cell (see
$N^{\text{inf}}$ in Table 1) because one cell could receive two successful
infectious virions which would only result in one rather than two new
successful cell infections.

###  2.2 Important biological quantities

Following Pearson et al. [28], let us define
$\mathcal{P}_{V\to\,\text{Extinction}}$ and
$\mathcal{P}_{I\to\,\text{Extinction}}$ as the probability of infection
extinction given an infection initiated with either only one infectious virion
or only one infectious cell, respectively. The extinction probability given
any initial number $V_{0}$ of infectious virions and $I_{0}$ of infectious
cells is then
$\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{V_{0}}\cdot\left(\mathcal{P}_{I\to\,\text{Extinction}}\right)^{I_{0}}$.

If initially there is only one infectious virion, the infection can fail to
spread if either that initial infectious virion fails to cause a successful
cell infection with probability $\mathcal{P}_{V\to\,\varnothing}$, or if it
does cause a successful cell infection with probability
$\mathcal{P}_{V\to\,I}=(1-\mathcal{P}_{V\to\,\varnothing})$ but that cell
infection subsequently leads to extinction with probability
$\mathcal{P}_{I\to\,\text{Extinction}}$. We can therefore write

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\cdot\mathcal{P}_{I\to\,\text{Extinction}}\
.$ (4)

If initially there is only one infectious cell, the cell will produce $m$
infectious virions over its lifespan with probability
$\mathcal{P}_{I\to\,m\,V}$. Each one of these produced infectious virions can
be treated as an independent infection event such that,

$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}=\sum_{m=0}^{\infty}\mathcal{P}_{I\to\,m\,V}\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}\
.$ (5)

Taken together, the extinction probability given an infection initiated with
only one infectious virion, $\mathcal{P}_{V\to\,\text{Extinction}}$, yields
the recursive relation

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\cdot\sum_{m=0}^{\infty}\mathcal{P}_{I\to\,m\,V}\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}\
.$ (6)

The probability that an infectious virion is successful at causing a
productive cell infection, $\mathcal{P}_{V\to\,I}$, can be expressed as the
ratio between the rate of successful cell infection per infectious virion
$\gamma\beta N_{\text{cells}}/S$ and the rate of virion loss through loss of
infectivity plus cell entry, $c+\beta N_{\text{cells}}/S$ (see Methods,
Section 4.4 for derivation), namely,

$\displaystyle\mathcal{P}_{V\to\,I}=(1-\mathcal{P}_{V\to\,\varnothing})=\frac{\gamma\beta
N_{\text{cells}}/S}{c+\beta N_{\text{cells}}/S}\ .$ (7)

The probability that an infectious cell produces $m$ infectious virions over
its lifespan, $\mathcal{P}_{I\to\,m\,V}$, can be expressed as the marginal
probability distribution of the probability that an infectious cell produces
$m$ infectious virions given a lifespan of length $t$,
$\text{Poisson}(m|\lambda=pt)$, and the probability that the lifespan is of
length $t$, $\text{Erlang}(t|k=n_{I},\lambda=n_{I}/\tau_{I})$. Therefore,

$\displaystyle\mathcal{P}_{I\to\,m\,V}$
$\displaystyle=\int_{0}^{\infty}\text{Poisson}(m|\lambda=pt)\cdot\text{Erlang}(t|k=n_{I},\,\lambda=n_{I}/\tau_{I})\
\text{d}t$
$\displaystyle=\frac{(m+r-1)!}{m!(r-1)!}(1-p_{B})^{r}(p_{B})^{m}=\text{NB}(m|r=n_{I},\,p_{B}=\mathcal{B}/(n_{I}+\mathcal{B}))$
(8)

where NB stands for the negative binomial (or Pascal) distribution and
describes the probability that $m$ successes will have occurred by the time
one has observed $r=n_{I}$ failures from a series of independent Bernoulli
trials with a probability of success $p_{B}$ (see Methods, Section 4.5 for
derivation). The mean of the distribution corresponds to the average burst
size, $\mathcal{B}=p\tau_{I}$, i.e. the average number of infectious virions
produced by a productively infected cell over its lifespan, with $p$ the rate
of infectious virion production per cell and $\tau_{I}$ the average infectious
cell lifespan.

Combining the mean number of infectious virions produced by a productively
infected cell, $\mathcal{B}$, and the probability that each of them causes the
infection of a cell, $\mathcal{P}_{V\to\,I}$ (Eqn. (7)), we obtain the average
basic reproductive number ($R_{0}$), defined as the average number of
productive secondary infections caused by an infected cell over its lifespan
when placed within a fully susceptible and uninfected cell population,

$\displaystyle
R_{0}=\mathcal{B}\cdot\mathcal{P}_{V\to\,I}=p\tau_{I}\cdot\frac{\gamma\beta
N_{\text{cells}}/S}{c+\beta N_{\text{cells}}/S}\ .$ (9)

Substituting Eqn. (7) and Eqn. (8) into Eqn. (6) results in the following
expression for $\mathcal{P}_{V\to\,\text{Extinction}}$

$\displaystyle 0$ $\displaystyle=1-\frac{1+c/(\beta
N_{\text{cells}}/S)}{\gamma}[1-\mathcal{P}_{V\to\,\text{Extinction}}]-\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}\
.$ (10)

While Eqn. (10) does not lead to an analytical solution for
$\mathcal{P}_{V\to\,\text{Extinction}}$, a solution can easily be obtained
numerically (see Methods, Section 4.6). The probability of infection
establishment given an infection initiated with only one infectious virion,
$\mathcal{P}_{V\to\,\text{Establishment}}$, is simply
$1-\mathcal{P}_{V\to\,\text{Extinction}}$.

###  2.3 Effectiveness of antivirals to reduce an infection’s establishment
probability

Prophylactic antiviral therapies are often characterized and compared in terms
of their ability to reduce an infection’s establishment probability. This is
because for natural infections in a host, the initial virus inoculum can be
sufficiently small that prophylactic antiviral therapy can prevent infection,
i.e. induce infection extinction.

Recently, Czuppon et al. [10] used a DSM to evaluate the effectiveness of
prophylactic treatment with antivirals acting on the rate of virus entry into
cells sometimes called the virus infectivity rate, herein $\beta$, or the
virus production rate, $p$, to reduce the establishment probability for a
SARS-CoV-2 infection in vivo initiated by a small number of infectious
virions. But an endosomal fusion inhibitor (e.g., cathepsin L inhibitors for
SARS-CoV-2 [25]) would lead to the removal of an infectious virion from the
medium (i.e., leave $\beta$ unaffected), and yet by causing fusion failure
post cell entry, would block the cell from progressing to a productively
infectious state, hence blocking 100% of viral progeny production. Using our
DSM, which can explicitly represent productive infection failure post viral
entry via parameter $\gamma$, we extend Czuppon’s investigation to evaluate
how such an antiviral would compare against those acting on $\beta$ or $p$.

Figure 1: Evaluating the ability of antivirals to reduce an infection’s
establishment probability. The establishment probability given an infection
initiated with only one infectious virion as a function of efficacy
($\varepsilon$) for antivirals acting either to reduce the virus entry rate,
$\beta\to(1-\varepsilon)\beta$ (blue solid line), the virus production rate,
$p\to(1-\varepsilon)p$ (orange dashed line), or the probability of a
successful cell infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$
(green dotted line). The line labelled 1 indicates a value of lower efficacy
$\varepsilon=0.8$. The line labelled 2 indicates a value of higher efficacy
where the establishment probability goes to zero for an antiviral acting on
$p$ or $\gamma$. The line labelled 3 indicates an even higher value of
efficacy needed for the establishment probability to go to zero for an
antiviral acting on $\beta$. The infection parameters are from Czuppon et al.
[10] (see Methods, Section 4.1).

Fig 1 shows the establishment probability given an infection initiated with
only one infectious virion as a function of antiviral efficacy ($\varepsilon$)
for antivirals acting on $\beta$, $p$ or $\gamma$. The parameter values used
to generate this and all other figures herein are provided in the Methods,
Section 4.1. The infection parameters were taken from Czuppon et al. [10], and
in particular assume an exponentially distributed duration for the eclipse and
infectious phases ($n_{E}=n_{I}=1$). For $n_{I}=1$, the expression for the
establishment probability in Eqn. (10) reduces to

$\displaystyle\mathcal{P}_{V\to\,\text{Establishment}}=\mathcal{P}_{V\to\,I}-\frac{1}{\mathcal{B}}=\frac{R_{0}}{\mathcal{B}}-\frac{1}{\mathcal{B}}=\frac{\gamma\beta
N_{\text{cells}}/S}{c+\beta N_{\text{cells}}/S}-\frac{1}{p\tau_{I}}$ (11)

where $\mathcal{B}=p\tau_{I}$ is the burst size (see Methods, Section 4.6).

At relatively low efficacy, e.g. $\varepsilon=0.8$ (see line 1 in Fig 1), for
the parameters chosen, an antiviral acting on $\gamma$ is more effective than
one acting on $\beta$ which is more effective than one acting on $p$ at
reducing the establishment probability. The infection parameters used by
Czuppon et al. [10] correspond to a sufficiently large burst size
($\mathcal{B}=p\tau_{I}=18.8$, see Methods Section 4.1) that
$\mathcal{P}_{V\to\,\text{Establishment}}\approx\mathcal{P}_{V\to\,I}$
($=0.409$). As such, the bottleneck to infection establishment is the
probability that the single initial infectious virion causes a productive cell
infection, $\mathcal{P}_{V\to\,I}$, which depends on $\beta$ and $\gamma$, but
not $p$. While an antiviral acting on $\beta$ reduces both the numerator and
denominator of $\mathcal{P}_{V\to\,I}$, one acting on $\gamma$ reduces only
the numerator and therefore has a greater effect. At this relatively low
antiviral efficacy ($\varepsilon\approx 0.8$), as the initial number of
infectious virions increases and as the establishment probability approaches
100%, differences in the efficacy of antivirals acting on $p$, $\beta$ or
$\gamma$ to reduce the establishment probability vanishes (see Supplementary
Material S1).

As the antiviral efficacy is increased further, it eventually reaches the
critical point where $R_{0}=1$, and the establishment probability,
$(R_{0}-1)/\mathcal{B}$, equals zero. The value of the antiviral efficacy at
which this is achieved is the same for antivirals acting on $p$ and $\gamma$
(see line 2 in Fig 1) since they affect $R_{0}$ (Eqn. (9)) identically. The
efficacy of an antiviral acting on $\beta$ must be much higher in order to
reach this critical point (see line 3 in Fig 1), because reducing $\beta$
reduces both the numerator and denominator of $R_{0}$.

Therefore, as Czuppon et al. [10] before us, and for their choice of
parameters, we find that at lower efficacies, an antiviral acting on $\beta$
is better than one acting on $p$ at reducing the establishment probability,
whereas at higher efficacies it is the opposite. However, here we show that an
antiviral acting on $\gamma$ at any efficacy is best, better than one acting
on $p$ or $\beta$, at reducing the establishment probability.

###  2.4 Distinction between infection extinction and establishment

What constitutes the establishment or extinction of an infection is commonly,
somewhat vaguely defined. For example, Czuppon et al. [10] state _“The larger
the initial inoculum of infectious virus, the less likely is the prevention of
an infection.”_ equating infection extinction as predicted by their DSM with
prevention of the infection. Pearson et al. [28] state _“Whether exposure to
virus leads to systemic infection or complete elimination of the virus can be
a matter of luck, particularly when exposure is to low levels of virus.”_ and
later define the extinction probability as _“the probability that the virus
and all infected cells are completely eliminated from the host”_. While it is
never explicitly stated, one could be forgiven for thinking that extinction
means that almost no cells are consumed while establishment means that all but
a negligible number of cells are left untouched by the infection. But to what
extent is this true? Is the distinction always biologically meaningful and
clear? Can one look at an infection outcome and clearly categorize it as an
extinct versus an established infection?

Figure 2: Infection extinction versus establishment. The distribution for the
frequency (black dots) or complementary cumulative frequency (solid red
curve), or $1-$(cumulative frequency), of the number of cells consumed by the
infection, $N_{\text{cells}}-T(\infty)$, out of $N_{\text{cells}}=$4\times
10^{4}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$, based on
$10^{6}$ DSM simulations. For the frequency distribution, each black dot
represents the fraction of the DSM simulations where exactly this discrete
number of cells were consumed by the infection (e.g. $0,1,2,...$). The
vertical solid (orange) line represents the MFM-predicted fraction of cells
consumed by an established infection. The 95% ($2\sigma$) CI for the number of
cells consumed by infections corresponds to the region left of the vertical
(grey) dotted line for extinct infections, or enclosed by a pair of vertical
(orange) dashed lines for established infections. Parameters were the same as
in Fig 1 for an antiviral acting on the virus production rate,
$p\to(1-\varepsilon)p$, at efficacy $\varepsilon=0.79$ such that
$T^{*}/N_{\text{cells}}=0.5$ (Eqn. (12)).

Fig 2 shows the frequency and complementary cumulative frequency
distributions, or $1-$(cumulative frequency), for the number of cells consumed
by $10^{6}$ DSM simulated infections. The DSM simulations appear to fall into
2 distinct categories: infections with relatively few (close to 0% on the
left) or many (around 80% on the right) cells consumed, naturally
corresponding to infections said to have gone extinct or to have established,
respectively, as indicated by red arrows. Extinct infections make up 85%
(848,045) of the $10^{6}$ DSM simulations, in agreement with the theoretical
value of the extinction probability, $\mathcal{P}_{V\to\,\text{Extinction}}$,
for the infection parameters used. Of the extinct infections, 69% (59%/85%)
result in no cells consumed, 93% ($(59+16+4)\%/85\%$) in fewer than 3 cells
consumed, and 95% ($2\sigma$) in fewer than
$5\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$ consumed, or
less than 0.01% of all susceptible cells. With so few cells infected, it could
be reasonable to expect that no significant immune response nor symptoms were
triggered. Of the established infections, all consumed at least 30,600 cells
or about 10,000$\times$ more than infections considered extinct. The
difference between extinction and establishment in this example is therefore
statistically and biologically significant.

For established infections, the MFM predicts that $\sim$80% (or 31,873 out of
40,000 cells) of cells will be consumed, given by
$1-T(\infty)/N_{\text{cells}}$ in Eqn. (23), and indicated by a vertical solid
line in Fig 2. This roughly corresponds to the median of the distribution of
cells consumed by established DSM infections — also its mean and mode when the
distribution is symmetric, as is the case here — thus providing an analytical
expression to track this key feature of the DSM distribution.

Let us consider how this MFM-predicted fraction of cells consumed by
established infections depends on infection parameters (see Methods, Section
4.3 for derivation). At the start of an infection, all cells are uninfected,
$T(0)=N_{\text{cells}}$, and the reproductive number, $R(0)$, i.e. the number
of successful secondary infections caused by an infected cell over its
lifespan, corresponds to the basic reproductive number ($R_{0}$), given by
Eqn. (9), where $R_{0}>1$ for an infection that has a non-zero establishment
probability. As the infection progresses and fewer uninfected cells remain,
each infectious virion has fewer opportunities to cause a successful cell
infection and $R(t)$ decreases. Eventually the number of uninfected cells
reaches a critical value, $T(t)=T^{*}$, such that

$\displaystyle R(t)=1\equiv\frac{p\tau_{I}\cdot\gamma\beta T^{*}/S}{c+\beta
T^{*}/S}$ $\displaystyle\frac{T^{*}}{N_{\text{cells}}}=\frac{c}{(\beta
N_{\text{cells}}/S)\cdot(\gamma p\tau_{I}-1)}\ .$ (12)

When the number of uninfected cells remaining $T(t)$ equals $T^{*}$, then by
definition $R(t)=1$ marking the infectious cell population peak, as shown in
Fig 3(A). Thereafter, the infectious cell population declines and the number
of cells that remain uninfected approaches its final value, $T(\infty)$.

Figure 3: Exploring the critical fraction of uninfected cells,
$T^{*}/N_{\text{cells}}$. (A) MFM-predicted time courses for the fraction of
uninfected (black) and infectious (red) cells, for an infection initiated with
10 infectious virions under antiviral therapy acting to reduce the virus entry
rate, $\beta\to(1-\varepsilon)\beta$, at efficacy $\varepsilon=0.81$, which
yields an establishment probability
($\mathcal{P}_{V\to\,\text{Establishment}}=48\%$) that is $\sim$50% of its
value without antivirals [10]. The point $(t,T^{*}/N_{\text{cells}})$ is
represented by a triangle; the time when $T=T^{*}$ by a dashed line; and
$T=T(\infty)$ by an $\times$ on the right vertical axis. The parameters are
provided in Methods, Section 4.1, but notably $n_{I}=1$. (B,C) The
reproductive number, $R(t)$, as a function of the fraction of cells that
remain uninfected, $T(t)$, over the course of an infection (time is implicit),
based on Eqn. (31) when either $R_{0}=2.2$ and $T^{*}/N_{\text{cells}}$ is
varied (B); or $T^{*}/N_{\text{cells}}=0.4$ and $R_{0}$ is varied (C). The
start of the infection is represented by a circle; the critical point where
$T=T^{*}$ and $R=1$ by a triangle; and the end of the infection where
$T=T(\infty)$ as given by Eqn. (25) by an $\times$.

For our case of interest where the initial inoculum consists of one or a few
infectious virions, the MFM-predicted fraction of cells consumed,
$1-T(\infty)/N_{\text{cells}}$, can be represented by a strictly decreasing
function of $T^{*}/N_{\text{cells}}$ (see Eqn. (25)). This makes sense because
if the infection parameters are such that fewer cells need to be consumed by
the infection to reach $R(t)=1$ (larger $T^{*}$), then fewer cells will have
been consumed by the time the infection ends (larger $T(\infty)$), as shown in
Fig 3(B). The MFM-predicted fraction of cells consumed is not, however, simply
a function of $R_{0}$. Fig 3(C) shows how an infection can start with a higher
$R_{0}$ but $R(t)$ can decrease more rapidly as the infection progresses such
that $R(t)=1$ with the same fraction of uninfected cells
($T^{*}/N_{\text{cells}}$), ultimately resulting in the same fraction of cells
consumed, $1-T(\infty)/N_{\text{cells}}$.

Figure 4: The effect of infection parameters on extinction and establishment.
The frequency or complementary cumulative frequency of the fraction or number
of cells consumed by the infection out of $N_{\text{cells}}=$4\times
10^{4}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$ where the
efficacy ($\varepsilon$) of an antiviral acting on the virus production rate,
i.e. $p\to(1-\varepsilon)p$, was varied such that (A)
$T^{*}/N_{\text{cells}}=0.6$; (B) $0.7$ or (C) $0.85$. Everything else is
generated, computed, and represented visually as described in the caption of
Fig 2.

Fig 4 shows the frequency and complementary cumulative frequency distributions
of the number of cells consumed by an infection, using parameter sets where
$T^{*}/N_{\text{cells}}$ is increasingly closer to $1$. It is achieved here by
increasing the efficacy of an antiviral acting on the virus production rate,
$p\to(1-\varepsilon)p$, thus increasing $T^{*}/N_{\text{cells}}$ (Eqn. (12)).
As $T^{*}/N_{\text{cells}}$ approaches 1, the mean number of cells consumed by
infections considered extinct increases while that by established infections
decreases, and the distinction blurs as the two distributions begin to merge.
In Fig 4(C), the location of the red arrows demarcating so-called extinct and
established infections was chosen so that $\sim$96% of the $10^{6}$ DSM
simulated infections, the probability of extinction given by Eqn. (10), are to
the left of the arrows. Therefore, when an antiviral is applied and the
establishment probability is non-zero ($R_{0}>1$), varying the antiviral
efficacy (hence varying $T^{*}/N_{\text{cells}}$) affects not only the
establishment probability but also the fraction of cells consumed by both
established and extinct infections. For higher antiviral efficacy (higher
$T^{*}/N_{\text{cells}}$), the distinction between extinction and
establishment becomes increasingly irrelevant.

Figure 5: The effect of the size of the cell population on extinction and
establishment. The frequency and complementary cumulative frequency of the
fraction or number of cells consumed by the infection out of
$N_{\text{cells}}$, as the latter is varied, while the cell concentration
($N_{\text{cells}}/S$), and therefore also the establishment probability and
$T^{*}/N_{\text{cells}}$, are fixed; (A) $N_{\text{cells}}=$4\times
10^{3}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$; (B)
$4\times 10^{5}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$ or
(C) $4\times
10^{7}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$. The
frequency distribution in (A) is shown in (B) in grey, and that in (B) is
shown in (C) in grey. As in Fig 4(C), the efficacy ($\varepsilon$) of an
antiviral acting to reduce the virus production rate, $p\to(1-\varepsilon)p$,
was such that $T^{*}/N_{\text{cells}}=0.85$. Everything else is generated,
computed, and represented visually as described in the caption of Fig 2.

Fig 5 explores the number of cells consumed by established and extinct
infections as the size of the cell population, $N_{\text{cells}}$, is
increased while keeping the concentration of cells ($N_{\text{cells}}/S$) and
thus the establishment probability and $T^{*}/N_{\text{cells}}$ fixed. In Fig
5(A) with the smallest cell population considered, the distributions for the
number of cells consumed by established versus extinct infections overlap.

In Fig 5(B,C), as the size of the cell population ($N_{\text{cells}}$) is
increased, the distribution for the number of cells consumed by extinct
infections (left side of the graph) is unaffected, and therefore the fraction
of cells consumed becomes increasingly small. The 95% CI upper bound remains
fixed at $\sim$11\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$
which corresponds to 0.3%, 0.003% and 0.00003% out of $N_{\text{cells}}$ as
the latter is increased from $4\times
10^{3}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$ to $4\times
10^{7}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$. In
contrast, in the established infections, the number of cells consumed
increases as $N_{\text{cells}}$ increases, while the median fraction of cells
consumed remains unchanged, $\sim 28\%$. Therefore, as the size of the cell
population is increased, there is a greater distinction between extinction and
establishment. This means that, with a larger cell population, a higher
antiviral efficacy (higher $T^{*}/N_{\text{cells}}$) is required for the
distinction between extinction and establishment to vanish.

###  2.5 Reduction in the number of cells consumed by infections under
antiviral therapy

Figure 6: Effect of different antivirals on the MFM-predicted fraction of
cells consumed by established infections. (A) The MFM-predicted fraction of
cells consumed by established infections initiated with only one infectious
virion as a function of efficacy ($\varepsilon$) for antivirals acting
prophylactically either to reduce the virus entry rate,
$\beta\to(1-\varepsilon)\beta$ (blue solid line), the virus production rate,
$p\to(1-\varepsilon)p$ (orange dashed line), or the probability of a
successful cell infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$
(green dotted line). (B) The MFM-predicted fraction of cells consumed by
established infections shown in (A) for an antiviral acting to reduce $\beta$
minus that for one acting to reduce either $p$ or $\gamma$. The dash line in
(B) indicates the value of efficacy where the fraction of cells consumed for
an antiviral acting on $p$ or $\gamma$ equals zero. Infection parameters are
the same as in Fig 1.

Prophylactic antiviral therapy not only reduces an infection’s establishment
probability, $\mathcal{P}_{V\to\,\text{Extinction}}$ in Eqn. (10), but as seen
above it also decreases the number of cells consumed by established infections
while increasing that consumed by extinct infections (see Fig 4). The median
of the distribution for the number of cells consumed by established infections
in the DSM simulations approximately corresponds to the MFM-predicted value,
$N_{\text{cells}}-T(\infty)$. Let us then explore the effect of antivirals on
the MFM-predicted fraction of cells consumed by an infection.

Fig 6(A) shows the MFM-predicted fraction of cells consumed by an established
infection, given an initial inoculum of only one infectious virion, as a
function of antiviral efficacy ($\varepsilon$) for antivirals acting on
$\beta$, $p$, or $\gamma$. Since an antiviral acting on $p$ or $\gamma$ have
the same effect on $T^{*}=(c\cdot S)/[\beta\cdot(\gamma p\tau_{I}-1)]$ (see
Eqn. (12)), they also cause the same MFM-predicted fraction of cells to be
consumed by established infections, $1-T(\infty)/N_{\text{cells}}$ (see
Methods, Section 4.3). On the other hand, an antiviral acting on $\beta$, for
the same $\varepsilon$, results in a smaller $T^{*}$ and thus larger MFM-
predicted fraction of cells consumed. Fig 6(B) illustrates how these
differences, and therefore their biological relevance, is highly dependent on
the antiviral efficacy under consideration. For example, the difference in the
MFM-predicted fraction of cells consumed by established infections for
antivirals acting on $\beta$ rather than $p$ or $\gamma$ is $<$0.1% at an
antiviral efficacy $\varepsilon<0.5$ or $>0.92$.

Figure 7: Antiviral with different modes of action cause different fraction of
cells to be consumed by infections. The number of cells consumed by the
infection out of $N_{\text{cells}}=$4\times
10^{4}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$ for
antivirals with efficacy $\varepsilon=0.86$ acting either to reduce (A) the
virus entry rate, $\beta\to(1-\varepsilon)\beta$, (B) the virus production
rate, $p\to(1-\varepsilon)p$, or (C) the probability of a successful cell
infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$. Antiviral modes
of action are represented by different colours: reducing $\beta$ (blue), $p$
(orange) or $\gamma$ (green). Everything else is generated, computed, and
represented visually as described in the caption of Fig 2.

Fig 7 shows the number of cells consumed by extinct and established
infections, given an initial inoculum of a single infectious virion, for
antivirals acting on $\beta$, $p$ or $\gamma$ at an antiviral efficacy of
$\varepsilon=0.86$. This efficacy was chosen so that, given the infection
parameters, the establishment probability is the same for an antiviral acting
on $\beta$ or on $p$ ($\mathcal{P}_{V\to\,\text{Extinction}}=4\%$), yet it
corresponds to a lower establishment probability for an antiviral acting on
$\gamma$ ($\mathcal{P}_{V\to\,\text{Extinction}}=0.5\%$). This can be seen
from Fig 1: it corresponds to the efficacy where the orange line (antiviral
acting on $p$) crosses the blue line (antiviral acting on $\beta$).

At this efficacy, and for the chosen infection parameters, biologically and
statistically significantly more cells are consumed by established infections
for an antiviral acting on $\beta$ than on $p$ or $\gamma$. For extinct
infections, which are a more likely outcome, the least number of cells are
consumed for an antiviral acting on $\beta$, followed by $\gamma$, and then
$p$. When averaging over all infection outcomes, i.e. extinct and established
infections, an antiviral acting on $\gamma$ will lead to an average of 56
cells consumed, followed by one acting on $p$ with 396 cells, and finally on
$\beta$ with 1,055 cells. Rather than looking at extinct versus established
infections, one might consider that triggering an immune response, and
therefore possibly also a wide range of symptoms, is the most important aspect
of infection control. This could conceivably be triggered once a certain
number of cells, e.g. $\sim 20$ or so, have been consumed by infection. In
this case, an antiviral acting on $\gamma$, more so than one acting on
$\beta$, followed by one acting on $p$, would help avoid triggering an immune
response, i.e. have a smaller proportion of infection outcomes where less than
$\sim 20$ cells are consumed.

We investigated a second parameter set also explored in Czuppon et al. [10],
which is characterized by a 10-fold decrease in the number of cells
$N_{\text{cells}}$, a 10-fold increase in the virus production rate $p$
(hence, of the average burst size $\mathcal{B}=p\tau_{I}$), and a
corresponding $\sim$10-fold decrease in the virus entry rate $\beta
N_{\text{cells}}/S$ (see Supplementary Material S2). For this second parameter
set, we found that an antiviral acting on $\gamma$ was comparable to one
acting on $\beta$, but better than one acting on $p$, for reducing the
establishment probability. All 3 modes of action had a similar effect on the
MFM-predicted fraction of cells consumed by established infections. As such,
the degree to which an antiviral acting on $\gamma$ is better to reduce the
establishment probability than other antivirals, or an antiviral acting on
$\beta$ will leave more cells consumed by established infections, will depend
on infection parameters. It is therefore critical that such parameters be
well-determined if such investigations are to yield meaningful predictions.

Lastly, as in Czuppon et al. [10], we also considered antiviral therapy for an
infection initiated with only one infectious cell which could be
representative of post-exposure antiviral therapy (see Supplementary Material
S3). In this case, we found that an antiviral acting on $p$ or $\gamma$ have
the same effectiveness, greater than one acting on $\beta$, in reducing the
establishment probability. There was no meaningful difference in the MFM-
predicted fraction of cells consumed by established infections for infections
initiated with one infectious cell rather than one infectious virions.

Overall, these results over different infection parameters and antiviral
efficacy indicates that, under certain conditions, the fraction of cells
consumed by both or either extinct and established infections can be an
important consideration when evaluating and comparing antivirals with
different modes of action. It also highlights the importance of properly
identifying biologically relevant base parameter values in order to provide
meaningful comparisons.

###  2.6 Continuous versus burst release of virus and the impact of the virus
burst size distribution

In the MFM and DSM used herein, virus is released continuously at a fixed rate
of $p$ infectious virion per hour by infectious cells. In the DSM, this fixed
rate maps to a Poisson-distributed, discrete, stochastic number of infectious
virions produced per time step by each infected cell over the duration of its
infectious lifespan. The duration of this infectious phase is represented by
an Erlang distribution characterized by shape parameter $n_{I}$ and average
duration $\tau_{I}$. As shown in Eqn. (8), this means that the DSM’s
stochastic virus burst size follows a negative binomial or Pascal distribution
with a mean corresponding to the MFM’s virus burst size,
$\mathcal{B}=p\tau_{I}$, where $n_{I}$ now corresponds to the distribution’s
integer-valued, stopping-time parameter. Fig 8(A) illustrates how the DSM’s
burst size distribution varies as a function of $n_{I}$, for a fixed average
burst size.

Figure 8: Effect of the infectious phase duration’s shape parameter, $n_{I}$,
on the extinction probability. (A) Negative binomial-distributed virus burst
size as a function of the shape parameter ($n_{I}$) of the Erlang-distributed
infectious cell lifespan. The shaded region highlight a region where the burst
size is small ($\leq 5$ infectious virions) and therefore more likely to
result in infection extinction. The vertical grey line corresponds to the
average burst size value
($\mathcal{B}=p\tau_{I}=$18.8\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$$),
which is constant as $n_{I}$ is varied. (B,C) The probability of infection
extinction as a function of $n_{I}$, given an infection initiated with only
one infectious virion (B); or one infectious cell (C). Note that the
extinction probability is shown on a linear scale in (B) but a logarithmic
scale in (C) to better capture the relationships. Unless otherwise specified,
the parameters are the same as in Fig 1.

Up to this point, as with most DSMs, the duration of the infectious phase has
been assumed to be exponentially-distributed ($n_{I}=1$ in Eqn. (2.1)). This
choice does not affect the MFM-predicted fraction of cells consumed by
established infections, since $n_{I}$ does not appear in Eqn. (23), but it
does affect the extinction probability, as per Eqn. (10). Previously, Yan et
al. [41] have shown that increasing $n_{I}$ leads to a decrease in the
extinction probability when infection is initiated with a small initial number
of infectious virions. Fig 8(B) shows this for the infection parameters used
so far for an infection initiated with a single infectious virion.

Recall that the extinction probability is a recursive expression with 2 main
terms,

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+(1-\mathcal{P}_{V\to\,\varnothing})\cdot\sum_{m=0}^{\infty}\underbrace{\text{NB}\left(m\left|r_{\text{fail}}=n_{I},p_{\text{success}}=\frac{\mathcal{B}}{n_{I}+\mathcal{B}}\right.\right)}_{\text{burst
size
distribution}}\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}\ .$

where the first term, $\mathcal{P}_{V\to\,\varnothing}$, is the probability
that the initial virion inoculum fails to productively infect a single cell,
and the second is the likelihood that it does actually cause a cell infection
which then itself fails to establish an infection. Fig 8(C) shows how the
contribution from the second term, i.e. the extinction probability for an
infection initiated with a single infected cell, decreases as $n_{I}$
increases. What is observed is Fig 8(B), therefore, is the shrinking
contribution to $\mathcal{P}_{V\to\,\text{Extinction}}$ by the second term as
$n_{I}$ increases.

This second term depends critically on the burst size distribution which, in
our DSM, is a result of the Erlang-distributed infectious phase duration. As
the latter distribution goes from an exponential distribution ($n_{I}=1$) to a
fat-tailed ($n_{I}\in[2,7]$), to a normal-like ($n_{I}=60$), tending
ultimately to a Dirac delta distribution ($n_{I}\to\infty$), it becomes less
probable that an infectious cell will have a very small burst size, as shown
in Fig 8(A). For example, the probability that an infectious cell will have a
burst size less than or equal to 5 infectious virions (shaded region in Fig
8(A)) is 27% when $n_{I}=1$ compared to 0.07% when $n_{I}=60$ ($>380\times$
less likely). This is why, as $n_{I}$ increases, the probability of infection
extinction once one cell has been infected decreases, and this effect is more
pronounced for larger viral burst sizes.

Figure 9: Visual representation of the burst size distributions explored in
Pearson et al. [28]. Burst size distribution for (A) $n_{I}=1$ (so-called
continuous production in Pearson et al. [28]); (B) a fixed value (so-called
burst production in Pearson et al. [28]); (C) $n_{I}\to\infty$ (so-called
random burst production in Pearson et al. [28]). The grey line represents the
average burst size value
$\mathcal{B}=$19\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$$.

This brings up a misconception, introduced by Pearson et al. [28], that has
been repeatedly stated by others [43, 35, 40, 32]. Pearson et al. [28] derived
the extinction probability for different DSMs: DSMs where infectious virions
are continuously released at a fixed rate over an infectious cell’s lifespan
(so-called continuous production); and DSMs where infectious virions are
released all at once, as a single burst, upon an infected cell’s death (so-
called burst production). In their work, Pearson et al. [28] conclude that
continuous production is more likely to lead to infection extinction than
burst production. But since each infection event, i.e. whether or not each
infectious virion produced results in an infection, can be treated
independently based on the models under consideration, the timing of the
release of each infectious virions cannot and indeed does not affect the
extinction probability.

The difference observed by Pearson et al. [28] in the extinction probabilities
for the DSMs explored is entirely a consequence of the different burst size
distributions used by the authors in the different DSMs, and not their mode of
release (continuous versus burst), as stated therein. The burst size for their
continuous production DSM is geometrically-distributed [10], as depicted in
Fig 9(A), where the geometric distribution is the discrete analogue of the
exponential distribution ($n_{I}=1$ in our DSM). The burst size for their
burst production DSM is a fixed value [28], shown in Fig 9(B), and that for
their random burst production DSM is Poisson-distributed [28], depicted in Fig
9(C). If their burst production DSM is modified so that the burst size
distribution follows the same geometric distribution as that for the
continuous production DSM, both DSMs have the same extinction probability.
With our DSM, the burst size is geometrically-distributed when $n_{I}=1$ and
Poisson-distributed when $n_{I}\to\infty$ (see Methods, Section 4.5). As such,
the comparison of the extinction probability for the so-called continuous
versus random burst production DSMs in Pearson et al. [28] is in fact a
comparison of our continuous production DSM with two extreme values of
$n_{I}$, i.e. $n_{I}=1$ vs. $n_{I}\to\infty$. In other words, what Pearson et
al. [28] actually investigated was the decrease in the extinction probability
as $n_{I}$ increases.

Having explored how the shape parameter of the infectious phase duration
distribution ($n_{I}$) affects the extinction probability, we can now consider
its impact in evaluating and comparing antivirals. For a higher value of
$n_{I}$, since the extinction probability is lower, the establishment
probability, or $1-$(extinction probability), is higher. This means that
although the effectiveness of antivirals in reducing the establishment
probability does not change qualitatively (the better ones remain better), it
does change quantitatively (see Supplementary Material Fig S4). For example,
the choice of $n_{I}$ affects $\varepsilon_{50}$, the efficacy at which the
establishment probability is 50% of its value without antivirals, i.e. at
$\varepsilon=0$ (see Table 2). Czuppon et al. [10] state that for an infection
initiated with 10 infectious virions, $\varepsilon_{50}$ is 81% for an
antiviral acting on $\beta$, and 85% for an antiviral acting on $p$, where the
authors have assumed $n_{I}=1$. When $n_{I}=60$, however, we find that
$\varepsilon_{50}$ is comparable for an antiviral acting on $\beta$ vs. $p$
(85% vs. 86%), and far less (77%), and therefore potentially easier to
achieve, for one acting on $\gamma$. This again highlights the importance of
properly estimating parameters before making quantitative comparisons of
antiviral regimens.

Table 2: Efficacy at which the establishment probability is 50% of its value without antivirals ($\varepsilon_{50}$). Mode of action | Actual $\varepsilon_{50}$ (relative to $\gamma$)
---|---
| for $n_{I}=1$ | for $n_{I}=60$
for an initial inoculum of $V_{0}=1\,$IV
Reducing $\gamma$ | 43% (0%) | 55% (0%)
Reducing $\beta$ | 57% (14%) | 67% (12%)
Reducing $p$ | 77% (34%) | 83% (28%)
for an initial inoculum of $V_{0}=10\,$IV
Reducing $\gamma$ | 71% (0%) | 77% (0%)
Reducing $\beta$ | 81% (10%) | 85% (8%)
Reducing $p$ | 85% (14%) | 86% (9%)

##  3 Discussion

Discrete, stochastic mathematical models (DSMs) of viral infection kinetics
usually do not represent failure of a virion to cause an infection post cell
entry. Yet biologically, there are many way in which a virion, post cell
entry, will fail to cause a cell infection that will yield infectious progeny.
Herein, we constructed a DSM of viral infection kinetics with an explicit
parameter ($\gamma$) to represent the probability that a virion will cause a
productive infection post cell entry. The DSM was first used to estimate the
extinction probability of an infection, i.e. the probability that an infection
will fail to take hold or spread significantly.

Previously, Czuppon et al. [10] evaluated prophylactic antivirals acting on
viral entry ($\beta$) or production ($p$) to reduce the establishment
probability, or $1-$(extinction probability), for a SARS-CoV-2 infection.
Extending this work, we investigated how an antiviral acting on $\gamma$ would
compare against antivirals acting on $\beta$ or $p$. We found that a
prophylactic antiviral acting on $\gamma$ was best at reducing the
establishment probability when infection is initiated with a small number of
infectious virions. When instead an infection is initiated with an initial
number of infectious cells, possibly representative of post-exposure antiviral
therapy, we found that an antiviral acting on $\gamma$ or $p$ caused the same
reduction in the establishment probability, better than that for an antiviral
acting on $\beta$. More generally, we found that the degree to which an
antiviral with a particular mode of action is better than another critically
depends on the chosen infection parameters.

In HIV antiviral therapy, reverse transcriptase inhibitors (RTIs) prevent the
transcription of viral DNA from viral RNA, a step that occurs after viral
entry and is necessary for viral replication. Conway et al. [8] have reported
that, under pre-exposure antiviral therapy, RTIs which they represented as
acting on their DSM’s cell infection rate ($\beta$) are more effective than
protease inhibitors (acting on the virus production rate, $p$), at reducing
the risk of HIV infection. Using our DSM, we can show that an antiviral acting
on $\gamma$, which better captures the mode of action of a RTI, is even more
effective than one acting on $\beta$ at reducing the risk of infection for
pre-exposure antiviral therapy, at least for some of the infection parameter
sets explored in Conway et al. [8] (Supplementary Material Fig S5(A)). These
findings echo the findings reported herein for SARS-CoV-2 parameters, as to
the equal or better performance of antivirals acting to reduce $\gamma$,
compared to $p$ or $\beta$. The introduction of parameter $\gamma$ to capture
productive cell infection failure _after_ an infectious virion has
successfully entered a cell is an important consideration when comparing
antivirals based on their mode of action.

We found that the distribution for the number of cells consumed by DSM-
simulated infections tended to fall into two patterns: those that consumed a
low vs a high number of cells, which one could identify as extinct and
established infections, respectively. We found that the fraction of infections
resulting in a low number of cells consumed matched the infection extinction
probability derived for our DSM, which depends on the infection parameters. We
also found that the median number of cells consumed by established infections
closely matched that predicted by the mean-field mathematical model (MFM),
also expressed in terms of the infection parameters. Therefore, as the
efficacy of an antiviral acting on a particular infection parameter is
increased, it will decrease both the infection’s establishment probability and
the number of cells consumed by infections identified as established. This
finding had 2 important implications. Firstly, since the infection’s
establishment probability and the number of cells consumed by established
infections depend differently on the DSM’s parameters, antivirals acting on
different parameters can decrease one quantity more effectively than the
other. Secondly, and perhaps more importantly, as antiviral efficacy
increases, the distinction in the number of cells consumed by established and
extinct infections vanishes. For example, at a drug efficacy that yields an
equal probability of infection extinction, we found that an antiviral acting
on $\beta$, compared to one acting on $p$, resulted in $\sim 3\times$ more
cells consumed by so-called established infections. At that same efficacy, an
antiviral acting on $\gamma$ yielded both a lower probability of infection
establishment, and resulted in fewer cells consumed overall. Under such
conditions, the average number of cells consumed by all infections,
established and extinct, becomes a more biologically relevant quantity to
track than the probability of infection extinction. Looking at the probability
that infections will consume more than some biologically critical number of
cells, e.g. that identified as sufficient to trigger an immune response and
its associated symptoms or minimally sufficient for transmission, could be
more appropriate.

To facilitate direct comparison of our work herein to past work by others,
most of our results were based on an infectious period, during which an
infected cell is releasing virus, whose duration is exponentially distributed
(Erlang-distributed with shape parameter $n_{I}=1$). But the infectious period
for many different viruses has been shown in vitro to be inconsistent with an
exponential distribution, and instead follow a more normal-like distribution
($n_{I}>\sim 10$). Yan et al. [41] have shown previously that increasing
$n_{I}$ leads to a decrease in the extinction probability. Herein we
established that this happens because the probability that an infectious cell
will stochastically produce a very small burst size becomes vanishingly small
as $n_{I}$ increases. We showed that while $n_{I}$ does not affect the
fraction of cells consumed by established infections, increasing $n_{I}$
decreases the probability of infection extinction, requiring higher antiviral
efficacy to achieve the same risk reduction. Importantly, we demonstrated that
continuous release of virus over time results in the same probability of
infection extinction as virus released at once as a burst, addressing a
misconception introduced in Pearson et al. [28], and since repeated [43, 35,
40, 32]. In the end, the differences observed by Pearson et al. [28] were a
consequence of the different burst size distributions used in comparing
continuous ($n_{I}=1$) and random burst ($n_{I}=\infty$) production, rather
than a consequence of the timing of viral release.

The DSM introduced herein is general and, like the ODE model on which it is
based, it should be applicable to a wide range of different viruses. A logical
improvement to this DSM would be the inclusion of semi-infectious and
defective interfering particles. Our DSM explicitly represents virion that
successfully enter a cell but fail to cause an infection. Our DSM assumes this
process leaves the cell in the same state as if entry had not occurred.
Biologically, repeated viral failure post cell entry would likely trigger
antiviral pathways within the cell, leaving it in a different state than its
naive, susceptible state. Additionally, the accumulation of failed virions, if
it occurs over a sufficiently short time, could eventually add up to one
functional productive infection or lead to defective interfering particle
production. The resulting, more complicated DSM would have more parameters to
identify, and would require a very rich, plentiful data set to validate and
parametrize. This poses a challenge that could be hard to overcome.

##  4 Methods

###  4.1 Parameter values used to generate all figures

The following are the parameters used to generate each figure found within the
manuscript.

Figs 1, 6, 11, 12, 13

$p=$(11.2/24)\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{(}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\cdot\mathrm{h}\mathrm{)}$$,
$N_{\text{cells}}=$4\times
10^{4}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$,
$R_{0}=7.69$, $\tau_{E}=$24/5\text{\,}\mathrm{h}$$,
$\tau_{I}=$(24/0.595)\text{\,}\mathrm{h}$$,
$c=$(10/24)\text{\,}\mathrm{h}^{-1}$$, $n_{E}=1$, $n_{I}=1$ and
$\beta=[cR_{0}/\tau_{I}]/[N_{\text{cells}}(p-R_{0}/\tau_{I})]$ [10]. Also,
$\gamma=$1\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{/}\mathrm{I}\mathrm{V}$$
and $S=$1\text{\,}\mathrm{m}\mathrm{L}$$.

Fig 2

As in Fig 1 but with $p\to(1-\varepsilon)p$, where $\varepsilon=0.79$.

Fig 3(A)

As in Fig 1 but with $\beta\to(1-\varepsilon)\beta$, where $\varepsilon=0.81$.

Fig 4

As in Fig 2 but with $\varepsilon=0.82,\,0.84,\,0.86$.

Fig 5

As in Fig 2 but with $\varepsilon=0.86$,
$N_{\text{cells}}=$4\times[10^{3},10^{5},10^{7}]\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$
and $N_{\text{cells}}/S$ fixed.

Fig 7

As in Fig 1 but with an antiviral acting either on $\beta$, $p$ or $\gamma$,
where $\varepsilon=0.86$.

Fig 8

As in Fig 1 but with $n_{I}=1,\,2,\,7,\,60$.

Fig 9

$\mathcal{B}=$19\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$$.

Fig 3(B,C) and Fig 10 are generated using Eqn. (31) and Eqn. (25)
respectively, neither of which depend directly on infection parameters.

###  4.2 MFM using the expected value of the DSM’s random variables

One way of defining a MFM is by replacing the random variables of the DSM by
their expected values, listed in Table 3. This is trivial for the random
variables that are drawn from well-defined probability distributions. But it
is not for $N^{\text{inf}}$, which is determined using a Python expression
that simulates randomly placing $V^{\text{suc}}$ infectious virions into
$T^{t}$ cells chosen at random with replacement, numpy.random.choice, and
counting the number of distinct cells that received one or more infectious
virions, len(numpy.unique(...)).

Consider then $e_{n}$, the expected number of empty cells after placing $n$
infectious virions into $T^{t}$ cells. After placing $n-1$ infectious virions
into $T^{t}$ cells, there is on average $e_{n-1}$ empty cells by definition.
There is then a $e_{n-1}/T^{t}$ probability to place the last infectious
virion into an empty cell. It follows that

$\displaystyle e_{n}$
$\displaystyle=\frac{e_{n-1}}{T^{t}}(e_{n-1}-1)+\bigg{(}1-\frac{e_{n-1}}{T^{t}}\bigg{)}e_{n-1}$
$\displaystyle e_{n}$
$\displaystyle=\frac{e^{2}_{n-1}}{T^{t}}-\frac{e_{n-1}}{T^{t}}+e_{n-1}-\frac{e^{2}_{n-1}}{T^{t}}$
$\displaystyle e_{n}$
$\displaystyle=e_{n-1}\bigg{(}\frac{T^{t}-1}{T^{t}}\bigg{)}$ (13)

By recursion and since trivially $e_{0}\,=\,T^{t}$,

$\displaystyle e_{n}$
$\displaystyle=T^{t}\bigg{(}\frac{T^{t}-1}{T^{t}}\bigg{)}^{n}$ (14)

The expected number of $N^{\text{inf}}$ cells after placing $V^{\text{suc}}$
infectious virions into $T^{t}$ cells is therefore represented as

$\displaystyle
T^{t}\Bigg{[}1-\bigg{(}\frac{T^{t}-1}{T^{t}}\bigg{)}^{V^{\text{suc}}}\Bigg{]}$
(15)

Table 3: Expected value of the random variables used by the DSM. Random variable | Expected value
---|---
$E_{i}^{\text{out}}$ |  $E_{i}^{t}\cdot\Delta tn_{E}/\tau_{E}$ where $i=1,2,...,n_{E}$
$I_{j}^{\text{out}}$ |  $I_{j}^{t}\cdot\Delta tn_{I}/\tau_{I}$ where $j=1,2,...,n_{I}$
$V^{\text{prod}}$ | $\Delta t\cdot p\,\sum_{j=1}^{n_{I}}I_{j}^{t}$
$V^{\text{decay}},V^{\text{enter}},V^{\text{remain}}$ | $V^{t}\cdot\Delta tc,V^{t}\cdot\Delta t\beta_{\text{}}T^{t}/S,V^{t}\cdot(1-\Delta tc-\Delta t\beta_{\text{}}T^{t}/S)$
$N^{\text{inf}}$ |  $T^{t}\big{[}1-[(T^{t}-1)/T^{t}]^{V^{\text{suc}}}\big{]}$ for $T^{t}\geq 1$ and 0 for $T^{t}<1$
|  where the expected value of $V^{\text{suc}}$ is
$V^{\text{enter}}\cdot\gamma$

###  4.3 MFM-predicted fraction of cells consumed by the infection

Let us derive the MFM-predicted fraction of cells consumed by the infection
given that the infection starts with a number of infectious virions $V(0)$.
Following [26], we have the following relations from Eqn. (2.1),

$\displaystyle\dot{T}+\sum_{i=1}^{n_{E}}\dot{E}_{i}+\sum_{j=1}^{k}\dot{I}_{j}=-n_{I}I_{k}/\tau_{I}$
$\displaystyle\Longrightarrow
I_{k}=-(\tau_{I}/n_{I})\left[\dot{T}+\sum_{i=1}^{n_{E}}\dot{E}_{i}+\sum_{j=1}^{k}\dot{I}_{j}\right]$
(16)
$\displaystyle\frac{\text{d}}{\text{d}t}\ln(T)=\frac{\dot{T}}{T}=-\gamma\beta
V/S$
$\displaystyle\Longrightarrow\frac{1}{\gamma\beta/S}\frac{\text{d}}{\text{d}t}\ln(T)=-V$
(17)

Substituting these relations into Eqn. (2.1) for $\dot{V}$, we obtain

$\displaystyle\dot{V}$
$\displaystyle=-(p\tau_{I}/n_{I})\left[n_{I}\left(\dot{T}+\sum_{i=1}^{n_{E}}\dot{E}_{i}\right)+\sum_{k=1}^{n_{I}}\sum_{j=1}^{k}\dot{I}_{j}\right]+\frac{c}{\gamma\beta/S}\frac{\text{d}}{\text{d}t}\ln(T)+\dot{T}/\gamma$
(18)

Integrating Eqn. (18) from $0$ to $+\infty$ and using the fact that
$E_{i}(0)=E_{i}(\infty)=I_{j}(0)=I_{j}(\infty)=V(\infty)=0$ for
$i=1,2,...,n_{E}$ and $j=1,2,...,n_{I}$, it follows that,

$\displaystyle-V(0)=-p\tau_{I}\left\\{T(\infty)-T(0)\right\\}+\frac{c}{\gamma\beta/S}\left\\{\ln[T(\infty)]-\ln[T(0)]\right\\}+(1/\gamma)\left\\{T(\infty)-T(0)\right\\}$
(19)

Dividing by $T(0)=N_{\text{cells}}$ we have

$\displaystyle-V(0)/N_{\text{cells}}$
$\displaystyle=(1/\gamma-p\tau_{I})[T(\infty)/N_{\text{cells}}-1]+\frac{c}{\gamma\beta
N_{\text{cells}}/S}\ln(T(\infty)/N_{\text{cells}})$
$\displaystyle-V(0)/N_{\text{cells}}+1/\gamma-p\tau_{I}$
$\displaystyle=(1/\gamma-p\tau_{I})T(\infty)/N_{\text{cells}}+\frac{c}{\gamma\beta
N_{\text{cells}}/S}\ln(T(\infty)/N_{\text{cells}})$ $\displaystyle-\gamma
V(0)/N_{\text{cells}}+1-\gamma p\tau_{I}$ $\displaystyle=(1-\gamma
p\tau_{I})T(\infty)/N_{\text{cells}}+\frac{c}{\beta
N_{\text{cells}}/S}\ln(T(\infty)/N_{\text{cells}})$
$\displaystyle-\frac{\gamma\beta/S\cdot V(0)}{c}-\frac{(\beta
N_{\text{cells}}/S)\cdot(\gamma p\tau_{I}-1)}{c}$ $\displaystyle=-\frac{(\beta
N_{\text{cells}}/S)\cdot(\gamma
p\tau_{I}-1)}{c}T(\infty)/N_{\text{cells}}+\ln(T(\infty)/N_{\text{cells}})$
(20)

Substituting the critical fraction of cells uninfected
$T^{*}/N_{\text{cells}}=c/[(\beta N_{\text{cells}}/S)\cdot(\gamma
p\tau_{I}-1)]$ (right-hand side of Eqn. (12)) into Eqn. (20), we obtain

$\displaystyle-\frac{\gamma\beta/S\cdot
V(0)}{c}-\frac{1}{T^{*}/N_{\text{cells}}}$
$\displaystyle=-\frac{T(\infty)/N_{\text{cells}}}{T^{*}/N_{\text{cells}}}+\ln(T(\infty)/N_{\text{cells}})$
$\displaystyle\text{e}^{-[\gamma\beta/S\cdot
V(0)]/c}\text{e}^{-1/(T^{*}/N_{\text{cells}})}$
$\displaystyle=T(\infty)/N_{\text{cells}}\cdot\text{e}^{-(T(\infty)/N_{\text{cells}})/(T^{*}/N_{\text{cells}})}$
$\displaystyle-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}\text{e}^{-[\gamma\beta/S\cdot
V(0)]/c}$
$\displaystyle=-\frac{T(\infty)/N_{\text{cells}}}{T^{*}/N_{\text{cells}}}\cdot\text{e}^{-(T(\infty)/N_{\text{cells}})/(T^{*}/N_{\text{cells}})}$
(21)

Eqn. (21) is of the form $z=w\exp(w)$. The Lambert $W$ function is the
function that gives the inverse relation, i.e. $W(z)=w$. Therefore, we have
the following,

$\displaystyle-\frac{T(\infty)/N_{\text{cells}}}{T^{*}/N_{\text{cells}}}=W_{0}\left(-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}\text{e}^{-[\gamma\beta/S\cdot
V(0)]/c}\right)$ $\displaystyle
T(\infty)/N_{\text{cells}}=-T^{*}/N_{\text{cells}}\cdot
W_{0}\left(-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}\text{e}^{-[\gamma\beta/S\cdot
V(0)]/c}\right)$ (22)

where here we make use of $W_{0}$ the upper branch of the Lambert $W$
function. The upper branch is used because $T(\infty)/T^{*}<1$ since
$T(\infty)$ is always reached after $T^{*}$ and therefore is always smaller.

The fraction of cells consumed by the infection,
$1-T(\infty)/N_{\text{cells}}$, is then given by

$\displaystyle 1-T(\infty)/N_{\text{cells}}=1+T^{*}/N_{\text{cells}}\cdot
W_{0}\left(-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}\text{e}^{-[\gamma\beta/S\cdot
V(0)]/c}\right)$ (23)

If instead there was initially a number of infectious cells but no infectious
virions ($I_{1}(0)\neq 0$, $V(0)=0$) then Eqn. (19) would be

$\displaystyle
0=-p\tau_{I}\left\\{T(\infty)-T(0)-I_{1}(0)\right\\}+\frac{c}{\gamma\beta/S}\left\\{\ln[T(\infty)]-\ln[T(0)]\right\\}+(1/\gamma)\left\\{T(\infty)-T(0)\right\\}$
(24)

which is equivalent to Eqn. (19) where $V(0)$ is replaced with
$p\tau_{I}I_{1}(0)$. This means that Eqn. (23) would be the same but with
$V(0)$ replaced with $p\tau_{I}I_{1}(0)$.

For $V(0)$ such that $[\gamma\beta/S\cdot V(0)]/c\approx 0$, Eqn. (23)
simplifies to

$\displaystyle 1-T(\infty)/N_{\text{cells}}=1+T^{*}/N_{\text{cells}}\cdot
W_{0}\left(-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}\right)$
(25)

The Taylor series of $W_{0}(x)$ around 0 is
$W_{0}(x)=x-x^{2}+3x^{3}/2-\dots\approx x$. It follows that, when
$T^{*}/N_{\text{cells}}\ll 1$, Eqn. (25) simplifies to

$\displaystyle 1-T(\infty)/N_{\text{cells}}\approx
1+T^{*}/N_{\text{cells}}\cdot-\frac{\text{e}^{-1/(T^{*}/N_{\text{cells}})}}{T^{*}/N_{\text{cells}}}$
$\displaystyle 1-T(\infty)/N_{\text{cells}}\approx
1-\text{e}^{-1/(T^{*}/N_{\text{cells}})}$ (26)

Figure 10: MFM-predicted fraction of cells consumed by the infection. The MFM-
predicted fraction of cells consumed by the infection
($1-T(\infty)/N_{\text{cells}}$, Eqn. (25), blue solid line) as a function of
the fraction of cells uninfected such that the reproductive number is 1
($T^{*}/N_{\text{cells}}$). The black dashed line is
$1-\exp[-1/(T^{*}/N_{\text{cells}})]$.

Fig 10 shows that the fraction of cells consumed by the infection,
$1-T(\infty)/N_{\text{cells}}$, is simply a strictly decreasing function of
$T^{*}/N_{\text{cells}}$ and that, when $T^{*}/N_{\text{cells}}\ll 1$,
$1-T(\infty)/N_{\text{cells}}\approx
1-\text{e}^{-1/(T^{*}/N_{\text{cells}})}$.

Let us now derive an expression for the reproductive number $R(t)$ as a
function of the fraction of cells uninfected $T(t)/N_{\text{cells}}$ for a
given basic reproductive number $R_{0}$ (Eqn. (9)) and critical fraction of
cells uninfected $T^{*}/N_{\text{cells}}$ (Eqn. (12)).

To begin, using Eqn. (9) and the left-hand side of Eqn. (12), we have the
following relation,

$\displaystyle\frac{1}{R_{0}}\frac{p\tau_{I}\cdot\gamma\beta
N_{\text{cells}}/S}{c+\beta N_{\text{cells}}/S}$
$\displaystyle=\frac{p\tau_{I}\cdot\gamma\beta T^{*}/S}{c+\beta T^{*}/S}$
$\displaystyle\frac{1}{R_{0}}\frac{N_{\text{cells}}}{c/(\beta/S)+N_{\text{cells}}}$
$\displaystyle=\frac{T^{*}}{c/(\beta/S)+T^{*}}$ $\displaystyle
N_{\text{cells}}[c/(\beta/S)]+N_{\text{cells}}T^{*}$
$\displaystyle=R_{0}[c/(\beta/S)]T^{*}+R_{0}N_{\text{cells}}T^{*}$
$\displaystyle c/(\beta/S)\cdot(N_{\text{cells}}-R_{0}T^{*})$
$\displaystyle=N_{\text{cells}}T^{*}(R_{0}-1)$ $\displaystyle c/(\beta/S)$
$\displaystyle=\frac{N_{\text{cells}}T^{*}(R_{0}-1)}{N_{\text{cells}}-R_{0}T^{*}}$
(27)

The left-hand side of Eqn. (12) can be rearranged to

$\displaystyle\gamma p\tau_{I}$ $\displaystyle=\frac{c+\beta T^{*}/S}{\beta
T^{*}/S}$ $\displaystyle\gamma p\tau_{I}$
$\displaystyle=\frac{c/(\beta/S)+T^{*}}{T^{*}}$ (28)

Substituting Eqn. (27) into Eqn. (28), we obtain

$\displaystyle\gamma p\tau_{I}$
$\displaystyle=\frac{1}{T^{*}}\left[\frac{N_{\text{cells}}T^{*}(R_{0}-1)}{N_{\text{cells}}-R_{0}T^{*}}+T^{*}\right]$
$\displaystyle\gamma p\tau_{I}$
$\displaystyle=\frac{N_{\text{cells}}(R_{0}-1)}{N_{\text{cells}}-R_{0}T^{*}}+1$
$\displaystyle\gamma p\tau_{I}$
$\displaystyle=\frac{R_{0}(N_{\text{cells}}-T^{*})}{N_{\text{cells}}-R_{0}T^{*}}$
(29)

The reproductive number can also be written as

$\displaystyle R(t)$ $\displaystyle=\frac{p\tau_{I}\cdot\gamma\beta
T(t)/S}{c+\beta T(t)/S}$ $\displaystyle R(t)$ $\displaystyle=\frac{\gamma
p\tau_{I}\cdot T(t)}{c/(\beta/S)+T(t)}$ (30)

Substituting Eqn. (27) and Eqn. (28) into Eqn. (30), we then have

$\displaystyle R(t)$
$\displaystyle=\frac{\frac{R_{0}(N_{\text{cells}}-T^{*})}{N_{\text{cells}}-R_{0}T^{*}}\cdot
T(t)}{\frac{N_{\text{cells}}T^{*}(R_{0}-1)}{N_{\text{cells}}-R_{0}T^{*}}+T(t)}$
$\displaystyle R(t)$
$\displaystyle=\frac{R_{0}T(t)\cdot(N_{\text{cells}}-T^{*})}{N_{\text{cells}}T^{*}(R_{0}-1)+T(t)\cdot(N_{\text{cells}}-R_{0}T^{*})}$
$\displaystyle R(t)$ $\displaystyle=\frac{R_{0}\cdot
T(t)/N_{\text{cells}}\cdot(1-T^{*}/N_{\text{cells}})}{T^{*}/N_{\text{cells}}\cdot(R_{0}-1)+T(t)/N_{\text{cells}}\cdot(1-R_{0}\cdot
T^{*}/N_{\text{cells}})}$ (31)

###  4.4 Probability that an infectious virion succeeds at causing a
productive cell infection

Let us derive the probability that an infectious virion is successful at
causing a productive cell infection, $\mathcal{P}_{V\to\,I}$, in a population
of fully susceptible, uninfected cells ($T=N_{\text{cells}}$).

In a time step $\Delta t$, the probability that an infectious virion neither
loses infectivity nor enters a cell is $(1-\Delta tc-\Delta t\beta
N_{\text{cells}}/S)$, the probability that an infectious virion enters a cell
is $\Delta t\beta_{\text{}}N_{\text{cells}}/S$ and the probability that an
infectious virion post cell entry will be successful at causing a cell
infection is $\gamma$. The probability that an infectious virion is successful
at causing a productive cell infection in time $t\,=\,k\Delta t$ is then given
by the following expression,

$\displaystyle(1-\Delta tc-\Delta
t\beta_{\text{}}N_{\text{cells}}/S)^{k-1}(\Delta
t\beta_{\text{}}N_{\text{cells}}/S)\gamma$ (32)

More generally, the probability that this happens in any time $t=k\Delta t$
where $0\leq t\leq t_{\text{incub}}$ and $t_{\text{incub}}\,=\,n\Delta t$ is
the incubation time, is then expressed as

$\displaystyle\sum_{k\,=\,1}^{n}\left[(1-\Delta tc-\Delta
t\beta_{\text{}}N_{\text{cells}}/S)^{k-1}(\Delta
t\beta_{\text{}}N_{\text{cells}}/S)\gamma\right]$ $\displaystyle=\left[\Delta
t\gamma\beta_{\text{}}N_{\text{cells}}/S\right]\sum_{k\,=\,1}^{n}\left[1-\Delta
t(c+\beta_{\text{}}N_{\text{cells}}/S)\right]^{k-1}$
$\displaystyle=\left[\Delta
t\gamma\beta_{\text{}}N_{\text{cells}}/S\right]\underbracket{\sum_{k\,=\,0}^{n-1}\left[1-\Delta
t(c+\beta_{\text{}}N_{\text{cells}}/S)\right]^{k}}_{\text{partial sum of
geometric series}}$ $\displaystyle=\left[\Delta
t\gamma\beta_{\text{}}N_{\text{cells}}/S\right]\left[\frac{1-[1-\Delta
t(c+\beta N_{\text{cells}}/S)]^{n}}{1-[1-\Delta t(c+\beta
N_{\text{cells}}/S)]}\right]$ $\displaystyle=\left[\Delta
t\gamma\beta_{\text{}}N_{\text{cells}}/S\right]\left[\frac{1-[1-\Delta
t(c+\beta N_{\text{cells}}/S)]^{n}}{\Delta t(c+\beta
N_{\text{cells}}/S)}\right]$
$\displaystyle=\left[\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta
N_{\text{cells}}/S}\right]\left[1-\left(1-\frac{t_{\text{incub}}(c+\beta
N_{\text{cells}}/S)}{n}\right)^{n}\right]$ (33)

Taking the limit of Eqn. (33) as $n\to\infty$ (i.e. $\Delta t\to 0$),

$\displaystyle\left[\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta
N_{\text{cells}}/S}\right]\left[1-\lim_{n\to\infty}\left(1-\frac{t_{\text{incub}}(c+\beta
N_{\text{cells}}/S)}{n}\right)^{n}\right]$ $\displaystyle=$
$\displaystyle\left[\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta
N_{\text{cells}}/S}\right]\left[1-\text{e}^{-t_{\text{incub}}(c+\beta
N_{\text{cells}}/S)}\right]$ (34)

As $t_{\text{incub}}\to\infty$, Eqn. (34) is then given by the following
expression,

$\displaystyle\mathcal{P}_{V\to\,I}=\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta_{\text{}}N_{\text{cells}}/S}$
(35)

Fig 11 shows that the frequency of success of an infectious virion to cause a
productive cell infection in a population of uninfected cells, generated from
$10^{5}$ DSM simulations, is in agreement with Eqn. (35), over a wide range of
infection parameters.

Figure 11: Probability that an infectious virion is successful at causing a
productive cell infection. Frequency of success of an infectious virion to
cause a productive cell infection in a population of uninfected cells for
$10^{5}$ DSM simulations (red scatter) compared to $\mathcal{P}_{V\to\,I}$ the
derived expression for the probability of success of an infectious virion to
cause a productive cell infection in a population of uninfected cells (black
curve, Eqn. (35)) while either varying (A) $c$, (B) $\gamma$ or (C)
$\beta_{\text{}}N_{\text{cells}}/S$. Unless otherwise specified, the
parameters were the same as in Fig 1.

###  4.5 Probability that an infectious cell produces $m$ infectious virions

Let us derive the probability that an infectious cell produces $m$ infectious
virions over its lifespan ($\mathcal{P}_{I\to\,m\,V}$).

Assuming that successive infectious virion productions are independent events
that happen under constant rate $p$, then the probability that an infectious
cell produces $m$ infectious virions over its lifespan $t$ is given by
$\text{Poisson}(m|\lambda=t\,p)$.

The negative binomial (NB) distribution represents the probability that there
are $k$ successes in a sequence of independent and identically distributed
Bernoulli trials with probability of success $p$ before a specified number of
failures $r$ occurs, namely,

$\displaystyle\text{NB}(k|r,\,p)={k+r-1\choose k}(1-p)^{r}p^{m}$ (36)

The probability that an infectious cell transition to the next compartment is
given by $\text{Binomial}(1|n=1,\,p_{I}=\Delta tn_{I}/\tau_{I})$. Therefore
the probability that an infectious cell transition to the next compartment for
$n_{I}$ time steps and does not for $n-n_{I}$ time steps is then given by
$\text{NB}(n_{I}|r=n-n_{I},\,p_{I}=\Delta t\,n_{I}/\tau_{I})$. It is possible
to show that the continuous analogue of this discrete probability distribution
is the Erlang distribution
($\text{Erlang}(t|k=n_{I},\,\lambda=n_{I}/\tau_{I})$). As a result, we make
use of this probability density function to express the probability that the
infectious lifespan has duration $t$.

The probability that an infectious cell produces $m$ infectious virions over
its lifespan, i.e. that the burst size of an infected cell is $m$, can then be
derived by integrating over time $t$, the joint probability distribution that
an infectious cell produces $m$ infectious virions given a lifespan of
duration $t$ and that an infectious cell has a lifespan of duration $t$,
namely,

$\displaystyle\mathcal{P}_{I\to\,m\,V}=\int_{0}^{\infty}\text{Poisson}(m|\lambda=pt)\cdot\text{Erlang}(t|k=n_{I},\,\lambda=n_{I}/\tau_{I})\text{d}t=\text{NB}(m|r=n_{I},\,p_{B}=\mathcal{B}/(n_{I}+\mathcal{B}))$
(37)

where $\mathcal{B}=p\tau_{I}$ is the average burst size. When $n_{I}=1$,
$\mathcal{P}_{I\to\,m\,V}=\text{NB}(m|r=1,\,p_{B}=\mathcal{B}/(1+\mathcal{B}))=\text{Geom}(m|p=1/(1+\mathcal{B}))$
where Geom represents the geometric distribution. When $n_{I}\to\infty$,
$\mathcal{P}_{I\to\,m\,V}=\text{NB}(m|r=n_{I},\,p_{B}=\mathcal{B}/(n_{I}+\mathcal{B}))=\text{Poisson}(m|\lambda=\mathcal{B})$.

Figure 12: Probability that an infectious cell will produce $m$ infectious
virions. Normalized histogram of the burst size, i.e. the total number of
infectious virions produced by one infectious cell over its lifespan, for
$10^{5}$ DSM simulations (blue x), compared to the derived expression for the
probability mass function of the burst size distribution (orange circle, Eqn.
(37)) while either varying (A–C) $\tau_{I}$, (D–F) $p$ or (G–I) $n_{I}$. The
black lines show the average burst size values $\mathcal{B}=p\tau_{I}$. Unless
otherwise specified, the parameters were the same as in Fig 1.

Fig 12 shows that the normalized histogram of the burst size, generated from
$10^{5}$ DSM simulations, is in agreement with Eqn. (37), over a wide range of
infection parameters.

###  4.6 Extinction probability

Let us derive the extinction probability of an infection. Following Pearson et
al. [28], the extinction probability of an infection with initially one
infectious virion ($\mathcal{P}_{V\to\,\text{Extinction}}$) can be written as

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\cdot\mathcal{P}_{I\to\,\text{Extinction}}$
(38)

since the infection can become extinct if the initial infectious virion fails
to cause a productive cell infection with probability
$\mathcal{P}_{V\to\,\varnothing}$ or if it does cause a productive cell
infection with probability
$\mathcal{P}_{V\to\,I}=(1-\mathcal{P}_{V\to\,\varnothing})$ but that cell
infection then leads to extinction with probability
$\mathcal{P}_{I\to\,\text{Extinction}}$.

Likewise, the extinction probability of an infection with initially one
infectious cell ($\mathcal{P}_{I\to\,\text{Extinction}}$) can be expressed as

$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}=\sum_{m=0}^{\infty}\mathcal{P}_{I\to\,m\,V}\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}$
(39)

since the initial infectious cell will produce some number $m$ of infectious
virions over its lifespan with probability $\mathcal{P}_{I\to\,m\,V}$ and each
one of these produced infectious virions can be seen as an independent
infection event that can lead to extinction with probability
$\mathcal{P}_{V\to\,\text{Extinction}}$.

Substituting our derived expression for $\mathcal{P}_{I\to\,m\,V}$ (Eqn. (37))
into Eqn. (39) yields

$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}$
$\displaystyle=\sum_{m=0}^{\infty}\text{NB}(m|r=n_{I},p_{B}=\mathcal{B}/(n_{I}+\mathcal{B}))\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}$
$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}$
$\displaystyle=\sum_{m=0}^{\infty}{m+n_{I}-1\choose
m}\left[1-\frac{\mathcal{B}}{n_{I}+\mathcal{B}}\right]^{n_{I}}\left[\frac{\mathcal{B}}{n_{I}+\mathcal{B}}\right]^{m}\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}$
$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}$
$\displaystyle=\left[\frac{n_{I}}{n_{I}+\mathcal{B}}\right]^{n_{I}}\sum_{m=0}^{\infty}{m+n_{I}-1\choose
m}\left[\frac{\mathcal{B}}{n_{I}+\mathcal{B}}\cdot\mathcal{P}_{V\to\,\text{Extinction}}\right]^{m}$
(40)

The following binomial series can be used to simplify Eqn. (40),

$\displaystyle\frac{1}{(1-x)^{s}}=\sum_{k}{s+k-1\choose k}x^{k}$ (41)

Let us derive some related expressions,

$\displaystyle x$
$\displaystyle\coloneqq\frac{\mathcal{B}}{n_{I}+\mathcal{B}}\cdot\mathcal{P}_{V\to\,\text{Extinction}}$
(42) $\displaystyle\Longrightarrow 1-x$
$\displaystyle=\frac{n_{I}+\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}+\mathcal{B}}$
$\displaystyle\Longrightarrow\frac{1}{(1-x)^{n_{I}}}$
$\displaystyle=\left[\frac{n_{I}+\mathcal{B}}{n_{I}+\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}\right]^{n_{I}}$
(43)

Eqn. (40) can then be simplified to

$\displaystyle\mathcal{P}_{I\to\,\text{Extinction}}$
$\displaystyle=\left[\frac{n_{I}}{n_{I}+\mathcal{B}}\right]^{n_{I}}\left[\frac{n_{I}+\mathcal{B}}{n_{I}+\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}\right]^{n_{I}}$
$\displaystyle=\left[\frac{n_{I}}{n_{I}+\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}\right]^{n_{I}}$
$\displaystyle=\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}$
(44)

Substituting Eqn. (44), our derived expression for $\mathcal{P}_{V\to\,I}$
(Eqn. (35)) and $\mathcal{P}_{V\to\,\varnothing}=1-\mathcal{P}_{V\to\,I}$ into
Eqn. (38) yields

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}$
(45) $\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}$
$\displaystyle=\left[1-\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta_{\text{}}N_{\text{cells}}/S}\right]+\frac{\gamma\beta_{\text{}}N_{\text{cells}}/S}{c+\beta_{\text{}}N_{\text{cells}}/S}\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}$
$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}$
$\displaystyle=1-\frac{\gamma}{c/(\beta
N_{\text{cells}}/S)+1}+\frac{\gamma}{c/(\beta
N_{\text{cells}}/S)+1}\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}$
$\displaystyle 0$ $\displaystyle=1-\frac{[1+c/(\beta
N_{\text{cells}}/S)]}{\gamma}[1-\mathcal{P}_{V\to\,\text{Extinction}}]-\left[\frac{\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}{n_{I}}+1\right]^{-n_{I}}$
(46)

This expression does not seem to have an analytical solution for
$\mathcal{P}_{V\to\,\text{Extinction}}$ but a solution can be found
numerically by finding the roots of the expression on the right-hand side of
Eqn. (46) using scipy.optimize.fsolve. Fig 13 shows that the frequency of
extinction given an infection initiated with only one infectious virion,
generated from $10^{5}$ DSM simulations, is in agreement with the probability
of extinction given an infection initiated with only one infectious virion,
over a wide range of infection parameters. The extinction probability given
any number of infectious virions $V_{0}$ and infectious cells $I_{0}$ is given
by
$\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{V_{0}}\cdot\left(\mathcal{P}_{I\to\,\text{Extinction}}\right)^{I_{0}}$.
It is important to note that, as others [28, 41], we also make the assumption
in this derivation that $T=N_{\text{cells}}$ is constant.

Figure 13: Extinction probability given an infection initiated with only one
infectious virion. Frequency of failure of an infection initiated with only
one infectious virion to cause more than 0.01% of cells to be infected by the
end for $10^{5}$ DSM simulations (red scatter) compared to
$\mathcal{P}_{V\to\,\text{Extinction}}$ the derived expression for the
extinction probability given an infection initiated with only one infectious
virion (black curve, Eqn. (46)) while either varying (A) $\tau_{I}$, (B) $p$,
(C) $n_{I}$, (D) $c$, (E) $\gamma$ or (F) $\beta N_{\text{cells}}/S$. Unless
otherwise specified, the parameters were the same as in Fig 1.

When $n_{I}=1$, Eqn. (45) simplifies to

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\left[\frac{1}{1+\mathcal{B}(1-\mathcal{P}_{V\to\,\text{Extinction}})}\right]$
(47)

In this case,
$\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
is a solution to the above expression as,

$\displaystyle\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\left[\frac{1}{1+\mathcal{B}[1-(\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B})]}\right]$
$\displaystyle\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\left[\frac{1}{\mathcal{B}(1-\mathcal{P}_{V\to\,\varnothing})}\right]$
$\displaystyle\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\left[\frac{1}{\mathcal{B}\cdot\mathcal{P}_{V\to\,I}}\right]$
$\displaystyle\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
$\displaystyle=\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
$\displaystyle\therefore\text{LHS}$ $\displaystyle=\text{RHS}$

$\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
is equivalent to the expression in [9] (Eq. (6) therein).
$\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+1/\mathcal{B}$
can also be written in terms of the basic reproductive number
$R_{0}=\mathcal{B}\cdot\mathcal{P}_{V\to\,I}$ (see Eqn. (9)), i.e.
$\mathcal{P}_{V\to\,\text{Extinction}}=1-(R_{0}-1)/\mathcal{B}$. Also,
$\mathcal{P}_{V\to\,\text{Establishment}}$ or
$1-\mathcal{P}_{V\to\,\text{Extinction}}$ is then given by
$\mathcal{P}_{V\to\,I}-1/\mathcal{B}$ or $(R_{0}-1)/\mathcal{B}$.

When $n_{I}\to\infty$,
$\mathcal{P}_{I\to\,m\,V}=\text{Poisson}(m|\lambda=\mathcal{B})$, therefore,
we obtain,

$\displaystyle\mathcal{P}_{V\to\,\text{Extinction}}=\mathcal{P}_{V\to\,\varnothing}+\mathcal{P}_{V\to\,I}\cdot\sum_{m=0}^{\infty}\text{Poisson}(m|\lambda=\mathcal{B})\cdot\left(\mathcal{P}_{V\to\,\text{Extinction}}\right)^{m}$
(48)

which is equivalent to the expression in Pearson et al. [28] for their random
burst production [DSM] (Eq. (26) therein).

##  5 Acknowledgements

This work was supported in part by Discovery Grants 355837-2013 and 2022-03744
(CAAB) from the Natural Sciences and Engineering Research Council of Canada
(www.nserc-crsng.gc.ca) and by the Kato Sechi Female PI Incentive
(www.riken.jp/en/careers/programs/kato_sechi) and Interdisciplinary
Theoretical and Mathematical Sciences (iTHEMS, ithems.riken.jp) programmes at
RIKEN (CAAB).

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Supplementary Material for
Stochastic failure of cell infection post viral entry: Implications for
infection outcomes and antiviral therapy

##  Appendix S1 Inoculum size

In the main text, the initial number of virions
$V_{0}=$1\text{\,}\mathrm{I}\mathrm{V}$$ and we explored varying the efficacy
$\varepsilon$. Now, we fix $\varepsilon=0.8$ and we explore varying $V_{0}$.
Fig S1(A) and Fig S1(B) shows the establishment probability or the MFM-
predicted fraction of cells consumed by the infection respectively as a
function of $V_{0}$ for antivirals with efficacy $\varepsilon=0.8$ acting on
$\beta$, $p$ or $\gamma$. As $V_{0}$ increases, the establishment probability
tends to 100% and differences in the establishment probability for the 3
antiviral modes of action disappear. For $V_{0}\ll N_{\text{cells}}=$4\times
10^{4}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}$$, the MFM-
predicted fraction of cells consumed by the infection is insensitive to
$V_{0}$. This suggests that this would also be true for the median of the DSM
simulated distribution of the fraction of cells consumed by established
infections.

Figure S1: Inoculum size. (A) The establishment probability or (B) the MFM-
predicted fraction of cells consumed by the infection as a function of the
initial number of infectious virions $V_{0}$ for antivirals with efficacy
$\varepsilon=0.8$ acting either to reduce the virus entry rate,
$\beta\to(1-\varepsilon)\beta$ (blue solid line), the virus production rate,
$p\to(1-\varepsilon)p$ (orange dashed line), or the probability of a
successful cell infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$
(green dotted line). The parameters were the same as in Fig 1.

##  Appendix S2 Parameter set with a higher burst size

Table S1: Parameter sets used in Czuppon et al. [10].

Parameter set |  $p$ [IV/($\text{cell}\cdot\text{h})$] |  $N_{\text{cells}}$ [cells] | ${R_{0}}^{*}$ |  $\tau_{I}$ [h] |  $c$ [h-1] | $n_{I}$
---|---|---|---|---|---|---
Lower burst size | 11.2/24 | $4\times 10^{4}$ | 7.69 | 24/0.595 | 10/24 | 1
Higher burst size | 112/24 | $4\times 10^{3}$ | 7.69 | 24/0.595 | 10/24 | 1

* $\beta=[cR_{0}/\tau_{I}]/[N_{\text{cells}}(p-R_{0}/\tau_{I})]$

Here, we investigate the other parameter set explored in Czuppon et al. [10].
The difference between the two parameter sets is a 10-fold decrease of the
number of cells $N_{\text{cells}}$, a 10-fold increase of the virus production
rate $p$ (hence, of the average burst size $\mathcal{B}=p\tau_{I}$) and a
corresponding $\sim$10-fold decrease of the virus entry rate $\beta
N_{\text{cells}}/S$ (see Table S1).

Fig S2 shows both the establishment probability and the MFM-predicted fraction
of cells consumed by the infection given that there is initially only one
infectious virion as a function of antiviral efficacy ($\varepsilon$) for
antivirals acting on $\beta$, $p$ or $\gamma$ for the parameter set with (A)
the lower burst size
($\mathcal{B}=$18.8\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$$)
or (B) the higher burst size
($\mathcal{B}=$188\text{\,}\mathrm{I}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}$$).

For the parameter set with the higher burst size, an antiviral acting on
$\gamma$ or $\beta$ have a similar effect on the establishment probability,
better than that for an antiviral acting on $p$. Due to the higher burst size,
the establishment probability is approximately given by the probability that
the initial infectious virion causes a productive cell infection,
$\mathcal{P}_{V\to\,\text{Establishment}}=\mathcal{P}_{V\to\,I}-1/\mathcal{B}\approx\mathcal{P}_{V\to\,I}$
for $n_{I}=1$ (see Methods, Section 4.6 for more details). The probability
that the initial infectious virion will cause a productive cell infection is
given by the ratio between the rate of successful cell infection per
infectious virion and the rate of virion loss $(\gamma\beta
N_{\text{cells}}/S)/(c+\beta N_{\text{cells}}/S)$. Since the rate of virion
entry into cells is much lower than the rate of virion loss of infectivity
($\beta N_{\text{cells}}/S\ll c$) the rate of virion loss is mostly governed
by the rate of virion loss of infectivity ($(c+\beta
N_{\text{cells}}/S)\approx c$). Therefore, the establishment probability is
approximately given by $(\gamma\beta N_{\text{cells}}/S)/c$ which is affected
by $\gamma$ or $\beta$ the same but not affected by $p$.

For the parameter set with the higher burst size, all antivirals affect
comparably the MFM-predicted fraction of cells consumed by the infection. The
MFM-predicted fraction of cells consumed is a function of the fraction of
cells uninfected such that the reproductive number is 1, i.e.
$T^{*}/N_{\text{cells}}=c/[(\beta N_{\text{cells}}/S)\cdot(\gamma
p\tau_{I}-1)]$ (see Methods, Section 4.3 for more details). Due to the higher
burst size ($\mathcal{B}=p\tau_{I}$), $\gamma p\tau_{I}-1\approx\gamma
p\tau_{I}$. Therefore, the MFM-predicted fraction of cells consumed is
approximately a function of $c/[(\beta N_{\text{cells}}/S)\cdot(\gamma
p\tau_{I})]$ which is affected by $\beta$, $p$ or $\gamma$ the same. This
suggests that all antivirals also have a similar effect on the median of the
DSM simulated distribution of the fraction of cells consumed by established
infections.

Figure S2: Lower vs. higher burst size parameter set. The establishment
probability (dark colours) or the MFM-predicted fraction of cells consumed by
the infection (pale colours) given that there is initially only one infectious
virion as a function of efficacy ($\varepsilon$) for antivirals acting either
to reduce the virus entry rate, $\beta\to(1-\varepsilon)\beta$ (blue solid
lines), the virus production rate, $p\to(1-\varepsilon)p$ (orange dashed
lines), or the probability of a successful cell infection post viral entry,
$\gamma\to(1-\varepsilon)\gamma$ (green dotted lines) for the parameter set
with (A) the lower burst size or (B) the higher burst size (see Table S1)
where our additional infection parameters were set to
$\gamma=$1\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{/}\mathrm{I}\mathrm{V}$$
and $S=$1\text{\,}\mathrm{m}\mathrm{L}$$.

##  Appendix S3 Post-exposure antiviral therapy

In the main text, we have considered pre-exposure antiviral therapy for an
infection initiated with a number of infectious virions. Now, as Czuppon et
al. [10] have done, let us also explore antiviral therapy for an infection
initiated with only one infectious cell. This may be representative of post-
exposure antiviral therapy as it is possible that by the time an antiviral has
been given after exposure, the virus has had time to cause some infectious
cells.

With $n_{I}=1$, the establishment probability given that there is initially
one infectious cell (see Methods, Section 4.6 for derivation) is given by

$\displaystyle\mathcal{P}_{I\to\,\text{Establishment}}=1-\frac{1}{R_{0}}=1-\frac{c+\beta
N_{\text{cells}}/S}{p\tau_{I}\cdot\gamma\beta N_{\text{cells}}/S}$ (S1)

Fig S3 shows the establishment probability and the MFM-predicted fraction of
cells consumed by the infection given that there is initially only (A) one
infectious virion or (B) one infectious cell as a function of antiviral
efficacy ($\varepsilon$) for antivirals acting on $\beta$, $p$ or $\gamma$.

Like Czuppon et al. [10], with initially one infectious cell, we find that an
antiviral acting on $p$ is better than an antiviral acting on $\beta$ to
reduce the establishment probability. Reducing $p$ affects only the
denominator in the expression for the establishment probability given that
there is initially one infectious cell (Eqn. (S1)). Whereas, reducing $\beta$
affects both the numerator and denominator (Eqn. (S1)). Unlike Czuppon et al.
[10], with initially one infectious cell, we find that an antiviral acting on
$p$ or $\gamma$ have the same effect on the establishment probability (Eqn.
(S1)).

In addition, there is no noticeable difference in the MFM-predicted fraction
of cells consumed by the infection given that there is initially one
infectious virion or one infectious cell. The initial number of infectious
virions or cells is small and hence has a negligible effect on the MFM-
predicted fraction of cells consumed by the infection (see Eqn. (25)). This
suggests that this would also be true for the median of the DSM simulated
distribution of the fraction of cells consumed by established infections.

Figure S3: Pre-exposure vs. post-exposure antiviral therapy. The establishment
probability (dark colours) or the MFM-predicted fraction of cells consumed by
the infection (pale colours) given that there is initially only (A) one
infectious virion or (B) one infectious cell as a function of efficacy
($\varepsilon$) for antivirals acting either to reduce the virus entry rate,
$\beta\to(1-\varepsilon)\beta$ (blue solid lines), the virus production rate,
$p\to(1-\varepsilon)p$ (orange dashed lines), or the probability of a
successful cell infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$
(green dotted lines). The parameters were the same as in Fig 1.

##  Appendix S4 Additional figures

Figure S4: Effect of infectious phase duration distribution on the ability of
antivirals to reduce the establishment probability. The establishment
probability given an infection initiated with (A) 1 infectious virion or (B)
10 infectious virions as a function of efficacy ($\varepsilon$) for antivirals
acting either to reduce the virus entry rate, $\beta\to(1-\varepsilon)\beta$
(blue solid lines), the virus production rate, $p\to(1-\varepsilon)p$ (orange
dashed lines), or the probability of a successful cell infection post viral
entry, $\gamma\to(1-\varepsilon)\gamma$ (green dotted lines) where either
$n_{I}=1$ (dark colours) or $n_{I}=60$ (pale colours). Unless otherwise
specified, the parameters were the same as in Fig 1. Figure S5: Mean infection
risk reduction. (A–C) Mean risk reduction as a function of efficacy
($\varepsilon$) for antivirals acting either to reduce the virus entry rate,
$\beta\to(1-\varepsilon)\beta$ (blue solid line), or the probability of a
successful cell infection post viral entry, $\gamma\to(1-\varepsilon)\gamma$
(green dotted line), for different infection parameter sets taken from Conway
et al. [8]. (D–F) Mean risk reduction in (A–C) for an antiviral acting to
reduce $\gamma$ minus that for an antiviral acting to reduce $\beta$.
Following Conway et al. [8], the risk of infection given that there is
initially $n$ virions is
$\text{risk}(n)=1-[1-Q_{c}(1-\mathcal{P}_{V\to\,\text{Extinction}})]^{n}$
where $Q_{c}$ is the fraction of infectious virions in the exposure inoculum.
The mean risk of infection is then
$\sum_{n=0}^{N_{\text{max}}}\text{risk}(n)/(N_{\text{max}}+1)$ where
$N_{\text{max}}$ is the maximum inoculum size determined such that the mean
risk of infection without antivirals is $\sim$0.3%. Finally, the mean risk
reduction for a given efficacy $\varepsilon\neq 0$ is calculated as
$100\%\cdot(\text{mean risk}_{\varepsilon=0}-\text{mean risk}_{\varepsilon\neq
0})/(\text{mean risk}_{\varepsilon=0})$. The infection parameters were taken
from Conway et al. [8], i.e. $\tau_{I}=$24\text{\,}\mathrm{h}$$,
$c=$23/24\text{\,}\mathrm{h}^{-1}$$, $n_{I}=1$, $p=pQ_{a}/24$, $R_{0}=8$,
$\beta N_{\text{cells}}=(cR_{0}/\tau_{I})/(p-R_{0}/\tau_{I})$,
$Q_{c}=10^{-3}$, and our additional infection parameters were set to
$\gamma=$1\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{/}\mathrm{I}\mathrm{V}$$
and $S=$1\text{\,}\mathrm{m}\mathrm{L}$$.