# Impact of heterogeneity on infection probability: Insights from single-hit
dose-response models

Francisco J. Pérez-Reche<EMAIL_ADDRESS>
###### Abstract

The process of infection of a host is complex, influenced by factors such as
microbial variation within and between hosts as well as differences in dose
across hosts. This study uses dose-response and microbial growth models to
delve into the impact of these factors on infection probability. It is
rigorously demonstrated that within-host heterogeneity in microbial
infectivity enhances the probability of infection. The effect of infectivity
and dose variation between hosts is studied in terms of the expected value of
the probability of infection. General analytical findings, derived under the
assumption of small infectivity, reveal that both types of heterogeneity
reduce the expected infection probability. Interestingly, this trend appears
consistent across specific dose-response models, suggesting a limited role for
the small infectivity condition. Additionally, the vital dynamics behind
heterogeneous infectivity are investigated with a microbial growth model which
enhances the biological significance of single-hit dose-response models.
Testing these mathematical predictions inspire new and challenging laboratory
experiments that could deepen our understanding of infections.

###### keywords:

Probability of infection , Heterogeneity , Dose-response models , Single-Hit
Model , Microbial growth dynamics

[inst1]organization=Institute for Complex Systems and Mathematical Biology,
School For Natural and Computing Sciences, SUPA,addressline=University of
Aberdeen, city=Aberdeen, postcode=AB24 3UE, country=United Kingdom

##  1 Introduction

A host exposed to a pathogenic microbe can become infected. However, infection
is a complex process that depends on numerous factors, including the number of
microbes to which the host has been exposed (i.e. the dose), the virulence and
genetic variability of the pathogens, as well as the susceptibility of the
host to such pathogens [1, 2, 3, 4, 5, 6, 7]. Given the typical uncertainty
about all these factors, it is useful to describe the occurrence of infection
as a random event that can occur with certain probability.

Estimating the probability of infection following exposure to a pathogenic
microbe is a crucial step in quantitative microbial risk assessment [8].
Heterogeneity associated with the host-microbe interaction within hosts and
differences between hosts in terms of the ingested dose and susceptibility,
are expected to influence the chances of infection. However, a systematic
understanding of the role of heterogeneity on infection is lacking.
Interpreting infection as a biological invasion could guide intuition on the
role of heterogeneity in the probability of infection. Different types of
heterogeneity have been considered by ecological and epidemiological models
for the probability of biological invasions [9, 10, 11, 12, 13]. Such models,
however, suggest that depending on the type of heterogeneity, the probability
of invasion could either be enhanced of reduced.

Here, a general and mathematically rigorous analysis of the role of several
types of heterogeneities on the probability of infection is presented within
the context of dose-response models. These are mathematical functions that map
a dose to the probability of infection [8]. Dose-response models have been
effectively utilised in previous works to characterise the probability of
infection across diverse microbial species and host populations [14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

Among several possible mechanistic approaches proposed to formulate dose-
response relationshipts, the single-hit framework is the most widely used [17,
18, 8]. It is based on the following two assumptions:

Assumption 1.

Ingestion of a single pathogenic microbe is sufficient to induce infection in
a host (i.e., no combined action of two or more microbes is required for
infection, provided proliferation from a single microbe occurs).

Assumption 2.

If a host ingests a dose containing several pathogenic microbes, the survival
of any microbe within the host is independent of the survival of any other
microbe.

A central quantity of single-hit models is the infectivity, which denotes the
probability that any microbe within an ingested dose will infect the host. The
necessity to account for heterogeneity in infectivity and dose has long been
recognised by the scientific community employing dose-response models [27, 15,
16, 14]. Three sources of heterogeneity can be naturally considered:

(I) Within-host heterogeneous infectivity.

If a host ingests several pathogenic microbes, the infectivity may be
different for different microbes.

(II) Between-host heterogeneous infectivity.

Two identical microbes may have different infectivity in different hosts.

(III) Between-host heterogeneous dose.

The ingested dose may be different for different hosts.

The systematic investigation of whether each type of heterogeneity enhances or
diminishes the probability of infection has not been systematically undertaken
in previous studies. Furthermore, a link between heterogeneous infectivity and
microbial growth mechanisms (e.g., microbial rates of birth and death) has yet
to be established. Advancing in these two directions constitutes the central
motivation of the present work.

To address the first point, two possible definitions of the probability of
infection will be considered to address the first point: The probability that
a given host becomes infected after ingesting a dose with specific
infectivities and the average probability of infection for a set of hosts.
Previous works have predominantly focused on the latter definition, as it is
well-suited for interpreting experimental data. In such cases, variability has
been addressed by treating infectivity and/or dose as random variables. This
has led to dose-response formulas that map the expected value of the dose to
the expected value of the probability of infection, as derived from
infectivity and dose probability distributions. Despite the widespread use of
this modeling approach, a comprehensive analysis of how the probability of
infection varies with infectivity dispersion remains lacking.

For instance, the exact beta-Poisson model or its widely-used approximation,
the “approximate” beta-Poisson formula, assume Poisson and beta distributions
for the dose and infectivity, respectively [16, 18]. The shape parameters of
the infectivity distribution, usually denoted as $\alpha$ and $\beta$, encode
the dependence of the probability of infection on the variance of infectivity.
However, the link between these parameters and the variance of infectivity has
not been exploited to understand the effect of such variance on the expected
probability of infection. The approximate beta-Poisson formula has also been
used to mathematically justify an ubiquitous flattening of the dose-response
curve that cannot be explained assuming homogeneous infectivity [8]. However,
an explicit link between the slope and the variance of infectivity was again
not established. Indeed, the demonstration was based on the so-called median
effective dose and parameter $\alpha$ in such a way that both the mean and
variance of the infectivity vary simultaneously. A complete proof of whether
the decrease of the slope is directly related to the infectivity variance is
still lacking.

Concerning the second direction mentioned above, the growth dynamics of
microbes have been described with birth-death models that have been related to
dose-response models [28, 29]. However, heterogeneous infectivity was not
incorporated into these models [30], and for instance, a mechanistic
interpretation of the parameters $\alpha$ and $\beta$ of the beta-Poisson
model has not been provided.

Overall, the current understanding of how variability in dose and/or
infectivity affects the probability of infection is limited, even for
relatively simple models like the approximate beta-Poisson formula. This paper
addresses this gap by presenting general results applicable to any single-hit
dose-response model with heterogeneous dose and/or infectivity. These findings
are then demonstrated using specific dose-response models.

The remaining of the paper is structured as follows: Section 2 introduces a
general single-hit response model and provides general results for the
probability of a host becoming infected by a dose of microbes with
heterogeneous infectivity (i.e., heterogeneity of type I). A particular case
of this model with homogeneous dose and infectivity is presented in Sec. 3.
Subsequently, Section 4 gradually extends the homogeneous model to analyse the
effect of all types of heterogeneity on the expected value of the probability
of infection (see a schematic representation in Fig. 1). Well-known dose-
response models like the exponential and beta-Poisson formulas [8, 30] are
special cases of such models. Sec. 5 presents a population growth model for
microbial infection, considering heterogeneous birth and death rates. Sec. 6
finalises the paper with a summary of the main findings, followed by a
discussion on the scientific significance of experimentally validating the
mathematical predictions in laboratory settings, along with the potential
challenges involved.

Figure 1: Single-hit models for the expected probability of infection. Each
axis represents a different source of heterogeneity: (I) Randomly distributed
infectivity within hosts, (II) randomly distributed infectivity for different
hosts and (III) randomly distributed dose for different hosts. Model $A$ is
located at the origin where both infectivity and dose are homogeneous. Models
of type $A^{\prime}$ are located along the axis corresponding to heterogeneous
doses. Models of type $B$ are located along the axis corresponding to random
infectivity within hosts and homogeneous between-hosts infectivity and dose.
Models of type $C$ are located on the horizontal plane with homogeneous dose
but heterogeneous infectivity within and between hosts. Models of type
$C^{\prime}$ incorporate all three types of heterogeneity and are located in
the bulk of the model space.

##  2 General single-hit dose-response model

Let as assume a population of $H$ hosts such that each host ingest (or is in
contact with) $n_{h}$ pathogenic microbes. The infectivity of a microbe $i$ is
quantified by the probability $x_{i,h}$ that the microbe grows and infects the
host. From assumption 1 in the introduction, any microbe with $x_{i,h}>0$ can
cause infection. Following assumption 2, the probability that a host is
infected is

$P_{h}\left(\\{x_{i,h}\\}_{i=1}^{n_{h}},n_{h}\right)=1-\prod_{i=1}^{n_{h}}(1-x_{i,h})~{}.$
(1)

Assuming that the infection of any host is independent of the rest of the
hosts, the number of infected hosts obeys a Poisson-Binomial distribution,
$I\sim\text{PoissonBin}(H,\\{P_{h}\\}_{h=1}^{H})$ [31]. The mean and variance
of the number of infected individuals are then given by

$\text{E}(I)=\sum_{h=1}^{H}P_{h}~{},\\\
\text{Var}(I)=\sum_{h=1}^{H}P_{h}(1-P_{h})~{}.$ (2)

Within this model, type I heterogeneity is captured by the dependence of
$x_{i,h}$ on the microbe $i$. Heterogeneity of type II is encoded by the
dependence of $x_{i,h}$ on the host $h$. Heterogeneity of type III is
accounted for by the dependence of $n_{h}$ on the host.

After deriving a useful formula for the probability of infection of a host in
Lemma 2.1, a general result is proven in Theorem 2.2 for the effect of within-
host heterogeneous infectivity on the probability of infection.

######  Lemma 2.1.

The probability of infection of a host that ingests a dose of $n$ microbes
with infectivities $\\{x_{i}\\}_{i=1}^{n}$ can be expressed as

$P_{h}\left(\\{x_{i}\\}_{i=1}^{n},n\right)=1-(1-\bar{x})^{n}\exp{\left(-n\sum_{k=2}^{\infty}\frac{M_{k}}{k(1-\bar{x})^{k}}\right)}~{},$
(3)

in terms of the sample mean, $\bar{x}=n^{-1}\sum_{i=1}^{n}x_{i}$, and central
moments, $M_{k}=n^{-1}\sum_{i=1}^{n}(x_{i}-\bar{x})^{k}$, of the infectivity.
Here, the notation has been simplified by dropping the explicit dependence of
infectivities and dose on the host $h$.

###### Proof.

To derive Eq. (3), it is convenient to express Eq. (1) as

$P_{h}\left(\\{x_{i}\\}_{i=1}^{n},n\right)=1-\left[\exp{\left(\frac{1}{n}\sum_{i=1}^{n}\ln(1-x_{i})\right)}\right]^{n}~{}.$
(4)

The identity

$\ln(1-x_{i})=\ln(1-\bar{x})+\ln\left(1-\frac{x_{i}-\bar{x}}{1-\bar{x}}\right)~{}$
(5)

allows Eq. (4) to be written as

$P_{h}\left(\\{x_{i}\\}_{i=1}^{n},n\right)=1-(1-\bar{x})^{n}\left[\exp{\left(\frac{1}{n}\sum_{i=1}^{n}\ln\left(1-\frac{x_{i}-\bar{x}}{1-\bar{x}}\right)\right)}\right]^{n}~{}.$
(6)

The argument of the exponential function in Eq. (6) can be expressed as

$\frac{1}{n}\sum_{i=1}^{n}\ln\left(1-\frac{x_{i}-\bar{x}}{1-\bar{x}}\right)=-\sum_{k=1}^{\infty}\frac{1}{n}\sum_{i=1}^{n}\frac{\left(\frac{x_{i}-\bar{x}}{1-\bar{x}}\right)^{k}}{k}=-\sum_{k=1}^{\infty}\frac{M_{k}}{k(1-\bar{x})^{k}}~{}.$
(7)

Here, the first equality follows from the series expansion
$\ln(1-z_{i})=-\sum_{k=1}^{\infty}z_{i}^{k}$, where
$z_{i}=(x_{i}-\bar{x})/(1-\bar{x})$.

Introducing Eq. (7) into Eq. (6), leads to Eq. (3). ∎

######  Theorem 2.2.

Assume that a host ingests a dose of $n$ microbes with infectivities
$\\{x_{i,h}\\}_{i=1}^{n}$ and let $\bar{x}=n^{-1}\sum_{i}^{n}x_{i,h}$ be the
mean infectivity. The probability of infection of such host is minimal when
the infectivity is homogeneous, i.e., when all microbes have the same
infectivity, $x_{i,h}=\bar{x}$.

###### Proof.

The function $\ln(1-z_{i})$ is non-convex for any $z_{i}\leq 1$ so that, by
Jensen’s inequality [32],

$\frac{1}{n}\sum_{i=1}^{n}\ln(1-z_{i})\leq\ln\left(1-\frac{1}{n}\sum_{i=1}^{n}z_{i}\right)~{},$
(8)

and this implies

$\frac{1}{n}\sum_{i=1}^{n}\ln\left(1-\frac{x_{i}-\bar{x}}{1-\bar{x}}\right)\leq
0,$ (9)

if $z_{i}=(x_{i}-\bar{x})/(1-\bar{x})$. From Eqs. (9) and (7),

$-\sum_{k=1}^{\infty}\frac{M_{k}}{k(1-\bar{x})^{k}}\leq 0~{},$ (10)

and, from Eq. (3), one obtains

$P_{h}\left(\\{x_{i}\\}_{i=1}^{n},n\right)\geq 1-(1-\bar{x})^{n}~{}.$ (11)

The theorem follows from Eq. (11). Indeed, the equality only holds for
situations with homogeneous within-host infectivity such that $M_{k}=0$ for
every $k\geq 2$. The inequality holds strictly for any infection with
heterogenous within-host infectivity with mean $\bar{x}$. ∎

The general model given by Eq. (1) is conceptually interesting. However,
practical applications of dose-response models often deal with limited data
consisting in a dose-response curve, i.e., the probability of infection
estimated for a set of different doses. These data do not allow the parameters
of the general model to be properly inferred (i.e. the infectivities for all
microbes ingested by a host, $\\{x_{i,h}\\}_{i=1}^{n_{h}}$). It is more
practically feasible to assume that the different sources of heterogeneity can
be described in terms of a smaller number of parameters for the probability
distribution of infectivity and dose. This motivates the adoption of models
for the expected value of the probability of infection from a pragmatic
standpoint. Such models are also crucial for comprehending the general effect
of heterogeneities of types II and III, and will be utilised for this purpose
in Sec. 4.

##  3 Model $A$: Homogeneous dose and infectivity

This model assumes that all hosts ingest the same number of microbes and that
all microbes have the same infectivity within all hosts, i.e., $n_{h}=n$ and
$x_{i,h}=x$ for any $h$ and $i$.

Under these assumptions, Eq. (1) reduces to

$P^{(A)}(x,n)=1-(1-x)^{n}~{},$ (12)

and the number of infected hosts obeys a binomial distribution,
$I\sim\text{Bin}(n,P^{(A)})$.

Consistent with intuition, this model suggests that the probability of
infection increases with both infectivity and dose.

##  4 Models for the expected value of the probability of infection

This section begins by presenting results for the expected value of the
probability of infection, applicable in scenarios with small infectivities
across any dose distribution. Subsequently, these findings will be
demonstrated through specific models $A^{\prime}-C^{\prime}$ introduced in
subsections 4.2-4.5.

###  4.1 General results for the expected probability of infection

Theorem 4.1 gives a general result for the dependence of the expected
probability of infection on the infectivity and/or dose variability across
different hosts.

######  Theorem 4.1.

Let us suppose that the following conditions hold:

(a) The probability of infection for a _given_ infectivity and dose is given
by $P^{(A)}(x,n)$ (Eq. 12).

(b) The between-host variation of infectivity is described by a random
variable with expectation $\mu_{x}\in[0,1]$ and variance $v_{x}\in[0,1/4]$.

(c) The dose ingested by different hosts is a random variable with expectation
$\mu_{n}\geq 1$ (i.e. at least one microbe is ingested) and variance
$v_{n}\geq 0$.

Then, in the limit of small infectivity, the expected probability of
infection, $\hat{P}$, is a decreasing function of both $v_{x}$ and $v_{n}$.

###### Proof.

The proof essentially uses a cumulant expansion procedure [33]. From
assumption (a), the expected probability of infection is given by

$\hat{P}=1-{\text{E}}_{x}({\text{E}}_{n}((1-x)^{n}))~{},$ (13)

where ${\text{E}}_{x}$ and ${\text{E}}_{n}$ are the expectations corresponding
to infectivity and dose, respectively.

The binomial power in Eq. (13) can be expanded to give
${\text{E}}_{x}({\text{E}}_{n}((1-x)^{n}))=1+y$, where

$y\equiv-\left(\mu_{x}+\frac{1}{2}(v_{x}+\mu_{x}^{2})+O(x^{3})\right)\mu_{n}+\frac{1}{2}(v_{x}+\mu_{x}^{2}+O(x^{3}))(v_{n}+\mu_{n}^{2})+O(x^{3}n^{3})~{}.$
(14)

Here, the identities ${\text{E}}_{x}(x^{2})=v_{x}+\mu_{x}^{2}$ and
${\text{E}}_{n}(n^{2})=v_{n}+\mu_{n}^{2}$ have been used to express $y$ as a
function of expected values and variances of $x$ and $n$.

It is now convenient to express the expected value in Eq. (13) as
${\text{E}}_{x}({\text{E}}_{n}((1-x)^{n}))=\exp(\ln(1+y))$ and use the
expansion $\ln{(1+y)}=y-y^{2}/2+O(y^{3})$ which, after neglecting $O(x^{3})$
terms, leads to the following approximation:

$\hat{P}\simeq f_{1}(\mu_{x},v_{x},\mu_{n},v_{n})\equiv
1-\exp\left(-\mu_{x}\mu_{n}+\frac{\mu_{x}^{2}}{2}(v_{n}-\mu_{n})+\frac{v_{x}}{2}(\mu_{n}^{2}-\mu_{n}+v_{n})\right)~{}.$
(15)

According to Eq. (15), $\hat{P}$ decreases with increasing $v_{n}$ since
$\mu_{x}^{2}>0$. $\hat{P}$ also decreases with increasing $v_{x}$ since
$\mu_{n}^{2}-\mu_{n}+v_{n}\geq 0$ for any reasonable dose with $\mu_{n}\geq 1$
and $v_{n}\geq 0$. ∎

Corollary 4.1.1 establishes a general link between the slope of the dose-
response curve and the variance of the infectivity. This gives mathematical
support to the ubiquitous flattening of the dose-response curve experimentally
observed and modelled by assuming heterogeneous infectivity [16, 8].

######  Corollary 4.1.1.

In the limit of small infectivity, the slope of $\hat{P}$ evaluated at the
median dose is an increasing function of $\mu_{x}$ and a decreasing function
of $v_{x}$ for any dose distribution.

###### Proof.

The median dose, $\mu_{50}$, corresponds to $\hat{P}=1/2$ which, in the limit
of small infectivity, reduces to $f_{1}(\mu_{x},v_{x},\mu_{50},v_{n})=1/2$
(cf. Eq. 15). From this condition, the rate of variation of $\hat{P}$ at
$\mu_{n}=\mu_{50}$ can be expressed as

$\left.\frac{\text{d}\hat{P}}{\text{d}\mu_{n}}\right|_{\mu_{n}=\mu_{50}}=\frac{\mu_{x}}{2}\left(1-\frac{\mu_{x}}{2}\right)-\frac{v_{x}}{2}\left(\mu_{50}-\frac{1}{2}\right)~{}.$
(16)

This equation trivially proves the corollary for any reasonable parameters
$\mu_{50}>1/2$ and $\mu_{x}\leq 1$. ∎

###  4.2 Models of type $A^{\prime}$: Homogeneous infectivity and heterogenous
dose

Models of type $A^{\prime}$ extend Model A to include heterogeneity of type
III (see Figure 1). Specifically, it is posited that the doses ingested by
different hosts are independent and identically distributed (_iid_) random
variables drawn from a probability distribution $p_{n}^{\text{d}}(\bm{\xi})$,
where $\bm{\xi}$ are the distribution parameters. The expected probability of
infection can then be expressed in terms of $P^{(A)}(x,n)$ as follows:

$P^{(A^{\prime})}(x,\bm{\xi})=\sum_{n=1}^{\infty}P^{(A)}(x,n)p^{\text{d}}(n;\bm{\xi})~{}.$
(17)

The subsequent part of this section illustrates the effect of heterogeneous
doses on $P^{(A^{\prime})}(x,\bm{\xi})$ for two commonly assumed scenarios for
$p^{\text{d}}(n;\bm{\xi})$.

####  4.2.1 Example $A^{\prime}_{1}$: Poisson-distributed dose

Assuming that the number of particles ingested by a host is a Poisson
distribution with mean $\mu_{n}$ (i.e. $n_{h}\sim\text{Pois}(\mu_{n})$) leads
to the well-known exponential dose-response model for the probability of
infection [17, 8]:

$P^{(A^{\prime}_{1})}(x,\mu_{n})=1-e^{-x\mu_{n}}~{}.$ (18)

This expression follows from Eq. (17) by using the probability mass function,
$p^{\text{d}}(n;\mu_{n})=\mu_{n}^{n}e^{-\mu_{n}}/n!$, for the Poisson
distribution.

####  4.2.2 Example $A^{\prime}_{2}$: Negative binomial dose distribution

The Poisson distribution assumes $\mu_{n}=v_{n}$ and does not allow the effect
of the effect of $v_{n}$ on the expected probability of infection to be
independently analysed from $\mu_{n}$. In fact, doses can be characterised by
a larger dispersion than that allowed by a Poisson distribution (i.e.
$v_{n}>\mu_{n}$) [34, 35, 21]. Over-dispersion can be modelled by assuming a
negative binomial distribution, $n_{h}\sim\text{NB}(r,p)$, with parameters
$r>0$ and $p\in[0,1]$. The corresponding probability function is

$p^{\text{d}}(n_{h};r,p)=\frac{\Gamma(n_{h}+r)}{n_{h}!\Gamma(r)}(1-p)^{n_{h}}p^{r}~{},$
(19)

which results in the following expected probability of infection:

$P^{(A^{\prime}_{2})}(x,r,p)=1-\left(\frac{p}{p+x(1-p)}\right)^{r}~{}.$ (20)

The parameters $r$ and $p$ are related to the mean and variance of the dose,
$\mu_{n}$ and $v_{n}$, as follows:

$r=\frac{\mu_{n}^{2}}{v_{n}-\mu_{n}}\text{ and }p=\frac{\mu_{n}}{v_{n}}~{}.$
(21)

Using these expressions in Eq. (20) allows the expected probability of
infection to be written as a function $P^{(A^{\prime}_{2})}(x,\mu_{n},v_{n})$
of the mean and variance of the dose.

Note that the condition $r>0$ implies $v_{n}\geq\mu_{n}$ for the negative
binomial distribution. The smallest possible dispersion for a negative
binomial distribution corresponds to $v_{n}=\mu_{n}$ which is the condition
defining a Poisson distribution. The following proposition shows that the
expected probability of invasion in examples $A^{\prime}_{1}$ and
$A^{\prime}_{2}$ are identical when $v_{n}=\mu_{n}$.

######  Proposition 4.1.

Model $A^{\prime}_{2}$ reduces to model $A^{\prime}_{1}$ when the variance of
the dose approaches the mean from above (i.e. in the limit
$v_{n}\searrow\mu_{n}$).

###### Proof.

To prove the proposition, it is convenient to express the expected probability
of infection in terms of $r$ and $\mu_{n}$. Combining Eqs. (20) and (21),
leads to

$P^{(A^{\prime}_{2})}(x,\mu_{n},r)=1-\left(\frac{r}{r+x\mu_{n}}\right)^{r}~{}.$
(22)

From Eq. (21), the limit $v_{n}\searrow\mu_{n}$ corresponds to
$r\rightarrow\infty$. Through simple algebraic manipulations, one obtains the
probability (18) for $r\rightarrow\infty$:

$\lim_{t\rightarrow\infty}P^{(A^{\prime}_{2})}(x,\mu_{n},r)=1-\lim_{t\rightarrow\infty}\left(1+\frac{-x\mu_{n}}{-r}\right)^{-r}=1-e^{-x\mu_{n}}~{}.$
(23)

∎

The decrease of the infection probability when increasing the variance
$v_{n}$, holds for $P^{(A^{\prime}_{2})}(x,r,p)$ even without the small
infectivity requirement of Theorem 4.1. This is shown in Proposition 4.2 after
proving the following lemma:

######  Lemma 4.2.

The function

$f(y,r)=y+(r+y)\ln\left(\frac{r}{r+y}\right)~{}$ (24)

satisfies $f(y,r)\leq 0$ for any $y\geq 0$ and $r>0$.

###### Proof.

This can be easily demonstrated by plotting $f(y,r)$ as a function of $y$ for
different values of $r$. Alternatively, a rigorous proof for any $r>0$ can be
given from the following two facts:

(i)

$f(y,r)$ has only one zero and is located at $y=0$ for any $r>0$. To show
this, it is convenient to define a new variable $z=\frac{r+y}{er}$ such that
$f(y,r)=0$ reduces to $z\ln z=e^{-1}$. The solution to this equation is
$z=-e^{-1}$ which corresponds to $y=0$.

(ii)

$f(y,r)$ is a decreasing function of $y$ in the interval $y>0$. Indeed,
$\partial f(y,r)/\partial y=-\ln\left(\frac{r+y}{r}\right)<0$.

Following these results for $f(y,r)$ must be non-positive for any $y>0$ and
$r>0$. ∎

######  Proposition 4.2.

For a fixed mean dose, the expected probability of invasion of model
$A^{\prime}_{2}$ is a decreasing function of the dose variance.

###### Proof.

It is again convenient to use expression (22) for
$P^{(A^{\prime}_{2})}(x;\mu_{n},r)$.

The variation of $P^{(A^{\prime}_{2})}$ with $v_{n}$ at fixed $\mu_{n}$ can be
calculated as

$\frac{\partial P^{(A^{\prime}_{2})}}{\partial v_{n}}=\frac{\partial
P^{(A^{\prime}_{2})}}{\partial r}\frac{\partial r}{\partial v_{n}}~{}.$ (25)

On the one hand, Lemma 4.2 ensures that

$\frac{\partial P^{(A^{\prime}_{2})}}{\partial
r}=-\frac{r^{r}}{(r+x\mu_{n})^{r+1}}f(x\mu_{n},r)\geq 0~{}$ (26)

for any $x\mu_{n}\geq 0$ and $r>0$. On the other hand, the following
inequality holds:

$\frac{\partial r}{\partial
v_{n}}=-\frac{\mu_{n}^{2}}{(v_{n}-\mu_{n})^{2}}<0~{}.$ (27)

From Eq. (25), the inequalities (26) and (27) imply that $\partial
P^{(A^{\prime}_{2})}/\partial v_{n}<0$ and this proves the proposition. ∎

The next proposition relates all the models of type $A$ and $A^{\prime}$ and
gives further evidence that increasing dose variability decreases the expected
probability of infection for any given infectivity.

######  Proposition 4.3.

The following inequalities hold for any infectivity $x$ and dose distribution
parameters $\mu_{n}$ and $v_{n}>\mu_{n}$:

$P^{(A)}(x,\mu_{n})\geq P^{(A^{\prime}_{1})}(x,\mu_{n})\geq
P^{(A^{\prime}_{2})}(x,\mu_{n},v_{n})~{}.$ (28)

###### Proof.

The first inequality trivially follows from the comparison of Eqs. (12) and
(18) and the fact that $e^{-x\mu_{n}}\geq(1-x)^{\mu_{n}}$ for any $\mu_{n}>0$.
The second inequality follows from Propositions 4.1 and 4.2. Fig. 2
demonstrates the inequalities for particular values of $\mu_{n}$ and $v_{n}$.
∎

Figure 2: Graphical illustration of the inequalities given in Proposition 4.3.
The dependence of the probability of infection on the infectivity $x$ for a
system with homogeneous infectivity (Model $A$, continuous black line) is
compared to the dependence on $x$ of the expected probability of infection of
models $A^{\prime}_{1}$ (dashed red line) and $A^{\prime}_{2}$ (dot-dashed
blue line). The dose was set to $n=5$ for model $A$. The mean and variance of
the dose were set to $\mu_{n}=5$ and $v_{n}=20$, respectively, for models
$A^{\prime}_{1}$ and $A^{\prime}_{2}$.

###  4.3 Models of type B: Heterogeneous infectivity within hosts and
homogeneous dose

Models of type B extend Model A to account for heterogeneity of type I, by
assuming that the infectivity depends on the microbe type but not on the host
(see Figure 1). Mathematically, this corresponds to $x_{i,h}=x_{i}$ within any
host, $h$. The infectivities $\\{x_{i}\\}_{i=1}^{n}$ of the $n$ microbes in a
dose are described as _iid_ random variables drawn from a distribution with
probability density function (PDF) $\rho_{\text{m}}(x;\bm{\xi}_{\text{B}})$.
Here, $\bm{\xi}_{\text{B}}$ are the parameters of the distribution. The
expected value of the probability of infection under these conditions is [35]:

$P^{(B)}(\mu_{m}(\bm{\xi}_{\text{B}}),n)=\prod_{i=1}^{n}\int_{0}^{1}P^{(A)}(x_{i},n)\rho_{\text{m}}(x_{i};\bm{\xi}_{\text{B}})\text{d}x_{i}=1-(1-\mu_{m}(\bm{\xi}_{\text{B}}))^{n}~{},$
(29)

where
$\mu_{m}(\bm{\xi}_{\text{B}})=\int_{0}^{1}x\rho_{\text{m}}(x;\bm{\xi}_{\text{B}})\text{d}x$
is the expected value of the infectivity within a host.

Interestingly, the expected probability $P^{(B)}$ depends solely on the
infectivity parameter $\mu_{m}$; the variability in $x$ does not influence
$P^{(B)}$. In other words, when a set of hosts ingest doses with a mean
infectivity of $\mu_{m}$, the mean probability of infection calculated across
different hosts is solely determined by $\mu_{m}$. This contrasts with the
probability that _an individual host_ becomes infected, which, for a given
mean infectivity $\bar{x}$, increases with greater heterogeneity in
infectivity (Theorem 2.2).

The relationship between $P^{(B)}$ and $\mu_{m}$ mirrors the relationship
between $P^{(A)}$ and $x$. Consequently, for models of type $B$, it can be
inferred that the expected probability of infection for a dose of $n$ microbes
with _iid_ random infectivities can effectively be represented by a set of $n$
microbes with homogeneous infectivity $\mu_{m}$. Following this premise, the
infectivity within hosts in models of type $C$ and $C^{\prime}$ will be
denoted as $x$, despite the possibility for such models to account for
variability in infectivity within hosts by simply substituting $x$ with
$\mu_{m}$.

###  4.4 Models of type $C$: Heterogeneous infectivity between hosts and
homogeneous dose

Models of type $C$ represent an expansion of models of type $B$, introducing a
host-dependent mean infectivity to accommodate heterogeneities of type I and
II (see Figure 1). The expansion of model $A$ to include heterogeneity of type
II represents a specific case wherein all microbes within a host share the
same infectivity, denoted as $x_{i,h}=x_{h}$ for every microbe $i$, yielding
$\mu_{m}=x_{h}$. Within a set of $H$ hosts, the mean infectivities,
$\\{x_{h}\\}_{h=1}^{H}$, are regarded as _iid_ random variables drawn from a
distribution with probability density function (PDF)
$\rho_{h}(x;\bm{\xi}_{\text{C}})$. The probability of infection for a randomly
selected host is expressed as follows:

$P^{C}(\bm{\xi}_{\text{C}},n)=\int_{0}^{1}P^{(A)}(x,n)\rho_{h}(x;\bm{\xi}_{\text{C}})\text{d}x~{}.$
(30)

The general results presented in Sec. 4.1 for small infectivity hold for
models of type $C$ which are particular cases of the model in Sec. 4.1 with
$v_{n}=0$.

An alternative family of models of type $C$ could be defined by assuming that
the parameters of the distribution of infectivity within hosts,
$\bm{\xi}_{\text{B}}$, vary across different hosts instead of assuming a
variation of $x_{h}$. A formulation along these lines will be used in Sec. 5
to establish a link between single-hit dose-response models and microbial
growth processes. From a practical viewpoint, however, assuming that the
infectivity is a random variable as in Eq. (30) has proven useful for many
applications of dose-response models and this is the option chosen to
illustrate models of type $C$ in the following example.

####  4.4.1 Example C1: Beta-distributed between-host infectivity

It is common to assume that the randomness of infectivity between hosts can be
described with a beta distribution [27, 16], i.e.
$x\sim\text{Beta}(\alpha,\beta)$, where $\alpha$ and $\beta$ are positive
parameters. The corresponding PDF is

$\rho_{h}^{(C_{1})}(x;\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}~{}.$
(31)

The beta distribution has support in the interval $x\in[0,1]$ and is therefore
a mathematically reasonable description of the infectivity which is ultimately
a probability.

Introducing the PDF from Eq. (31) into Eq. (30), yields an expected
probability of infection

$P^{(C_{1})}(\alpha,\beta,n)=1-\frac{\Gamma(\alpha+\beta)\Gamma(n+\beta)}{\Gamma(\beta)\Gamma(n+\alpha+\beta)}~{}.$
(32)

Expressing $\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ and
$P^{(C_{1})}(\alpha,\beta,n)$ in terms of $\alpha$ and $\beta$ is typical in
dose-response studies assuming a beta-distributed infectivity. To understand
the impact of heterogeneity, however, it is useful to express $\alpha$ and
$\beta$ as a function of the expected value, $\mu_{x}$, and variance, $v_{x}$,
of the infectivity as follows:

$\alpha=\mu_{x}\left(\frac{\mu_{x}(1-\mu_{x})}{v_{x}}-1\right)~{},\quad\beta=(1-\mu_{x})\left(\frac{\mu_{x}(1-\mu_{x})}{v_{x}}-1\right)~{}.$
(33)

Note that the conditions $\alpha,\beta>0$ impose an upper bound for the
variance $v_{x}<v_{\max}(\mu_{x})\equiv\mu_{x}(1-\mu_{x})$.

Four regimes can be defined in terms of the dependence of
$\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ on $x$ (see Fig. 3). The regimes depend
on whether each of the parameters $\alpha$ and $\beta$ is larger or smaller
than one:

Regime (i) - Highly heterogeneous infectivity:

This regime corresponds to $\alpha,\beta<1$. The infectivity PDF
$\rho_{h}^{(C_{1})}$ diverges for $x=0$ and $x=1$ (see Fig. 3(a)). This
corresponds to populations of hosts that tend to have a bimodal distribution
of infectivities: The microbe can be highly infective for some hosts whereas
it tends to be harmless for others. The relative proportion of hosts at risk
of infection is larger when $\beta<\alpha$ as this makes the divergence of
$\rho_{h}^{(C_{1})}$ at $x=1$ dominant over the one at $x=0$.

In this regime, $\mu_{x}\in(0,1)$ and the variance takes relatively high
values, $v_{x}\in(\max\\{b_{1}(\mu_{x}),b_{2}(\mu_{x})\\},v_{\max}(\mu_{x}))$
(see Fig. 4). Here,

$\displaystyle
b_{1}(\mu_{x})\equiv\frac{\mu_{x}^{2}(1-\mu_{x})}{1+\mu_{x}}~{},b_{2}(\mu_{x})\equiv\frac{\mu_{x}(1-\mu_{x})^{2}}{2-\mu_{x}}~{},$
(34)

are boundary functions for $v_{x}$ corresponding to the conditions $\alpha=1$
and $\beta=1$ (derived from Eqs. (33)). It is important to note that regime
(i) is described as a high heterogeneity regime since it allows the largest
possible values of the variance $v_{\max}$ for given $\mu_{x}$. However, this
does not imply that $v_{x}$ will necessarily take large values since, for
instance, $v_{\max}(\mu_{x})$ tends to zero for $\mu_{x}\rightarrow 0$ or
$\mu_{x}\rightarrow 1$.

Regime (ii) - High infectivity with intermediate heterogeneity:

$\alpha>1$ and $\beta<1$. The PDF $\rho_{h}^{(C_{1})}$ is zero at $x=0$ and
monotonically increases with increasing $x$ to exhibit a divergence at $x=1$
(see Fig. 3(b)). The mean infectivity takes high values, $\mu_{x}\in(1/2,1)$,
and the variance takes intermediate values,
$v_{x}\in(b_{2}(\mu_{x}),b_{1}(\mu_{x}))$ (Fig. 4).

Regime (iii) - Low infectivity with intermediate heterogeneity:

$\alpha<1$ and $\beta>1$. This regime is complementary to regime (ii) in the
sense that $\rho_{h}^{(C_{1})}$ diverges at $x=0$ and monotonically decreases
with increasing $x$ until $\rho_{h}^{(C_{1})}=0$ is reached at $x=1$ (see Fig.
3(c)). This regime corresponds to situations with relatively small mean
infectivity, $\mu_{x}\in(0,1/2)$, and intermediate values of the variance,
$v_{x}\in(b_{1}(\mu_{x}),b_{2}(\mu_{x}))$.

Regime (iv) - Low heterogeneity:

$\alpha>1$ and $\beta>1$. In this regime, $\rho_{h}^{(C_{1})}$ is zero for
$x=0,1$ and takes positive values for any $x\in(0,1)$ (see Fig. 3(d)). The
mean infectivity can take any admissible value, $\mu_{x}\in(0,1)$, and the
variance takes low values,
$v_{x}\in(0,\min\\{b_{1}(\mu_{x}),b_{2}(\mu_{x})\\})$.

In all regimes, the proportion of hosts at high risk of infection increases
for decreasing $\beta$ and increasing $\alpha$.

Figure 3: Probability density function of the infectivity for model $C_{1}$
(red dashed line, Eq. (31)) and the population growth model proposed in Sec.
5.0.2 (black line, Eq. (56)). Different panels illustrate the dependence of
$\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ on infectivity in each of the regimes
described in the text: (a) Regime (i) with $(\alpha,\beta)=(0.5,0.25)$, (b)
regime (ii) with $(\alpha,\beta)=(1.2,0.36)$, (c) regime (iii) with
$(\alpha,\beta)=(0.89,1.32)$, and (d) regime (iv) with
$(\alpha,\beta)=(1.5,1.8)$. When parametrised by the mean and variance of the
effective microbe mortality considered in the population growth model of Sec.
5.0.2, the four panels correspond to: (a)
$(\mu_{\lambda},v_{\lambda})=(0.5,1.0)$, (b)
$(\mu_{\lambda},v_{\lambda})=(0.3,0.25)$, (c)
$(\mu_{\lambda},v_{\lambda})=(1.5,1.7)$ and (d)
$(\mu_{\lambda},v_{\lambda})=(1.2,0.8)$. Figure 4: Location of the four
regimes for the shape of the PDF $\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ in the
space spanned by the mean and variance of the infectivity.

A fully analytical study of the trends of the probability of infection
$P^{(C_{1})}$ when varying $\mu_{x}$ and $v_{x}$ is challenging and a
graphical analysis will be presented instead. Fig. 5(a) shows the dependence
of $P^{(C_{1})}$ on the mean infectivity for two values of the variance (see
red and black continuous lines). For given $v_{x}$, the condition
$v_{x}<\mu_{x}(1-\mu_{x})$ implies that $P^{(C_{1})}$ is only defined for
$\mu_{x}\in(\mu_{x}^{-},\mu_{x}^{+})$, where
$\mu_{x}^{\pm}(v_{x})=(1\pm\sqrt{1-4v_{x}})/2$. At the boundaries
$\mu_{x}^{\pm}$, the probability of infection satisfies

$\lim_{\mu_{x}\rightarrow\mu_{x}^{\pm}}P^{(C_{1})}=\mu_{x}^{\pm}~{}.$ (35)

Consequently, the endpoints of the curves for $P^{(C_{1})}$ versus $\mu_{x}$
lie along a diagonal line with a slope of 1 in the $(\mu_{x},P^{(C_{1})})$
space (as indicated by the dotted line in Fig. 5(a)). When $\mu_{x}$ takes
relatively small values, the probability $P^{(C_{1})}$ increases with
$\mu_{x}$ for any fixed value of $v_{x}$. This behavior aligns with our
expectations based on Theorem 4.1, specifically Eq. (15) with $v_{n}=0$.

In contrast, for sufficiently high values of $\mu_{x}$, $P^{(C_{1})}$
decreases. This might appear counterintuitive at first glance. However, this
decrease is a consequence of keeping $v_{x}$ fixed. Indeed, when $\mu_{x}$
becomes large enough, the system enters regime (i) for any $v_{x}>0$ (refer to
Fig. 4). In this regime, the divergence of
$\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ becomes increasingly pronounced at both
$x=0$ and $x=1$ as $\mu_{x}$ increases. The prominence of divergence at $x=0$
is essential for maintaining the fixed value of $v_{x}$. In more qualitative
terms, as the mean infectivity increases, the likelihood of microbes having
either high or low infectivity also rises.

The graphical representation of $P^{(C_{1})}$ vs. $v_{x}$ suggests that
$P^{(C_{1})}$ monotonically decreases with increasing $v_{x}$ for any $n$ and
$\mu_{x}$ (see the examples for two values of $\mu_{x}$ shown by the
continuous curves in Fig. 5(b)). The general decrease of $P^{(C_{1})}$ with
$v_{x}$ predicted for small infectivities in Theorem 4.1 seems to hold for any
mean infectivity in this example. In the limit of small variance for the
infectivity,

$\lim_{v_{x}\searrow 0}P^{(C_{1})}=1-(1-\mu_{x})^{n}~{},$ (36)

i.e. model A is recovered. In the limit of high variance, one can use Euler’s
infinite product formula for $1/\Gamma(z)$ [36] to show that

$\lim_{v_{x}\nearrow v_{\max}}P^{(C_{1})}=\mu_{x}~{}.$ (37)

This is graphically illustrated by the examples in Fig. 5(b) which show that
the curves of $P^{(C_{1})}$ vs. $v_{x}$ end at the dotted line corresponding
to $v_{x}\nearrow v_{\max}(\mu_{x})$.

Figure 5: Expected probability of infection for examples $C_{1}$ with dose
$n=5$ (continuous lines, Eq. (32)) and $\tilde{C_{1}}$ (dashed lines, Eq.
(38)). Panel (a) shows the dependence of $P^{(C_{1})}$ and
$P^{(\tilde{C_{1}})}$ on the mean infectivity, $\mu_{x}$, for two values of
the variance, $v_{x}$, as marked by the legend. The dotted diagonal line
indicates the bounds and (b) $v_{x}$ .

The validity of Corollary 4.1.1 for model $C_{1}$ can be analytically studied
for small and relatively homogeneous infectivity, as shown in Proposition 4.4.

######  Proposition 4.4.

If the infectivity distribution satisfies $\mu_{x}\ll 1$ and
$v_{x}\ll\mu_{x}$, the slope of $P^{(C_{1})}(\alpha,\beta,n)$ evaluated at the
median dose decreases when increasing $v_{x}$.

###### Proof.

From Eq. (33), the conditions $\mu_{x}\ll 1$ and $v_{x}\ll\mu_{x}$ correspond
to $\beta\gg\alpha$ and $\beta\gg 1$. Under these conditions, Stirling’s
formula for the gamma function, $\Gamma(z)\sim e^{-z}z^{z-1/2}\sqrt{2\pi}$,
allows $P^{(C_{1})}$ to be approximated as

$P^{(C_{1})}(\alpha,\beta,n)\simeq P^{(\tilde{C_{1}})}(\alpha,\beta,n)\equiv
1-\left(1+\frac{n}{\beta}\right)^{-\alpha}~{}.$ (38)

The median infectious dose corresponding to $P^{(\tilde{C_{1}})}=1/2$ is
$n_{50}=\beta(2^{1/\alpha}-1)$ and this yields the following slope for the
infection curve:

$\left.\frac{\text{d}P^{(\tilde{C_{1}})}}{\text{d}n}\right|_{n=n_{50}}=\frac{\alpha}{\beta}2^{1-1/\alpha}~{}.$
(39)

Expressing the slope as a function of $\mu_{x}$ and $v_{x}$ through Eqs. (33)
and taking the binary logarithm gives

$\log_{2}\left.\frac{\text{d}P^{(\tilde{C_{1}})}}{\text{d}n}\right|_{n=n_{50}}=f^{(\tilde{C_{1}})}(\mu_{x},v_{x})\equiv
1+\log_{2}\frac{\mu_{x}}{1-\mu_{x}}-\frac{v_{x}}{\mu_{x}(\mu_{x}(1-\mu_{x})-v_{x})}~{}.$
(40)

Differentiating the logarithmic slope with respect to $v_{x}$ yields

$\frac{\partial f^{(\tilde{C_{1}})}(\mu_{x},v_{x})}{\partial
v_{x}}=-\frac{(1-\mu_{x})}{(v_{x}-\mu_{x}(1-\mu_{x}))^{2}}<0~{},$ (41)

which proves the proposition.

∎

The approximate probability of infection $P^{(\tilde{C_{1}})}$ derived in the
proof of Proposition 4.4 approaches zero at the boundaries
$\mu_{x}^{\pm}(v_{x})$ for the infectivity (see the dashed curves in Fig.
(5)(a)). This significantly deviates from the behaviour of the exact
probability $P^{(C_{1})}$ given by Eq. (5). This failure of the approximation
is expected: On the one hand, the necessary condition $\mu_{x}\ll 1$ for
$P^{(\tilde{C_{1}})}$ to approximate $P^{(C_{1})}$ is not satisfied when
$\mu_{x}\nearrow\mu_{x}^{+}$. On the other hand, the condition
$v_{x}\ll\mu_{x}$ is violated when $\mu_{x}\searrow\mu_{x}^{-}$.

Comparing the continuous and dashed lines in Fig. (5)(b) reveals that
$P^{(\tilde{C_{1}})}$ offers a better approximation to $P^{(C_{1})}$ for small
values of $v_{x}$ and $\mu_{x}$ (compare the good agreement between the red
curves for small $v_{x}$ with the poorer agreement between the black curves
for small $v_{x}$). In the limit of vanishing variance, the approximate
probability approaches the value

$\lim_{v_{x}\searrow 0}P^{(\tilde{C_{1}})}=1-e^{-n\mu_{x}/(1-\mu_{x})}~{}.$
(42)

This approximate value exceeds the exact probability $P^{(C_{1})}$ (see Eq.
(36)) since $e^{-\mu_{x}/(1-\mu_{x})}>1-\mu_{x}$ for any $\mu_{x}\in(0,1)$.

Fig. (5)(b) also shows that the deviation of $P^{(\tilde{C_{1}})}$ from
$P^{(C_{1})}$ worsens as $v_{x}$ increases. For given $\mu_{x}$, this
deviation is more pronounced when $v_{x}$ approaches the maximal variance,
$v_{\max}(\mu_{x})$. In such cases, $P^{(\tilde{C_{1}})}$ approaches zero and
significantly deviates from the corresponding limit for $P^{(C_{1})}$ given by
Eq. (37).

###  4.5 Models of type $C^{\prime}$: Between-host heterogeneous infectivity
and dose

Models of type $C^{\prime}$ incorporate the three types of heterogeneity
described in Fig. 1. These models can either be regarded as extensions of
models of type $A^{\prime}$ to incorporate heterogeneities of type I and II or
extensions of models of type $C$ to account for type III heterogeneity.
Following this, the expected probability of infection for models of type
$C^{\prime}$ can be calculated in several equivalent ways:

$\displaystyle P^{(C^{\prime})}(\bm{\xi},\bm{\xi}_{\text{C}})=$
$\displaystyle\sum_{n=1}^{\infty}p^{\text{d}}(n;\bm{\xi})\int_{0}^{1}P^{(A)}(x,n)\rho_{h}(x;\bm{\xi}_{\text{C}})\text{d}x~{},$
(43) $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}P^{(C)}(n,\bm{\xi}_{\text{C}})p^{\text{d}}(n;\bm{\xi})~{},$
(44) $\displaystyle=$
$\displaystyle\int_{0}^{1}P^{(A^{\prime})}(x,\bm{\xi})\rho_{h}(x;\bm{\xi}_{\text{C}})\text{d}x~{}.$
(45)

The general results for the expected probability of infection in the limit of
small infectivity presented in Sec. 4.1 directly apply to models of type
$C^{\prime}$.

The next subsections present two models of type $C^{\prime}$ that have been
proposed in previous works. Both assume beta-distributed infectivity (i.e.,
PDF given by Eq. (31)). As a consequence, both examples will exhibit the four
regimes described in Sec. 4.4.1.

####  4.5.1 Example $C_{1}^{\prime}$: Beta-distributed infectivity and
Poisson-distributed dose

Here, it is assumed that the dose is $n_{h}\sim\text{Pois}(\mu_{n})$ so that
model $C_{1}^{\prime}$ can be interpreted as an extension of model
$A_{1}^{\prime}$ (Sec. 4.2.1) to include a beta-distributed infectivity. Using
Eq. (18) to set $P^{(A^{\prime})}(x,\bm{\xi})=P^{(A^{\prime}_{1})}(x,\mu_{n})$
in Eq. (45) leads to the following expected probability of infection:

$P^{(C_{1}^{\prime})}(\alpha,\beta,\mu_{n})=1-~{}_{1}F_{1}(\alpha,\alpha+\beta,-\mu_{n})~{}.$
(46)

Here, ${}_{1}F_{1}$ is the Kummer confluent function [36]. This is the exact
beta-Poisson model proposed by Furumoto and Mickey [16]. In the same work, the
authors derived the beta-Poisson approximation,

$P^{(C_{1}^{\prime})}(\alpha,\beta,\mu_{n})\simeq
P^{(\tilde{C_{1}}^{\prime})}(\alpha,\beta,\mu_{n})\equiv
1-\left(1+\frac{\mu_{n}}{\beta}\right)^{-\alpha}~{},$ (47)

which is valid for $\beta\gg\alpha$ and $\beta\gg 1$, i.e. for $\mu_{x}\ll 1$
and $v_{x}\ll\mu_{x}$ (cf. Eq. (33)).

The functional relationship between $P^{(\tilde{C_{1}}^{\prime})}$ and
$\mu_{n}$ mirrors that of $P^{(\tilde{C_{1}})}$ and $n$ (compare Eqs. (47) and
(38)). Consequently, Proposition 4.4, established for model $C_{1}$, directly
extends to model $C_{1}^{\prime}$ through the expression of the median dose in
terms of $\mu_{n}$. While this proposition may bear resemblance to previous
demonstrations expressing $P^{(\tilde{C_{1}}^{\prime})}$ as a function of
$\alpha$ and the median dose [8], it is essential to highlight a crucial
distinction. Proposition 4.4 elucidates the decline in the slope of
$P^{(\tilde{C_{1}}^{\prime})}$ with increasing $v_{x}$ while holding $\mu_{x}$
constant. Such delineation between the variance and mean of infectivity cannot
be achieved by parameterising $P^{(\tilde{C_{1}}^{\prime})}$ in terms of
$\alpha$ and the median dose.

Despite the formal similarity between Eqs. (38) and (47) and the fact that
both are valid in the limit of low and relatively homogeneous infectivity,
$P^{(\tilde{C_{1}}^{\prime})}$ does not result by using $P^{(\tilde{C_{1}})}$
and the Poisson probability mass function in Eq. (44). Irrespective of that,
the deviations of $P^{(\tilde{C_{1}}^{\prime})}$ from the exact probability
$P^{(C_{1}^{\prime})}$ are analogous to those described in Sec. 4.4.1 for
$P^{(\tilde{C_{1}})}$ compared to $P^{(C_{1})}$.

####  4.5.2 Example $C_{2}^{\prime}$: Beta-distributed infectivity and
negative binomial dose distribution

This model assumes a negative binomial distribution for the dose,
$n_{h}\sim\text{NB}(r,p)$. It can be viewed as an extension of model
$A_{2}^{\prime}$ (Sec. 4.2.2) to incorporate a beta-distributed infectivity.
Setting $P^{(A^{\prime})}(x,\bm{\xi})=P^{(A^{\prime}_{2})}(x,r,p)$ in Eq. (45)
gives the probability of infection [35],

$P^{(C_{2}^{\prime})}(\alpha,\beta,r,p)=1-~{}_{2}F_{1}(r,\beta,\alpha+\beta,1-p)p^{r}~{},$
(48)

where ${}_{2}F_{1}$ is the Gaussian hypergeometric function [36].

The following Corollaries about the dependence of model $C_{2}^{\prime}$ on
the variability of the dose follow from the results presented in Sec. 4.2.2
for model $A_{2}^{\prime}$.

######  Corollary 4.2.1.

The expected infection probability of model $C_{2}^{\prime}$ reduces to that
of model $C_{1}^{\prime}$ when the variance of the dose approaches the mean
from above (i.e. in the limit $v_{n}\searrow\mu_{n}$).

###### Proof.

Expressing the expected probability of infection for model $C_{2}^{\prime}$ as

$P^{(C_{2}^{\prime})}(\alpha,\beta,r,p)=\int_{0}^{1}P^{(A_{2}^{\prime})}(x,r,p)\rho^{(C_{1})}_{h}(x;\alpha,\beta)\text{d}x~{},$
(49)

the corollary easily follows from Proposition 4.1. ∎

######  Corollary 4.2.2.

For a fixed mean dose, the expected probability of invasion of model
$C_{2}^{\prime}$ is a decreasing function of the dose variance.

###### Proof.

This is a straightforward consequence of Proposition 4.2 when
$P^{(C_{2}^{\prime})}(\alpha,\beta,r,p)$ is expressed as in Eq. (49). ∎

The following result establishes a hierarchy for the expected probability of
infection predicted by model $A$ and all models with heterogeneous infectivity
of types $C$ and $C^{\prime}$. This highlights the fact that increasing
heterogeneity in infectivity and dose leads to a decrease of the expected
probability of infection.

######  Proposition 4.5.

The following inequalities hold for any infectivity and dose distribution
parameters $\mu_{x}$, $v_{x}$, $\mu_{n}$ and $v_{n}$:

$P^{(A)}(\mu_{x},\mu_{n})\geq P^{(C_{1})}(\mu_{x},v_{x},\mu_{n})\geq
P^{(C_{1}^{\prime})}(\mu_{x},v_{x},\mu_{n})\geq
P^{(C_{2}^{\prime})}(\mu_{x},v_{x},\mu_{n},v_{n})~{}.$ (50)

###### Proof.

The first inequality arises from the non-convex nature of $P^{(A)}(x,\mu_{n})$
with respect to $x$. Jensen’s inequality implies that
$P^{(A)}({\text{E}}_{x}(x),\mu_{n})\geq{\text{E}}_{x}(P^{(A)}(x,\mu_{n}))$.
This equivalently yields $P^{(A)}(\mu_{x},\mu_{n})\geq
P^{(C_{1})}(\mu_{x},v_{x},\mu_{n})$ since $\mu_{x}={\text{E}}_{x}(x)$ and
$P^{(C_{1})}(\mu_{x},v_{x},\mu_{n})$ can be expressed as
${\text{E}}_{x}(P^{(A)}(x,\mu_{n}))$ through Eq. (30).

The second inequality stems from the non-convexity of
$P^{(C_{1})}(\mu_{x},v_{x},n)$ with respect to $n$. Applying Jensen’s
inequality,
$P^{(C_{1})}(\mu_{x},v_{x},{\text{E}}_{n}(n))\geq{\text{E}}_{n}(P^{(C_{1})}(\mu_{x},v_{x},n))$
is derived. This reduces to $P^{(C_{1})}(\mu_{x},v_{x},\mu_{n})\geq
P^{(C_{1}^{\prime})}(\mu_{x},v_{x},\mu_{n})$ using $\mu_{n}={\text{E}}_{n}(n)$
and
$P^{(C_{1}^{\prime})}(\mu_{x},v_{x},\mu_{n})={\text{E}}_{n}(P^{(C_{1})}(\mu_{x},v_{x},n))$
(cf. Eq. (44)).

The third inequality follows from the Corollaries 4.2.1 and 4.2.2.

Fig. 6 illustrates the inequalities given in Eq. (50) for fixed $v_{x}$,
$\mu_{n}$ and $v_{n}$ and variable $\mu_{x}$.

∎

Figure 6: Graphical illustration of the inequalities given in Proposition 4.5.
The dependence of the probability of infection on the mean infectivity for a
system with homogeneous infectivity (Model $A$, continuous black line) is
compared to the dependence on $\mu_{x}$ of the expected probability of
infection of models $C_{1}$ (dashed red), $C^{\prime}_{1}$ (dot-dashed blue)
and $C^{\prime}_{2}$ (long dashed green). The dose was set to $n=5$ for model
$A$. The mean and variance of the dose were set to $\mu_{n}=5$ and $v_{n}=20$,
respectively, for models $C_{1}$, $C^{\prime}_{1}$ and $C^{\prime}_{2}$.

##  5 A population growth model with random between-host infectivity

This section provides a biologically explicit interpretation of single-hit
dose-response models of type $C$ in which the between-host heterogeneity of
infectivity is described as a deterministic population growth process with
heterogeneity between hosts.

####  5.0.1 General formulation of the model

This section derives a biologically motivated probability distribution for the
infectivity based on a simple growth model [37] for the microbial population
within a host. In particular, it is assumed that microbes duplicate at a rate
$\beta$ and die at a rate $\delta$. At the early stages of microbial growth,
the number $N$ of microbes is assumed to obey the equation
$\text{d}N/\text{d}t=(\beta-\delta)N$. A necessary condition for a microbial
population to grow within a host is that the basic reproductive ratio,
$R=\beta/\delta$, is larger than 1. In this case, microbes produce more than
one offspring on average during their average lifetime, $1/\delta$. In
contrast, a microbial population dies out when $R<1$.

Let us assume that the basic reproductive ratio $R_{i,h}$ of a microbe $i$ in
host $h$ is a random variable with PDF $\rho_{R}(R_{i,h};\bm{\xi}_{h})$, where
$\bm{\xi}_{h}$ is a set of host-dependent parameters. The infectivity of a
randomly chosen microbe within host $h$ is the probability that such microbe
grows within the host, i.e. the probability that $R_{i,h}>1$:

$x_{h}=f(\bm{\xi}_{h})\equiv\int_{1}^{\infty}\rho_{R}(R;\bm{\xi}_{h})\text{d}R~{}.$
(51)

The variability of infectivity between hosts is described by assuming that the
parameters $\bm{\xi}_{h}$ are random variables with joint PDF
$\rho_{\xi}(\bm{\xi}_{h};\bm{\xi}^{\prime})$. Here, $\bm{\xi}^{\prime}$ are
parameters for the PDF of $\bm{\xi}_{h}$. Under this assumption, the
probability density function for the infectivity within a randomly chosen host
is given by [33]:

$\rho_{h}(x;\bm{\xi}^{\prime})=\int_{\text{supp}(\bm{\xi}_{h})}\text{d}\bm{\xi}_{h}\rho_{\xi}(\bm{\xi}_{h};\bm{\xi}^{\prime})\delta(f(\bm{\xi}_{h})-x)~{},$
(52)

where $\text{supp}(\bm{\xi}_{h})$ is the support of the distribution of
$\bm{\xi}_{h}$ and $\delta(\cdot)$ is the Dirac delta function.

In the particular case in which the PDF for $R$ involves a single parameter
$\lambda$ (i.e. $\bm{\xi}_{h}=\lambda$), Eq. (52) reduces to

$\rho_{h}(x;\bm{\xi}^{\prime})=\left|\frac{\text{d}f^{-1}(x)}{\text{d}x}\right|\rho_{\xi}(\lambda=f^{-1}(x);\bm{\xi}^{\prime})~{}.$
(53)

####  5.0.2 Example: Linking heterogeneous infectivity and microbial birth and
death

If the basic reproductive ratio is exponentially distributed with rate
parameter $\lambda$, Eq. (51) yields an infectivity

$x=\int_{1}^{\infty}\lambda e^{-\lambda R}\text{d}R=e^{-\lambda}\equiv
f(\lambda)~{}.$ (54)

The parameter $\lambda$ can be interpreted as an effective microbial death
rate since the chances for a microbe to grow decrease for increasing
$\lambda$.

Since $\lambda>0$, it is reasonable to assume that its value is drawn from a
gamma distribution with PDF

$\rho_{\xi}(\bm{\xi}_{h};\bm{\xi}^{\prime})=\rho_{\xi}(\lambda;\hat{\alpha},\hat{\beta})=\frac{\hat{\alpha}^{\hat{\beta}}}{\Gamma(\hat{\beta})}\lambda^{\hat{\beta}-1}e^{-\hat{\beta}\lambda}~{},$
(55)

where $\hat{\alpha}$ and $\hat{\beta}$ are the rate and shape parameters,
respectively.

Introducing now Eqs. (54) and (55) into Eq. (55) leads to the following PDF
for the infectivity:

$\rho_{h}(x;\hat{\alpha},\hat{\beta})=\frac{\hat{\alpha}^{\hat{\beta}}}{\Gamma(\hat{\beta})}(-\ln
x)^{\hat{\beta}-1}x^{\hat{\alpha}-1}~{}.$ (56)

This PDF does not seem to correspond to any well-known probability
distribution. It will be referred to as the _exponential gamma distribution_
(denoted as $x\sim\text{ExpGamma}(\hat{\alpha},\hat{\beta})$) since it
corresponds to a random variable $x$ whose exponential is gamma distributed.

The influence on infectivity of randomness in the effective microbe mortality
$\lambda$ can be elucidated by expressing the parameters $\hat{\alpha}$ and
$\hat{\beta}$ in terms of the mean and variance of the effective death rate,
$\mu_{\lambda}$ and $v_{\lambda}$, as follows:

$\displaystyle\hat{\alpha}$
$\displaystyle=\frac{\mu_{\lambda}}{v_{\lambda}}~{}$ (57)
$\displaystyle\hat{\beta}$
$\displaystyle=\frac{\mu_{\lambda}^{2}}{v_{\lambda}}~{}.$ (58)

These identities can also be used as approximate relations between the moments
of the effective microbe mortality and the parameters of the beta distribution
used in models $C$, $C_{1}^{\prime}$ and $C_{2}^{\prime}$. Indeed, the
$\text{ExpGamma}(\hat{\alpha},\hat{\beta})$ distribution reduces to the
$\text{Beta}(\alpha,\beta)$ distribution with $\alpha=\hat{\alpha}$ and
$\beta=\hat{\beta}$ for $x$ close to 1. This directly follows from the
comparison of Eqs. (31) and (56) for $x\nearrow 1$ which yields $-\ln
x\simeq(1-x)$. Fig. 3 shows the similarity of the PDFs
$\rho_{h}(x;\hat{\alpha},\hat{\beta})$ and
$\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ (Eq. (31)) for examples with
$\alpha=\hat{\alpha}$ and $\beta=\hat{\beta}$.

Eq. (57) shows that $\hat{\alpha}$ is the inverse of the relative variance of
$\lambda$ ($\text{RV}_{\lambda}=v_{\lambda}/\mu_{\lambda}$), meaning that it
decreases as the microbe mortality becomes more heterogeneous between hosts.
This approximately applies to the coefficient $\alpha$ of the beta
distribution.

Eq. (58) shows that $\hat{\beta}$ is the square of the inverse of the
coefficient of variation of $\lambda$
($\text{CV}_{\lambda}=v_{\lambda}^{1/2}/\mu_{\lambda}$). Accordingly,
$\hat{\beta}$ (and approximately $\beta$) also decreases with increasing
heterogeneity in $\lambda$.

Fig. 3 graphically demonstrates that the dependence of
$\rho_{h}(x;\hat{\alpha},\hat{\beta})$ on $x$ falls into four different
regimes which are analogous to those defined for
$\rho_{h}^{(C_{1})}(x;\alpha,\beta)$ in Sec. 4.4.1. The regions of each regime
in the space $(\mu_{\lambda},v_{\lambda})$ are defined as follows:

Regime (i) - Highly heterogeneous infectivity:

It is observed for any mean effective mortality rate $\mu_{\lambda}>0$
provided the effective mortality variance is high enough,
$v_{\lambda}>\max\\{\mu_{\lambda},\mu_{\lambda}^{2}\\}$ (see the boundaries
marked with continuous and dotted lines in Fig. 7).

Regime (ii) - High infectivity with intermediate heterogeneity:

This corresponds to low microbe mortality, $\mu_{\lambda}<1$, and intermediate
effective mortality variance, $\mu_{\lambda}^{2}<v_{\lambda}<\mu_{\lambda}$.

Regime (iii) - Low infectivity with intermediate heterogeneity:

It corresponds to situations with relatively high effective mortality,
$\mu_{\lambda}>1$ and intermediate variance,
$\mu_{\lambda}<v_{\lambda}<\mu_{\lambda}^{2}$.

Regime (iv) - Low heterogeneity:

This regime is observed for any $\mu_{\lambda}>0$, provided the effective
mortality variance is below a certain level, i.e. for
$v_{\lambda}<\min\\{\mu_{\lambda},\mu_{\lambda}^{2}\\}$.

Figure 7: Contour plots depict the interdependencies between (a) the mean
($\mu_{x}$) and (b) the variance ($v_{x}$) of infectivity, with the mean
($\mu_{\lambda}$) and variance ($v_{\lambda}$) of microbial mortality in the
growth model proposed in Sec. 5.0.2. The four regimes for the ExpGamma
infectivity PDF ($\rho_{h}(x;\hat{\alpha},\hat{\beta})$, Eq. (56)) are
indicated in both panels, together with the regime boundaries corresponding to
$v_{\lambda}=\mu_{\lambda}$ (continuous line) and
$v_{\lambda}=\mu_{\lambda}^{2}$ (dotted line).

The expectation and variance of the infectivity can be calculated from Eq.
(55) and expressed as a function of $\mu_{\lambda}$ and $v_{\lambda}$ as
follows:

$\displaystyle\mu_{x}$
$\displaystyle=(\frac{\hat{\alpha}}{1+\hat{\alpha}})^{\hat{\beta}}=\left(\frac{\mu_{\lambda}}{\mu_{\lambda}+v_{\lambda}}\right)^{\mu_{\lambda}^{2}/v_{\lambda}}~{},$
(59) $\displaystyle v_{x}$
$\displaystyle=(\frac{\hat{\alpha}}{2+\hat{\alpha}})^{\hat{\beta}}-(\frac{\hat{\alpha}}{1+\hat{\alpha}})^{2\hat{\beta}}=\left(\frac{\mu_{\lambda}}{\mu_{\lambda}+2v_{\lambda}}\right)^{\mu_{\lambda}^{2}/v_{\lambda}}-\left(\frac{\mu_{\lambda}}{\mu_{\lambda}+v_{\lambda}}\right)^{2\mu_{\lambda}^{2}/v_{\lambda}}~{}.$
(60)

As expected, the mean infectivity of the microbe, $\mu_{x}$, decreases with
$\mu_{\lambda}$ (see the contour levels in Fig. 7(a)), i.e. higher mortality
of the microbe leads to lower infectivity. Conversely, there is a positive
association between $\mu_{x}$ and $v_{\lambda}$, indicating that greater
variability in microbe mortality tends to enhance infectivity on average.

Regions characterised by low infectivity variance, $v_{x}$, are evident in
regimes (i) and (vi), visible as dark areas in Fig. 7(b). In these contexts,
models featuring homogeneous infectivity (such as type $A$ and $A^{\prime}$)
can offer a reasonable description of the expected probability of infection.
In regime (i), the low infectivity variance region corresponds to situations
with high mean infectivity, which forces the maximum infectivity variance
$v_{\max}(\mu_{x})$ to be small. Conversily, regime (iv) corresponds to
systems characterised by relatively homogeneous infectivity, irrespectively of
the mean infectivity.

Elevated values of $v_{x}$ manifest along a ridge situated within regime (i),
where neither $\mu_{\lambda}$ nor $v_{\lambda}$ are negligible. This
observation implies that systems characterised by both substantial expectation
and variance in effective mortality are more effectively characterised by
models of type $C$ and $C^{\prime}$, which incorporate variability in
infectivity.

##  6 Conclusion

This work has delved into the impact of variation in dose and microbial
infectivity on infection probability within the framework of single-hit dose-
response models. It has been rigorously shown that heterogeneity in the
infectivity of microbial populations ingested by a host enhances the chances
of infection (heterogeneity of type I, Theorem 2.2). This finding should be
differentiated from the observation that the expected probability of infection
solely depends on the mean infectivity, $\mu_{m}$. In essence, the expected
probability represents the average infection likelihood across a group of
hosts exposed to doses with heterogeneous infectivity, rather than the
infection probability of an individual.

The presence of heterogeneity of type II, where microbial infectivity varies
across hosts, has been demonstrated to reduce the expected probability of
infection. General results have been given for microbial populations with
small infectivity (Theorem 4.1). Moreover, the ubiquitous flattening of the
dose-response curve observed in experiments has also been shown to be
associated with an increase in the variance of infectivity. The validity of
these results has been illustrated with several models which assume random
infectivity between hosts. Across these models, trends indicating heightened
infectivity variance remain consistent regardless of the infectivity
expectation, as summarised in Proposition 4.5. This suggests that the small
infectivity limit required in Theorem 4.1 may not be overly restrictive.
Investigating a more general theorem which relaxes this condition might be
interesting.

Heterogeneity of type III, associated with variations in dose size among
hosts, has similarly been demonstrated to reduce the expected probability of
infection. Theorem 4.1 establishes this result in the limit of small
infectivity. Specific examples of models show that an increase of the expected
probability of infection with the variance of the dose holds for any mean
infectivity (summarised in Proposition 4.3). Once again, this indicates a mild
role played by the required small infectivity to probe Theorem 4.1.

The last results section has proposed a microbial growth model with randomly
distributed reproductive ratio within a host. An example of this model has
been presented, utilising the effective microbial death rate as a parameter,
which essentially acts as the inverse of the reproductive ratio. The model
provides interesting predictions: near-uniform infectivity is anticipated when
either the mean or variance of the microbial death rate is small. Conversely,
systems characterised by substantial mean and variance in the effective death
rate will exhibit heterogeneous infectivity. Furthermore, this model offers a
biological interpretation for the parameters of the beta-Poisson dose-response
model (referred to as Example $C_{1}^{\prime}$ within the framework presented
here). Specifically, it has been found that both $\alpha$ and $\beta$ decrease
with increasing heterogeneity in the effective microbial death rate. In
contrast, both $\alpha$ and $\beta$ increase when increasing the expectation
of the effective microbial death rate.

Exploring experimental validations to assess the anticipated impacts of
diverse forms of heterogeneity is of significant scientific interest, but it
also presents considerable challenges. For instance, consider the prediction
of Theorem 2.2, which suggests that the infection probability of a host is
minimal for doses with homogeneous infectivity. Strictly speaking, testing
this would require a counterfactual approach [38], where the probability of a
host being infected by a dose with homogeneous infectivity is compared with
the probability of the same host being infected by heterogeneous doses with
the same mean infectivity. However, ensuring consistent hosts across different
experiments poses challenges, as hosts may change over time or differ between
tests. Using very similar hosts might mitigate this issue.

Even if confounding associated with host differences is addressed, significant
challenges remain regarding microbial heterogeneity and dose variability.
Accurately quantifying and controlling the infectivity of microbial
populations is inherently difficult. Furthermore, variations in dose delivery
methods, host responses, and environmental factors can introduce additional
variability, complicating the interpretation of results.

## Acknowledgements

The author acknowledges fruitful discussions with Norval Strachan and Ovidiu
Rotariu as well as funding support from a Medical Research Council Fellowship
(MR/W021455/1).

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