# Propagation properties in a multi-species SIR reaction-diffusion system.

Romain Ducasse Université Paris Cité and Sorbonne Université, CNRS,
Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France  Samuel Nordmann
Department of Applied Mathematics, Tel Aviv University

###### Abstract

We consider a multi-species reaction-diffusion system that arises in
epidemiology to describe the spread of several strains, or variants, of a
disease in a population. Our model is a natural spatial, multi-species,
extension of the classical SIR model of Kermack and McKendrick. First, we
study the long-time behavior of the solutions and show that there is a
“selection via propagation” phenomenon: starting with $N$ strains, only a
subset of them - that we identify - propagates and invades space, with some
given speeds that we compute. Then, we obtain some qualitative properties
concerning the effects of the competition between the different strains on the
outcome of the epidemic. In particular, we prove that the dynamic of the model
is not well characterized by the usual notion of basic reproduction number,
which strongly differs from the classical case with one strain.

Keywords: reaction-diffusion systems, SIR models, propagation properties,
spreading speed, multi-species models, epidemiology, threshold phenomenon.

MSC: 35B36, 35B40, 35K10, 35K40, 35K57, 92D30.

##  1 Introduction

###  1.1 Problem under consideration

In $1927$, Kermack and McKendrick [15, 16, 17] introduced several
deterministic models designed to describe the temporal development of a
disease in a population, among which the celebrated _SIR_ model, which became
a corner stone of mathematical epidemiology. In this model, the population
under consideration is divided into three compartments: the _susceptible_ (who
are not infected), the infectious (who have the disease and can transmit it to
the susceptibles) and the recovered (who had the disease and are now immune,
or dead). The original SIR model is the following system of ODEs:

$\left\\{\begin{array}[]{rlll}\dot{S}(t)&=-\alpha SI,&&t>0,\\\
\dot{I}(t)&=\alpha SI-\mu I,&&t>0,\\\ \dot{R}(t)&=\mu
I,&&t>0.\end{array}\right.$ (1.1)

The functions $S(t),I(t),R(t)$ represent the number of susceptible, infectious
or recovered individuals respectively, at time $t>0$. The susceptible are
contaminated by the infectious, according to a law of mass-action, with a rate
$\alpha I$, where $\alpha>0$ is a constant parameter that accounts for the
transmission efficiency of the disease. The infected individuals cease to be
infectious with some recovery rate $\mu$ (then, $\frac{1}{\mu}$ is the mean
duration of the infection). They then become recovered. The recovered
individual do not become susceptible again and therefore do not play any role
in the dynamic of the system.

We review the main results on (1.1) in Section 1.2. For now, let us mention
that a serious drawback of this model is that it is purely temporal: spatial
effects (migrations and localisation of individuals for instance) are not
taken into account. Spatial effects are now recognised as being of first
importance in the description of epidemics. Many models were introduced to
bridge this gap. A very classical way to build a spatial model is to add
diffusion terms on the equations. The simplest diffusive SIR model is the
following (originally introduced by Källén [12]):

$\left\\{\begin{array}[]{rll}\partial_{t}S(t,x)&=-\alpha SI,&t>0,\
x\in\mathbb{R},\\\ \partial_{t}I(t,x)&=d\Delta I+\alpha SI-\mu I,&t>0,\
x\in\mathbb{R},\\\ \partial_{t}R(t,x)&=\mu I,&t>0,\
x\in\mathbb{R}.\end{array}\right.$ (1.2)

Now, $S,I,R$ do not only depend on the time $t$ but also on a space variable
$x\in\mathbb{R}$. The presence of a diffusion operator on the infected
individuals account for the fact that they diffuse, they move randomly, and
therefore can bring the disease into places were it was not present before,
potentially leading to a spatial spread of the disease. Many other spatial
models were introduced, see Section 1.2 for a brief review. One may find
natural to add diffusion on the $S$ and $R$ individuals also. It turns out
that doing so in the equation for $R$ does not change much, but doing so in
the equation for $S$ makes the analysis much more intricate, we give more
details about that in Section 1.2.

The goal of this paper is to answer the following natural question:

What happens when there are several strains?

To answer this, we consider the following spatial and multi-species SIR model:

$\left\\{\begin{array}[]{lll}\partial_{t}S&=-S\left(\sum_{k=1}^{N}\alpha_{k}I_{k}\right),&t>0,\
x\in\mathbb{R},\\\ \partial_{t}I_{k}&=d_{k}\Delta
I_{k}+(\alpha_{k}S-\mu_{k})I_{k},&t>0,\ x\in\mathbb{R},\ k\in\llbracket
1,N\rrbracket,\\\ \partial_{t}R_{k}&=\mu_{k}I_{k},&t>0,\ x\in\mathbb{R},\
k\in\llbracket 1,N\rrbracket.\end{array}\right.$ (1.3)

Here $N\in\mathbb{N}$ is a positive integer, and the system is therefore made
of $2N+1$ equations. The parameters
$\alpha_{1},\ldots,\alpha_{N},\mu_{1},\ldots,\mu_{N},d_{1},\ldots,d_{N}$ are
strictly positive constants. The unknowns are the $2N+1$ functions
$(S(t,x),I_{1}(t,x),\ldots,I_{N}(t,x),R_{1}(t,x),\ldots,R_{N}(t,x))$. Being an
evolution problem, (1.3) has to be completed with $2N+1$ initial data.

This paper is concerned with the long-time behavior of solutions to (1.3) -
our main interests are the convergence to equilibrium and the speed of
propagation of solutions.

System (1.3) describes the spread of $N$ traits, or strains, or variants of a
disease in a population. In the following, we will always use the generic term
_trait_. For $k\in\llbracket 1,N\rrbracket$, $I_{k}$ represents the number of
individuals infected by the trait $k$. They can transmit this trait to the
susceptible individuals with rate $\alpha_{k}$, they recover with rate
$\mu_{k}$, and when recovering they turn into individuals of type $R_{k}$. The
individuals $I_{k}$ diffuse with rate $d_{k}$.

The model (1.3) is probably the simplest and most natural spatial multi-
species SIR model one could come up with: the only phenomena taken into
account are contamination and remission. There is no waning of immunity, there
are no vital dynamics (arrival of new individuals, birth and death of
individuals), and there can not be multiple infections: individuals who were
infected by the trait $k$ will not be infected by any other trait. Yet, as we
shall prove, this model exhibits some interesting behaviors.

The main result of this paper, presented in Section 1.3 as Theorem 1.3, is
that there is a selection via propagation phenomenon that appears: only a
given subset of all the traits (possibly empty) propagates, all the other
traits vanish. Such a phenomenon can be seen as an emergent property of the
model: there is no a priori selection mechanism in the equations.

In addition to finding the traits that survive, and to computing their limits,
we also identify how these traits spread across space. As we will see, the
system develops spatial traveling patterns such as pulses and terraces.

This result at hand, we use it to obtain some qualitative properties on the
model. More precisely, we study how the parameters of the model (the
contamination, remission, and diffusion rates, and the initial densities),
influence the outcome of the epidemic. In the present paper, we consider only
three properties, that illustrate how this model with several strains differs
from the case with one strain.

First, we show that the so-called basic reproduction number (whose definition
is recalled after) is not a good predictor of the dynamic of the model, in the
sense that some strains that have large basic reproduction number may not
propagate while some with low basic reproduction number would. This is
somewhat counter-intuitive, as this basic reproduction number is usually
considered as the criterion to compute the evolution of the disease.

The second property we investigate concerns the contamination rates of each
trait that propagates. More precisely, we show that each trait that spreads is
less contagious and has a lower recovery rate that all the traits that have
propagated before.

Third, we study how the number of surviving individuals depends on the initial
number of susceptible individuals. When there is only one strain that
propagates, it is known that the number of surviving individuals is a
decreasing function of the number of initially susceptible individuals. When
there are several strains, we will see that this may be false.

Let us mention that some models with several strains were previously
introduced in different contexts, sometimes in connection with evolutionary
issues, we refer for instance to [2, 21] and the references therein. However,
these models do not exhibit the same behaviors as the one we consider here:
either only one of the trait propagates, or all the traits propagate - there
is either no selection, or a very strong one. This is because the models
usually considered in the literature take into account other effects (as
internal dynamics, such as reproduction, for instance), which somehow impose a
given dynamic to the system and strongly change its outcome, and also may
simplify the analysis.

To conclude this introduction, let us mention that the system we consider,
(1.3), is a reaction-diffusion system. Although there is a wide literature on
such topics, there are very few results concerning general systems with more
than $2$ components. A key difficulty is the lack of comparison principles for
such systems. Also, the convergence to equilibrium for reaction-diffusion
systems is in general a very intricate question.

Before presenting our results, we first give a quick review on the one-species
SIR models (1.1) and (1.2). This will prove useful in order to develop the
intuition to understand the dynamic of (1.3).

###  1.2 SIR models: review and basic results

As we mentioned already, the original SIR model is the ODE system (1.1). The
main result on this system, proved by Kermack and McKendrick in [15], is that
there is a threshold phenomenon:

  * •

If $\frac{\alpha S_{0}}{\mu}>1$, the disease propagates, in the sense that
there is $\eta>0$ such that, no matter how small $I_{0}>0$ is,
$\lim_{t\to+\infty}S(t)<S_{0}-\eta$.

In other words, even the presence of a very small amount of infectious
individuals leads to a strictly positive number of infections.

  * •

If $\frac{\alpha S_{0}}{\mu}\leq 1$, the disease fades off, i.e.,
$\lim_{t\to+\infty}S(t)$ can be made as close as we want to $S_{0}$ by
decreasing $I_{0}>0$.

This celebrated result led to introduce the so-called basic reproduction
number

$\mathcal{R}_{0}:=\frac{\alpha S_{0}}{\mu},$

whose value dictates the behavior of the disease. This notion is now a
cornerstone of mathematical epidemiology, and was extended to several other
contexts.

As we mentioned, a natural way to turn the ODE system into a spatial system is
to add diffusion terms in it. This leads to the system

$\left\\{\begin{array}[]{rll}\partial_{t}S(t,x)&=d_{S}\Delta S-\alpha
SI,&t>0,\ x\in\mathbb{R},\\\ \partial_{t}I(t,x)&=d_{I}\Delta I+\alpha SI-\mu
I,&t>0,\ x\in\mathbb{R},\\\ \partial_{t}R(t,x)&=\mu I,&t>0,\
x\in\mathbb{R}.\end{array}\right.$ (1.4)

We refer to [20] for more details on this model. Observe that we do not write
diffusion on the $R$ individuals, this is because they do not really play any
role in the dynamic. Adding diffusion on them would not change much the
analysis and the results. The diffusion on the $S$ individuals, on the other
hand, rises many complications. For instance, the boundedness of solutions is
not clear at all (when $d_{S}=d_{I}$, one could sum the equations for $S$ and
$I$ and apply the parabolic comparison principle, but in general, the question
is rather intricate, see [8] for instance).

For this reason, we consider here only the case where $d_{S}=0$, that is, the
susceptible do not move, and then (1.4) boils down to (1.2). In this case, the
threshold phenomenon of Kermack and McKendrick can be extended in the
following form: consider (1.2) and take for initial data
$S(0,x)=S_{0}\in\mathbb{R}_{+}$ (i.e., the susceptible population is initially
uniformly distributed), $I(0,x)=I_{0}(x)$ where $I_{0}$ is non-negative, non-
zero, and compactly supported (the infectious individuals and initially
localized), and take $R(0,x)=0$.

First, if $\frac{\alpha S_{0}}{\mu}\leq 1$, then the epidemic fades off in the
sense that, for every $I_{0}$ non-negative and compactly supported, one has

$\lim_{|x|\to+\infty}\lim_{t\to+\infty}R(t,x)=0\quad\text{and}\quad\lim_{|x|\to+\infty}\lim_{t\to+\infty}S(t,x)=S_{0}.$

This means that the number of recovered and susceptible individuals are
unchanged after the contamination (in the limit $t\to+\infty)$, at least far
away in space (in the limit $|x|\to+\infty)$: far away from the initial focus
of infection (the support of $I_{0}$ say), no contamination occurred. Of
course, in the vicinity of the initial focus of infection, even when the
disease does not propagate, $\lim_{t\to+\infty}S$ and $\lim_{t\to+\infty}R$
can be arbitrarily small (resp. large), depending on the initial datum
$I_{0}$. This is natural: even if a disease does not propagate, if half of the
population of a city is initially contaminated, then at least half of the
population will become recovered.

Now, if $\frac{\alpha S_{0}}{\mu}>1$, the disease propagates with speed
$c^{\star}:=2\sqrt{d(\alpha S_{0}-\mu)}$ and has asymptotic value $\rho>0$, in
the sense that

$\sup_{|x|>ct}R(t,x)\underset{t\to+\infty}{\longrightarrow}0,\quad\forall
c>c^{\star}\quad\text{and}\quad\sup_{\delta<|x|<ct}|R(t,x)-\rho|\underset{\begin{subarray}{c}t\to+\infty\\\
\delta\to+\infty\end{subarray}}{\longrightarrow}0,\quad\forall c<c^{\star}.$

This means that infections occur everywhere in space asymptotically in time,
even far away from the support of $I_{0}$. The number $\rho$ represents the
number of casualties, and one can show that $\rho$ is the unique positive
solution of the transcendent equation
$S_{0}(1-e^{-\frac{\alpha}{\mu}\rho})=\rho$. Observe that it is independent of
$I_{0}$.

The propagation can also be seen on the evolution of the number of susceptible
$S(t,x)$:

$\sup_{|x|>ct}|S(t,x)-S_{0}|\underset{t\to+\infty}{\longrightarrow}0,\quad\forall
c>c^{\star}\quad\text{and}\quad\sup_{\delta<|x|<ct}|S(t,x)-S_{1}|\underset{\begin{subarray}{c}t\to+\infty\\\
\delta\to+\infty\end{subarray}}{\longrightarrow}0,\quad\forall c<c^{\star},$

where $S_{1}:=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho}<S_{0}$.

Again, saying that the disease propagates is relevant only for large values of
$|x|$, hence the $\delta$ in the above suprema.

A crucial observation is that the dynamic far away from the support of $I_{0}$
will be independent of $I_{0}$ itself.

The above result means that, when $\frac{\alpha S_{0}}{\mu}>1$, then $R$
develops into two traveling waves that connect $0$ to $\rho$, one traveling to
the right and one to the left, and $S$ develops into two waves that connect
$S_{0}$ to $S_{1}$. The $I$ density develops into two traveling pulses (one
going to the right, one going to the left). The dynamic at a given time is
represented in the following drawing, where $I$ is represented with dashed
line, and $S,R$ with solid lines.

The proof of this result relies on the fact that $R(t,x)$ solves

$\partial_{t}R(t,x)=d_{I}\Delta R(t,x)+f(R(t,x))+\mu I_{0}(x),\quad t>0,\
x\in\mathbb{R},$ (1.5)

where $f(z)=\mu S_{0}(1-e^{-\frac{\alpha}{\mu}z})-\mu z$.

This can be obtained by doing some easy algebraic manipulations on the
equations (we will present these in the sequel). The resulting equation (1.5)
is a KPP reaction-diffusion equation, whose name refers to Kolmogorov,
Petrovski and Piskunov, who proved some crucial results on such equations, see
[18]. The threshold phenomenon and the speed of propagation we mentioned
before are then direct consequences of some now basic results from reaction-
diffusion equation theory (that we will present below in the course of our
proof). In particular, $R$ converges to a stationary solution of (1.5), that
is, $R(t,x)\underset{t\to+\infty}{\longrightarrow}R_{\infty}(x)$, where
$R_{\infty}$ solves

$d_{I}\Delta R_{\infty}(x)+f(R_{\infty}(x))+\mu I_{0}(x)=0,\quad
x\in\mathbb{R}.$

Moreover, one can show that $\lim_{|x|\to+\infty}R_{\infty}(x)=\rho$, where
$\rho\in\mathbb{R}$ satisfies the stationary equation “far away” in space,
i.e., where $I_{0}$, which is compactly supported, has no influence. More
precisely, $\rho$ is solution of

$f(\rho)=0.$

It is easy to see that there is a unique positive $\rho$ solution of this
equation if and only if $f^{\prime}(0)=\alpha S_{0}-\mu>0$.

For the reader interested in the case where $d_{S}>0$, we refer to the work of
Hosono and Ilyas [11], who proved the existence of traveling fronts for (1.4),
and also [8], where the Cauchy problem is considered (under a more general
form).

Let us briefly mention some related works. Here, we build spatial models from
the ODE model by adding diffusion terms. One could also consider models where
contaminations occur at distance. This was first suggested by Kendall [13,
14], and such models were studied in the homogeneous case by Diekmann and
Thieme, see [3, 4, 23, 24, 25, 26], and in the heterogeneous case by the first
author, see [5]. Heterogeneous diffusive SIR models were also considered, see
[6, 7]. Let us also mention the work [1] where the authors consider the
influence of networks on similar models.

###  1.3 Results of the paper

We now present here our main results. Before doing so, let us mention that the
existence and uniqueness of classical solutions for (1.3) (that is, solutions
that are $C^{1}_{t}C^{2}_{x}$ for $I,R$ and $C^{1}_{t}C^{0}_{x}$ for $S$) is
standard, see [10] for instance.

We start with defining what it means for a trait to propagate, and what is the
speed of propagation.

######  Definition 1.1 (Propagation).

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$, where $S_{0}$ is a positive
constant and $I_{1}^{0},\ldots,I_{N}^{0}$ are continuous, non-zero, non-
negative and compactly supported.

  * •

We define the asymptotic value of the $k$-th trait, $k\in\llbracket
1,N\rrbracket$, by

$\rho_{k}:=\liminf_{|x|\to+\infty}\left(\lim_{t\to+\infty}R_{k}(t,x)\right).$

  * •

We say that the $k$-th trait propagates if and only if

$\rho_{k}>0.$

If $\rho_{k}=0$, we say that the trait vanishes or goes extinct.

  * •

When the $k$-th trait, $k\in\llbracket 1,N\rrbracket$, propagates, we say that
it propagates with speed $c^{\star}>0$ if

$\left(\sup_{|x|>ct}R_{k}(t,x)\right)\underset{t\to+\infty}{\longrightarrow}0,\quad\text{
for every }\ c>c^{\star},$

and

$\left(\sup_{\delta<|x|<ct}|R_{k}(t,x)-\rho_{k}|\right)\underset{\begin{subarray}{c}t\to+\infty\\\
\delta\to+\infty\end{subarray}}{\longrightarrow}0,\quad\text{ for every }\
c<c^{\star}.$

Observe that, in the system (1.3), we have that $\partial_{t}R_{k}(t,x)\geq
0$, hence, $\lim_{t\to+\infty}R_{k}(t,x)$ exists. Then $\rho_{k}$ is always
well defined.

We mention that $\lim_{t\to+\infty}R_{k}(t,x)$ depends on the values of the
initial data $I_{1}^{0},\ldots,I_{N}^{0}$, but $\rho_{k}$ does not, as we
shall see. Whether a trait propagates or not will not depend on the initial
data, provided they all satisfy the hypotheses of the theorem. In particular,
if some initial datum is not compactly supported, the result is false. Also,
if some $I_{k}^{0}$ is everywhere equal to $0$, one has to remove it and
consider the problem with $N-1$ traits.

Let us now introduce some notations. For $N$ fixed (the number of traits that
exist at initial time), for $(\alpha_{k})_{\llbracket
1,N\rrbracket},(\mu_{k})_{\llbracket 1,N\rrbracket},(d_{k})_{\llbracket
1,N\rrbracket}$ given, we define some functions
$c_{k}(\sigma),f_{k}(\rho,\sigma)$ and $\rho_{k}(\sigma)$, of variables
$\rho\in\mathbb{R}$ and $\sigma\in\mathbb{R}^{+}$.

  * •

The _speed function_ $c_{k}(\sigma)$ is defined for $\sigma\geq 0$ by

$c_{k}(\sigma):=\begin{cases}2\sqrt{d_{k}(\alpha_{k}\sigma-\mu_{k})},\quad&\text{
when }\ \alpha_{k}\sigma-\mu_{k}\geq 0,\\\
0,\quad&\text{otherwise}.\end{cases}$

  * •

The _reaction_ function $f_{k}(\rho,\mu)$, defined for
$\rho\in\mathbb{R},\sigma\geq 0$, is given by

$f_{k}(\rho,\sigma):=\sigma\mu_{k}(1-e^{-\frac{\alpha_{k}}{\mu_{k}}\rho})-\mu_{k}\rho.$

  * •

For $\sigma>0$ given, the equation $f_{k}(\cdot,\sigma)=0$ has a unique
positive solution if and only if $\frac{\alpha_{k}\sigma}{\mu_{k}}>1$. When it
exists, we denote this positive solution

$\rho_{k}(\sigma).$

According to the reminders we made in the previous section concerning the
model (1.2) with one strain, we see that $c_{k}(\sigma),f_{k}(\rho,\sigma)$
and $\rho_{k}(\sigma)$ would be respectively the speed of propagation, the
nonlinear reaction term and the asymptotic value for the $k$-th trait,
provided it were the only trait present in the model and if the initial number
of susceptible were $\sigma$.

We can now explain the heuristic underlying the results we are about to
present. When there is only one trait, the result we mention in the previous
section says that either it fades off, or it propagates with some speed. It it
propagates, the trait consumes a certain proportion of the susceptible
individuals. If we have several traits, then we can expect the competition
between the traits to be negligible at first, and then each trait would start
to propagate with the speed it would have provided it were alone: the trait
$k$ would propagate with speed $c_{k}(S_{0})$. The idea is that the trait that
would propagate faster will indeed do so. But, doing so, it will consume a
proportion of the susceptible, and therefore all the other traits will have to
propagate in an environment where the number of susceptible is reduced - and
the speed of propagation of each trait depends on the number of susceptible
individuals.

For instance, assume that $c_{1}(S_{0})>c_{k}(S_{0})$ for all $k\neq 1$. Then,
the trait $1$ would start to propagate, and it will leave behind a number of
susceptible equal to $S_{1}:=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}}$
(this is at least what happens if it were alone, as explained in the previous
section). Then all the other traits will be left in an environment where the
number of susceptible is equal to $S_{1}$, not $S_{0}$. Then, one can expect
the above dynamic to continue this way: the trait $j\neq 1$ that maximizes
$c_{j}(S_{1})$ will propagate, and consume a proportion of the susceptible,
and the dynamic should go on until either there are not enough susceptible
left for the remaining traits to propagate, or all the traits have propagated.

At a given time, the situation should look like this, when $2$ traits
propagate (we only represent the right hand side of space, the dynamic is
similar on the left hand side):

We will prove that the above heuristic is indeed true, but only under some
extra hypotheses. Basically, we will have to require the speeds of each trait
that propagate are not “too close”. This will ensure that we control the
interactions between the traits. Crucially, if this is not true, then the
heuristic and the result may be false, see the remark after the statement of
Theorem 1.3.

To formalise the heuristic, we introduce the following:

######  Definition 1.2.

Let $N\in\mathbb{N}^{\star}$ and $S_{0}>0$ be fixed, and let
$(d_{k},\alpha_{k},\mu_{k})_{k\in\llbracket 1,N\rrbracket}$ be strictly
positive real numbers. We define the _propagation sequences_ as four (finite)
sequences $(k_{i})_{i\in\llbracket 1,p\rrbracket}$, $(c_{i})_{i\in\llbracket
1,p\rrbracket}$, $(\rho_{i})_{i\in\llbracket 1,p\rrbracket}$ and
$(S_{i})_{i\in\llbracket 0,p\rrbracket}$ as follows: first, we take
$S_{i=0}=S_{0}$, and then we define recursively:

  * •

$k_{i}:=\text{argmax}_{z\in\llbracket
1,N\rrbracket\backslash\\{k_{1},\ldots,k_{i-1}\\}}\left\\{c_{z}(S_{i-1})\
\text{such that}\ c_{z}(S_{i-1})>0\right\\}$. For now, we assume that the
argmax is well defined (that is, there is a single $z$ that maximizes
$c_{z}(S_{i-1})$). When stating our results, we will actually need a stronger
hypothesis.

If the above set is empty, then $k_{i}$ is not defined and the sequence stops
here.

  * •

$c_{i}:=c_{k_{i}}(S_{i-1})$.

  * •

$\rho_{i}:=\rho_{k_{i}}(S_{i-1})$.

  * •

$S_{i}:=S_{i-1}e^{-\frac{\alpha_{k_{i}}}{\mu_{k_{i}}}\rho_{i}}$.

The heuristic we explained before then says that the traits
$k_{1},\dots,k_{p}$ are the one that should propagate, with speed
$c_{1},\ldots,c_{p}$ and each one will have asymptotic value
$\rho_{1},\ldots,\rho_{p}$ respectively.

For any trait $k_{i}$ that propagates, with $i=1,\ldots,p$, then there will be
$S_{i-1}$ susceptible just before this trait propagates, and $S_{i}$
susceptible after.

One could define the final number of susceptible individuals

$S_{\infty}:=\lim_{|x|\to+\infty}\left(\lim_{t\to+\infty}S(t,x)\right).$

It represents the number of individuals that have survived to all the traits.
When our heuristic holds true, we will have

$S_{\infty}=S_{p},$

where $p$ is given by the propagation sequences.

Our main result is the following :

######  Theorem 1.3.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$, where
$S_{0}\in\mathbb{R}^{+}$ and the $I_{i}^{0}$ are continuous non-negative, non-
zero, compactly supported initial data, for all $i\in\llbracket
1,N\rrbracket$.

Let $(k_{i})_{i\in\llbracket 1,p\rrbracket},(c_{i})_{i\in\llbracket
1,p\rrbracket},(\rho_{i})_{i\in\llbracket
1,p\rrbracket},(S_{i})_{i\in\llbracket 0,p\rrbracket}$ be the propagation
sequences given in Definition 1.2.

Assume that, for all $i\in\llbracket 1,p-1\rrbracket$,

$c_{k}(S_{i-1})+c_{k}(S_{i})<c_{i},\quad\text{for all}\ k\neq
k_{1},\ldots,k_{i},$ (1.6)

and that

$\alpha_{k}S_{p}-\mu_{k}<0,\quad\text{for all}\ k\neq k_{1},\ldots,k_{p}.$
(1.7)

Then, the traits $k_{1},\ldots,k_{p}$ propagate in the sense of Definition
1.1, with speeds $c_{1},\ldots,c_{p}$, and have asymptotic values
$\mathcal{\rho}_{1},\ldots,\rho_{p}$. All the other traits do not propagate
and fade off.

Let us give some remarks on this theorem. First, if, when computing the
propagation sequences, one finds that $p=0$ or $p=1$, then the hypothesis
(1.6) is void, and the theorem only requires that (1.7) to hold true.
Similarly, if one finds that $p=N$, then the hypothesis (1.7) is void and only
(1.6) is required to apply the theorem.

When $N=1$ and $p=0$, observe that our theorem does not completely boils down
to the result on the model with a single trait recalled in Section 1.2.
Indeed, in this case, we need (1.7) to be able to say that the (single) trait
does not propagate, but as we mentioned in Section 1.2 it is sufficient to
have $\alpha_{1}S_{0}-\mu_{1}\leq 0$.

Now, as we can see, Theorem 1.3 proves that our heuristic is true provided
that (1.6) and (1.7) hold true. These restrictions are actually necessary in
the sense that the result may be false without them. Indeed, hypothesis (1.6)
says that when a trait propagates, the speed of the other traits that could
propagate are “sufficiently” smaller. If two traits have almost equal speeds,
then the two fronts that should arise can actually interfere, and the slower
front could accelerate. This phenomenon was observed for instance in the
papers [9, 19], in the case of competition models. Actually, in the case where
there are two traits (i.e., $N=2$ in (1.3)), then we believe that one could
use the results presented in these papers to get rid of (1.6) and (1.7) in
Theorem 1.3 (but then the traits that propagate are not given bu our
propagation sequence). When $N>2$, the situation seems more intricate, and we
leave it for further study.

Once that we have Theorem 1.3, we can use it to obtain some qualitative
properties of our model. Indeed, this theorem tells us how the solutions
behave according to the values of the parameters: if we want to know which
traits propagate, or how many individuals are contaminated or left untouched
after the epidemic, then we can compute the propagation sequences. However,
these can be rather intricate to compute. We do not plan here to give a
complete picture of the qualitative properties of the model, we leave it for
further study, but we will nevertheless show three properties who shed light
on some aspects of the system (1.3).

First, we shall see that, when several strains can propagate, then the basic
reproduction number of each strain does not indicate which one can propagate.

More precisely, we shall construct examples of situations with two traits, one
which has a large basic reproduction number, and one with a small reproduction
number, and where the trait with the small basic reproduction number
propagates, while the other fades off. This is the object of Corollary 3.1
below.

Second, we prove that, when several traits propagate, each one is less
contagious (i.e., it has a lower contamination rate $\alpha_{i}$) and is less
“mortal” in the sense that its remission rate $\mu_{i}$ is lesser, than each
trait that propagates before. This result is the Corollary 3.2 below.

Third, we study how the number of individuals left untouched after the
epidemic depends on the number of susceptible individuals initially present.
We will consider the case with two traits. The picture here can be rather
different from the case with a single trait. This is the object of Corollary
3.3.

We give the precise statements and the proofs of this three properties, which
are corollaries of Theorem 1.3, in Section 3. First, we dedicate the next
Section 2, to the proof of Theorem 1.3.

##  2 Proof of Theorem 1.3

###  2.1 Outline of the proof

This section is dedicated to the proof of our main result, Theorem 1.3. In
this whole section, the parameters $S_{0},(\alpha_{i})_{i\in\llbracket
1,N\rrbracket},(\mu_{i})_{i\in\llbracket
1,N\rrbracket},(d_{i})_{i\in\llbracket 1,N\rrbracket}$ are fixed, strictly
positive constants, and $(k_{i})_{i\in\llbracket 1,p\rrbracket}$,
$(c_{i})_{i\in\llbracket 1,p\rrbracket}$, $(\rho_{i})_{i\in\llbracket
1,p\rrbracket}$, $(S_{i})_{i\in\llbracket 0,p\rrbracket}$ denote the
propagation sequences given in Definition 1.2. We also assume that the initial
data for the $(I_{i})_{i\in\llbracket 1,N\rrbracket}$, denoted
$(I^{0}_{i})_{i\in\llbracket 1,N\rrbracket}$, are continuous, non-zero, non-
negative and compactly supported.

We prove Theorem 1.3 by (finite) induction. For $j\in\llbracket
0,p\rrbracket$, we denote $(H_{j})$ the induction hypothesis:

$\text{ The traits }\ k_{1},\ldots,k_{j}\ \text{propagate with speeds}\
c_{1},\ldots,c_{j}\ \text{and asymptotic values}\ \rho_{1},\ldots,\rho_{j}.$
($H_{j}$)

When $j=0$, we use the convention that the hypothesis $(H_{0})$ is empty (that
is, when we write: assume ($H_{j}$), when $j=0$, we have nothing to assume).

To prove Theorem 1.3, we will prove first that $(H_{p})$ holds true, and then
that every trait $k\neq k_{1},\ldots,k_{p}$ does not propagate. A key point in
the proof will be to obtain suitable exponential estimates on the functions
$I_{k},R_{k}$, for all $k$. This is the object of the following:

######  Proposition 2.1.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

Assume that the separation of speeds hypothesis (1.6) holds true and that the
induction hypothesis ($H_{j}$) is verified for some $j\in\llbracket
0,p\rrbracket$.

Then, for all $k\neq k_{1},\ldots,k_{j}$, for all $\varepsilon>0$ small
enough, there are $A^{\varepsilon}_{k,j},B^{\varepsilon}_{k,j}>0$ such that

$I_{k}(t,x)\leq
A^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}\big{(}|x|-(c_{k}(S_{j})+\varepsilon)t\big{)}},\quad\text{for
all }\ t>0,\ x\in\mathbb{R},$

and

$R_{k}(t,x)\leq
B^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}\big{(}|x|-(c_{k}(S_{j})+\varepsilon)t\big{)}},\quad\text{for
all }\ t>0,\ x\in\mathbb{R}.$

This proposition will allow us to obtain upper bounds on the speed of
propagation.

Observe that the estimates on the functions $R_{k}$ are a consequence of the
estimates on the $I_{k}$. Indeed, we have $\partial_{t}R_{k}=\mu_{k}I_{k}$,
integrating the estimate on $I_{k}$ yields the same estimate on $R_{k}$ with
$B^{\varepsilon}_{k,j}=\mu_{k}A^{\varepsilon}_{k,j}\frac{2d_{k}}{(c_{k}(S_{k})+\varepsilon)^{2}}$.

We repeat that the hypothesis (1.6) is indeed necessary to have this result:
it would be false otherwise.

Let us mention that, in the above Proposition 2.1, it is possible to have
$c_{k}(S_{j})=0$ for some $k,j$. In this case, the exponential in the
proposition travels with a speed $\varepsilon>0$ as small as we want.

The proof of Proposition 2.1 is done in several steps, each one being a lemma.
First, we will rewrite the system (1.3) under a different form, more
precisely, we will write it as a system implying only the functions $R_{k}$,
$k\in\llbracket 1,N\rrbracket$. This is the object of Lemma 2.2.

Then, we will see that this lemma implies that the functions $I_{k},R_{k}$ are
respectively subsolutions and supersolutions of some equations, this is done
in Lemma 2.3. Combining these two lemmas with another technical result (Lemma
2.4), we will finally get Proposition 2.1.

We then use this proposition to show that, if $(H_{j})$ is verified for some
$j\in\llbracket 0,p-1\rrbracket$, then $(H_{j+1})$ also holds true. This is
the object of Proposition 2.5 below. Eventually, this yields that $(H_{p})$
holds true.

Finally, we prove in Proposition 2.7 that the traits $k\neq
k_{1},\ldots,k_{p}$ do not propagate, and this will conclude the proof of
Theorem 1.3.

###  2.2 Proof of the exponential estimates, Proposition 2.1

We start with rewriting the system (1.3). The next lemma strongly relies on
the fact that there is no diffusion on the susceptible individuals.

######  Lemma 2.2.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

  * •

The function $S(t,x)$ can be written as

$S(t,x)=S_{0}\prod_{i=1}^{N}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)},\quad\forall
t>0,\ x\in\mathbb{R}.$

  * •

For any $k\in\llbracket 1,N\rrbracket$, for $t>0,x\in\mathbb{R}$, we have

$\partial_{t}R_{k}=d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}-\mu_{k}R_{k}+\alpha_{k}S_{0}\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}\left(\prod_{i\neq
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)d\tau.$

###### Proof.

The proof follows from the simple observation that the first equation in
(1.3), giving $S(t,x)$, can be rewritten, using the equations for
$R_{1},\ldots,R_{N}$, as

$\frac{\partial_{t}S}{S}=-\sum_{i=1}^{N}\frac{\alpha_{i}}{\mu_{i}}\partial_{t}R_{i}.$

Integrating this relation directly gives the result.

To get the second point, we simply replace $S(t,x)$ by the expression we just
found in the equation giving $I_{k}$, thus obtaining

$\partial_{t}I_{k}=d_{k}\Delta
I_{k}+\left(\alpha_{k}S_{0}\prod_{i=1}^{N}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}-\mu_{k}\right)I_{k}.$

Then, we multiply by $\mu_{k}$ and we integrate over the $t$ variable, from
$0$ to some $t>0$. Remembering that $\partial_{t}R_{k}=\mu_{k}I_{k}$, we find
the result. ∎

The next lemma builds on the previous one, and shows that the functions
$I_{k},R_{k}$, $k\in\llbracket 1,N\rrbracket$, are respectively
subsolutions/supersolutions of some parabolic equations on some domains.

######  Lemma 2.3.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

Assume that the induction hypothesis ($H_{j}$) holds true for some
$j\in\llbracket 0,p\rrbracket$. Then, the following inequalities hold true:

  * •

For every $\theta>1$, $\varepsilon>0$, there is $T>0$ such that

$\partial_{t}I_{k}\leq d_{k}\Delta
I_{k}+(\alpha_{k}S_{j}\theta-\mu_{k})I_{k},\quad\text{for}\ t>T,\
|x|<(c_{j}-\varepsilon)t,\ k\in\llbracket 1,N\rrbracket.$

  * •

For $k\neq k_{1},\ldots,k_{j}$, for any $\omega\in(0,1)$, there is $L>0$ such
that

$\partial_{t}R_{k}\geq d_{k}\Delta R_{k}+f_{k}(R_{k},\omega
S_{j})-\mu_{k}S_{j}\left(\sum_{i\neq
k,k_{1},\ldots,k_{j}}\frac{\alpha_{i}}{\mu_{i}}R_{i}\right)R_{k},\quad\text{for
}\ t>0,|x|>L.$

When $j=0$, the first point in the above lemma is not well-written ($c_{j}$ is
not defined when $j=0$). In this case, we use the convention that the first
point should read: for every $\theta>1$, $\varepsilon>0$, there is $T>0$ such
that

$\partial_{t}I_{k}\leq d_{k}\Delta
I_{k}+(\alpha_{k}S_{0}\theta-\mu_{k})I_{k},\quad\text{for}\ t>T,\ \
k\in\llbracket 1,N\rrbracket,$

that is, there is no restriction on $x$.

The second point reads similarly when $j=0$, the sum in the right hand term
simply runs on all $i\in\llbracket 1,N\rrbracket\backslash k$.

###### Proof.

Let $j\in\llbracket 0,p\rrbracket$ be fixed and assume that ($H_{j}$) holds
true. We denote $K_{j}:=\\{k_{1},\ldots,k_{j}\\}$ and $K_{j}^{c}:=\llbracket
1,N\rrbracket\backslash K_{j}$ (if $j=0$, then $K_{j}=\emptyset$).

_First point._ We already know from Lemma 2.2 that, for all $k\in\llbracket
1,N\rrbracket$,

$\partial_{t}I_{k}=d_{k}\Delta I_{k}+\alpha_{k}S_{0}\left(\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)\left(\prod_{i\in
K_{j}^{c}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)-\mu_{k}\right)I_{k}.$

Owing to ($H_{j}$), the traits $i\in K_{j}$ all propagate, with speed at least
$c_{j}$ (because $c_{1}>c_{2}>\ldots>c_{j}$), hence, for every
$\eta,\varepsilon>0$, there is $T>0$ such that

$R_{i}(t,x)\geq\rho_{i}-\eta,\quad\text{ for all }\ i\in K_{j},\ t>T,\
|x|<(c_{j}-\varepsilon)t.$

Then, for $t>T,|x|<(c_{j}-\varepsilon)t$, we have

$\prod_{i\in K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}\leq\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}\prod_{i\in
K_{j}}e^{\frac{\alpha_{i}}{\mu_{i}}\eta}$

Take $\eta>0$ small enough so that $\prod_{i\in
K_{j}}e^{\frac{\alpha_{i}}{\mu_{i}}\eta}\leq\theta$. On the other hand, we
bound from above $\prod_{i\in
K_{j}^{c}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}$ by $1$ and we recall
that, by definition, $S_{j}=S_{0}\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}$, to get

$\partial_{t}I_{k}\leq d_{k}\Delta
I_{k}+(\alpha_{k}S_{j}\theta-\mu_{k})I_{k},\quad\text{for }\ t>T,\
|x|<(c_{j}-\varepsilon)t.$

_Second point._ Let us take $k\neq k_{1},\ldots,k_{j}$. Owing to Lemma 2.2, we
have

$\partial_{t}R_{k}=d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}-\mu_{k}R_{k}+\alpha_{k}S_{0}\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}\left(\prod_{i\neq
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)d\tau.$

Because all the $R_{i}$ are non-decreasing with respect to $t$, we find that

$\displaystyle\partial_{t}R_{k}$ $\displaystyle\geq d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}-\mu_{k}R_{k}+\alpha_{k}S_{0}\left(\prod_{i\neq
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}\right)\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}d\tau$
$\displaystyle\geq d_{k}\Delta
R_{k}-\mu_{k}R_{k}+\mu_{k}S_{0}\left(\prod_{i\neq
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}\right)(1-e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}).$

Now, owing to ($H_{j}$), for $i\in K_{j}$, we know that $R_{i}$ propagates and
has asymptotic value $\rho_{i}$. Therefore (because $R_{i}$ is time-
nondecreasing), for every $\eta>0$, there is $L>0$ such that

$R_{i}(t,x)\leq\rho_{i}+\eta,\quad\text{for}\ t>0,\ |x|>L,\ i\in K_{j}.$

This allows us to derive a lower bound on $\prod_{i\neq
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}$ for $t>0$ and $|x|>L$ (recall
that $k\not\in K_{j}$):

$\displaystyle\prod_{i\neq k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}$
$\displaystyle=\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}\prod_{i\in K_{j}^{c}\backslash
k}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)}$
$\displaystyle\geq\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}\right)\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\eta}\right)\left(1-\sum_{i\in
K_{j}^{c}\backslash k}\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)\right).$

Hence, up to taking $\eta$ small enough so that $\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\eta}>\omega$, we get, for $t>0$, $|x|>L$,

$\displaystyle\partial_{t}R_{k}$ $\displaystyle\geq d_{k}\Delta
R_{k}-\mu_{k}R_{k}+\mu_{k}S_{0}\omega\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}\left(1-\sum_{i\in
K_{j}^{c}\backslash
k}\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)\right)\left(1-e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\right)$
$\displaystyle\geq d_{k}\Delta R_{k}-\mu_{k}R_{k}+\mu_{k}\omega
S_{j}\left(1-\sum_{i\in K_{j}^{c}\backslash
k}\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)\right)\left(1-e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\right)$
$\displaystyle\geq d_{k}\Delta R_{k}+f_{k}(R_{k},\omega
S_{j})-\mu_{k}S_{j}\omega\left(\sum_{i\in K_{j}^{c}\backslash
k}\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)\right)\left(1-e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\right)$
$\displaystyle\geq d_{k}\Delta R_{k}+f_{k}(R_{k},\omega
S_{j})-\alpha_{k}S_{j}\left(\sum_{i\in K_{j}^{c}\backslash
k}\frac{\alpha_{i}}{\mu_{i}}R_{i}(t,x)\right)R_{k},$

hence the result. ∎

Now, in order to prove Proposition 2.1, we need the following technical lemma:

######  Lemma 2.4.

Let $d,r>0$ be chosen, and let $u(t,x)\geq 0$ be $C^{1}_{t}C^{2}_{x}$,
bounded, such that:

  * •

There are $c_{1},A>0$ such that, for all $t>0$, $x\in\mathbb{R}$, we have

$u(t,x)\leq Ae^{-\frac{c_{1}}{2d}(x-c_{1}t)}.$

  * •

There are $T,c_{2}>0$ such that, for all $t>T,|x|<c_{2}t$, we have

$\partial_{t}u-d\Delta u\leq ru.$

In addition, define

$c_{3}:=2\sqrt{rd}.$

Assume that

$c_{1}>c_{3}\ \text{and}\ c_{2}>c_{1}+c_{3}.$ (2.8)

Then, there is $B>0$ such that, for $t>0$, $x\in\mathbb{R}$,

$u(t,x)\leq Be^{-\frac{c_{3}}{2d}(x-c_{3}t)}.$

Observe that in this lemma, we need an hypothesis on $c_{1},c_{2},c_{3}$,
namely (2.8). When applying this lemma later, this requirement will force us
to assume the separation hypothesis (1.6) on the speeds of each trait.

###### Proof.

For $B>\max\\{A,\sup_{t,x}u\\}$, to be chosen large enough after, define

$w(t,x):=Be^{-\frac{c_{3}}{2d}(x-c_{3}t)}.$

Observe first that, for $t>0,x>c_{2}t$, we indeed have
$Ae^{-\frac{c_{1}}{2d}(x-c_{1}t)}\leq w(t,x)$, and then $w\geq u$ there. For
$t>0,x<-c_{2}t$, we have $w(t,x)\geq B$, and then $w\geq u$ also in this
region. Up to taking $B$ large enough, we can also ensure that $w\geq u$ in
the bounded domain $\\{(t,x)\ :t\in(0,T),\ |x|<c_{2}t\\}$.

It remains to prove that $w\geq u$ on the domain $E:=\\{(t,x)\ :t>T,\
|x|<c_{2}t\\}$. To do so, we use the parabolic comparison principle (see
[22]).

We have

$\partial_{t}w-d\Delta w=rw,\quad\text{ for }t>0,x\in\mathbb{R}.$

By hypothesis, we know that $u$ is subsolution of this equation on the domain
$E$.

To apply the parabolic comparison principle, we need that $w\geq u$ on
$\partial E$, that is, we need to have $w(T,x)\geq u(T,x)$ for $|x|\leq
c_{2}T$ and $w(t,x)\geq u(t,x)$ for $t>T$ and $|x|=c_{2}t$.

Because $u$ is bounded, taking $B$ large enough guarantees that the first
requirement is verified. For the second requirement, it is sufficient to have

$Be^{-\frac{c_{3}}{2d}(c_{2}-c_{3})t}\geq
Ae^{-\frac{c_{1}}{2d}(c_{2}-c_{1})t},\quad\text{for}\ t>T,$

that is, up to increasing $B$, this is true if we have
$\frac{c_{3}}{2d}(c_{2}-c_{3})<\frac{c_{1}}{2d}(c_{2}-c_{1})$, but this
condition is guaranteed by (2.8). Then, we can indeed apply the parabolic
comparison theorem to $u,w$ on $E$ to get the result. ∎

We can now finally turn to the proof of Proposition 2.1.

###### Proof of Proposition 2.1.

We will prove that, under the hypotheses of Proposition 2.1, we have, for
$\varepsilon>0$ small enough,

$I_{k}(t,x)\leq
A^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}(x-(c_{k}(S_{j})+\varepsilon)t)},\quad\text{for
all }\ t>0,\ x\in\mathbb{R}.$

This is not exactly was is required, because we do not have the absolute value
on the $x$ in the exponential. To get it, one can either redo the same
arguments with supersolution moving toward the left (that is, supersolutions
of the form $e^{\lambda(x+ct)}$), or, equivalently, one can do a change of
variable $x\to-x$, that does not change the equations in (1.3) and apply the
results to these new equations.

We argue by induction. In this proof, the induction hypothesis is denoted
$(P_{j})$, for $j\in\llbracket 0,p\rrbracket$ and is the following:

$(H_{j})\implies\forall k\neq k_{1},\ldots,k_{j},\ \forall\varepsilon>0\
\text{small enough},\ \exists A^{\varepsilon}_{k,j}>0\ \text{such that}\\\
I_{k}(t,x)\leq
A^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}(x-(c_{k}(S_{j})+\varepsilon)t)},\quad\text{for
all }\ t>0,\ x\in\mathbb{R}.$ (2.8)

When $j=0$, by convention $(H_{0})$ is the void hypothesis, it is always true.
Therefore, $(P_{0})$ would hold true if we can prove the upper bounds on the
$I_{k}$ functions, for all $k\in\llbracket 1,N\rrbracket$, unconditionally.

To do so, start with observing that $S(t,x)\leq S_{0}$. This implies that, for
every $k\in\llbracket 1,N\rrbracket$, $I_{k}$ satisfies

$\partial_{t}I_{k}\leq d_{k}\Delta I_{k}+(\alpha_{k}S_{0}-\mu_{k})I_{k},\quad
t>0,\ x\in\mathbb{R}.$

For any $\varepsilon>0$, the function

$w(t,x)=Ae^{-\frac{(c_{k}(S_{0})+\varepsilon)}{2d_{k}}(x-(c_{k}(S_{0})+\varepsilon)t)}$

is supersolution of this equation. Indeed, $\partial_{t}w-d_{k}\Delta
w=\frac{(c_{k}(S_{0})+\varepsilon)^{2}}{4d_{k}}w\geq(\alpha_{k}S_{0}-\mu_{k})w$.
Hence, provided $A$ is large enough so that $w(0,\cdot)\geq I_{k}^{0}$, the
parabolic comparison principle yields the result.

Now, assume the property (2.8) is verified for some $j\in\llbracket
0,p-1\rrbracket$, and let us show that $(P_{j+1})$ is true.

To do so, we assume that $(H_{j+1})$ holds true. Therefore, $(H_{j})$ is
verified, and then $(P_{j})$ tells us that for all $k\neq k_{1},\ldots,k_{j}$,
for $\varepsilon>0$ small enough, there is $A^{\varepsilon}_{k,j}>0$ such that

$I_{k}(t,x)\leq
A^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}(x-(c_{k}(S_{j})+\varepsilon)t)},\quad\text{for
all }\ t>0,\ x\in\mathbb{R}.$

In addition, because $(H_{j+1})$ is assumed to hold, we can apply Lemma 2.3,
to get that, for $\theta>1$, $\eta>0$, there is $T>0$ such that

$\partial_{t}I_{k}\leq d_{k}\Delta
I_{k}+(\alpha_{k}S_{j+1}\theta-\mu_{k})I_{k},\quad t>T,\ |x|<(c_{j+1}-\eta)t,\
k\in\llbracket 1,N\rrbracket.$

Define $r:=\frac{(c_{k}(S_{j+1})+\varepsilon)^{2}}{4d_{k}}$. Then, up to
taking $\theta$ close enough to $1$, we can ensure that
$\alpha_{k}S_{j+1}\theta-\mu_{k}\leq r$, and then

$\partial_{t}I_{k}\leq d_{k}\Delta I_{k}+rI_{k},\quad t>T,\
|x|<(c_{j+1}-\eta)t,\ k\in\llbracket 1,N\rrbracket.$

Now, define

$c_{1}=c_{k}(S_{j})+\varepsilon,c_{2}=c_{j+1}-\eta,c_{3}=c_{k}(S_{j+1})+\varepsilon.$

Then, up to taking $\varepsilon,\eta>0$ small enough, we can ensure that

$c_{1}+c_{3}<c_{2},$

thanks to the hypothesis (1.6). We always have $c_{1}>c_{3}$, because
$S_{j+1}<S_{j}$. Then, we can apply Lemma 2.4 and this yields the result. ∎

###  2.3 Conclusion. Proof of Theorem 1.3

We now use Proposition 2.1 to prove that the hypothesis $(H_{p})$ is true,
that is, the traits $k_{1},\ldots,k_{p}$ computed by the propagation sequences
indeed propagate, with their prescribed speeds to their prescribed asymptotic
values. To prove this, we argue by induction again, this is the object of the
following:

######  Proposition 2.5.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

Assume that ($H_{j}$) holds true for some $j\in\llbracket 0,p-1\rrbracket$.
Then the trait $k_{j+1}$ propagates with speed $c_{j+1}$ and has asymptotic
value $\rho_{j+1}$, that is, $(H_{j+1})$ also holds true.

We start with the following lemma, that uses Proposition 2.1 to improve Lemma
2.3.

######  Lemma 2.6.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

  * •

Assume that ($H_{j}$) holds true for some $j\in\llbracket 0,p-1\rrbracket$.
Let $\tilde{c}=\max_{i\neq k_{1},\ldots,k_{j+1}}c_{i}(S_{j})$. For any
$\eta>0$, $\omega\in(0,1)$ and for every $c>\tilde{c}$, there is $T>0$ such
that, for $t>T$ and $|x|>ct$,

$\partial_{t}R_{k_{j+1}}\geq d_{k_{j+1}}\Delta
R_{k_{j+1}}+f_{k_{j+1}}(R_{k_{j+1}},\omega S_{j})-\eta R_{k_{j+1}}.$ (2.9)

  * •

Assume that ($H_{j}$) holds true for some $j\in\llbracket 0,p\rrbracket$. For
any $\theta>1$, for any $k\neq k_{1},\ldots,k_{j}$, for any $c\in(0,c_{j})$,
there are $C,q>0$ such that, for $|x|<ct$,

$\partial_{t}R_{k}\leq d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}+Ce^{-q|x|}+f_{k}(R_{k},\theta S_{j}).$ (2.10)

###### Proof.

_First point. $R_{k_{j+1}}$ is supersolution of a reaction-diffusion
equation._

Let $j\in\llbracket 0,p-1\rrbracket$ be fixed and assume that ($H_{j}$) holds
true. Owing to Lemma 2.3, we know that, for any $\omega\in(0,1)$, there is
$L>0$ such that $R_{k_{j+1}}$ satisfies, for $t>0$, $|x|>L$,

$\partial_{t}R_{k_{j+1}}\geq d_{k_{j+1}}\Delta
R_{k_{j+1}}+f_{k_{j+1}}(R_{k_{j+1}},\omega
S_{j})-\mu_{k_{j+1}}S_{j}\left(\sum_{i\neq
k_{1},\ldots,k_{j},k_{j+1}}\frac{\alpha_{i}}{\mu_{i}}R_{i}\right)R_{k_{j+1}}.$

For $i\neq k_{1},\ldots,k_{j}$, for $\varepsilon>0$ small enough, the bounds
provided by Proposition 2.1 tell us that $R_{i}(t,x)\leq
B^{\varepsilon}_{i,j}e^{-\frac{(c_{i}(S_{j})+\varepsilon)}{2d_{i}}(|x|-(c_{i}(S_{j})+\varepsilon)t)}$,
for some $B^{\varepsilon}_{i,j}>0$. Hence, for $t>0$, $|x|>L$,

$\partial_{t}R_{k_{j+1}}\geq d_{k_{j+1}}\Delta
R_{k_{j+1}}+f_{k_{j+1}}(R_{k_{j+1}},\omega S_{j})-\\\
\mu_{k_{j+1}}S_{j}\left(\sum_{i\neq
k_{1},\ldots,k_{j},k_{j+1}}\frac{\alpha_{i}}{\mu_{i}}B^{\varepsilon}_{i,j}e^{-\frac{(c_{i}(S_{j})+\varepsilon)}{2d_{i}}(|x|-(c_{i}(S_{j})+\varepsilon)t)}\right)R_{k_{j+1}}.$

Let $c>\max_{i\neq k_{1},\ldots,k_{j+1}}c_{i}(S_{j})=\tilde{c}$. Up to taking
$\varepsilon$ smaller if needed, we ensure that $\tilde{c}+\varepsilon<c$.
Hence, for $|x|>ct$, for all $i\neq k_{1},\ldots,k_{j+1}$, we have

$e^{-\frac{(c_{i}(S_{j})+\varepsilon)}{2d_{i}}(|x|-(c_{i}(S_{j})+\varepsilon)t)}\leq
e^{-\frac{(c_{i}(S_{j})+\varepsilon)}{2d_{i}}(c-\tilde{c}-\varepsilon)t}.$

Therefore, for any $\eta>0$, there is $T>0$ large enough so that

$\partial_{t}R_{k_{j+1}}\geq d_{k_{j+1}}\Delta
R_{k_{j+1}}+f_{k_{j+1}}(R_{k_{j+1}},\omega S_{j})-\eta
R_{k_{j+1}},\quad\text{for}\ t>T,\ ct<|x|.$

_Second point. $R_{k}$ is subsolution of a reaction-diffusion equation._

Let $j\in\llbracket 0,p\rrbracket$ be fixed and assume that ($H_{j}$) holds
true. Let $k\not\in K_{j}=\\{k_{1},\ldots,k_{j}\\}$ be chosen. Owing to Lemma
2.2, we know that $R_{k}$ satisfies

$\partial_{t}R_{k}\leq d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}-\mu_{k}R_{k}+\alpha_{k}S_{0}\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)d\tau$

Let $c<c_{j}$ be chosen. We take $\overline{c}$ such that
$\overline{c}\in(\max\\{c,c_{j+1}\\},c_{j})$ (with the convention that, if
$j=p$, then $c_{p+1}:=0$).

The hypothesis ($H_{j}$) yields that, for any $\eta>0$, for all $i\in
K_{j}=\\{k_{1},\ldots,k_{j}\\}$, we have $R_{i}(t,x)\geq\rho_{i}-\eta$ for
$|x|<\overline{c}t$.

Denote $\theta=\prod_{i\in K_{j}}e^{\frac{\alpha_{i}}{\mu_{i}}\eta}$. Assume
that $|x|<ct$. Then, $|x|<\overline{c}t$ and we have

$\displaystyle\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}$
$\displaystyle\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)d\tau$
$\displaystyle\leq\int_{0}^{\frac{|x|}{\overline{c}}}\partial_{t}R_{k}d\tau+\int_{\frac{|x|}{\overline{c}}}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}R_{i}}\right)d\tau$
$\displaystyle\leq\int_{0}^{\frac{|x|}{\overline{c}}}\partial_{t}R_{k}d\tau+\theta\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}\right)\int_{\frac{|x|}{\overline{c}}}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}d\tau$
$\displaystyle\leq
R_{k}\left(\frac{|x|}{\overline{c}},x\right)+\theta\left(\prod_{i\in
K_{j}}e^{-\frac{\alpha_{i}}{\mu_{i}}\rho_{i}}\right)\int_{0}^{t}e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}}\partial_{t}R_{k}d\tau.$

Therefore,

$\partial_{t}R_{k}\leq d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}-\mu_{k}R_{k}+\mu_{k}\theta
S_{j}(1-e^{-\frac{\alpha_{k}}{\mu_{k}}R_{k}})+\alpha_{k}S_{0}R_{k}\left(\frac{|x|}{\overline{c}},x\right).$

Now, using again Proposition 2.1, for $\varepsilon>0$ small enough, we have
$R_{k}(t,x)\leq
B^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}}(|x|-(c_{k}(S_{j})+\varepsilon)t)}.$
Up to taking $\varepsilon$ smaller if needed so that
$\overline{c}>c_{k}(S_{j})+\varepsilon$, this implies that

$R_{k}\left(\frac{|x|}{\overline{c}},x\right)\leq
B^{\varepsilon}_{k,j}e^{-\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}\overline{c}}(\overline{c}-(c_{k}(S_{j})+\varepsilon))|x|}.$

In the end, taking
$q:=\frac{(c_{k}(S_{j})+\varepsilon)}{2d_{k}\overline{c}}(\overline{c}-(c_{k}(S_{j})+\varepsilon))$
and $C:=\alpha_{k}S_{0}B_{k,j}^{\varepsilon}$, we indeed find (2.10). ∎

We are now in position to prove Proposition 2.5.

###### Proof of Proposition 2.5..

We assume that $(H_{j})$ holds true. Let us prove that $(H_{j+1})$ is
verified, that is, we want to prove that, for all $c>c_{j+1}$,

$\sup_{ct<|x|}|R_{k_{j+1}}(t,x)|\underset{t\to+\infty}{\longrightarrow}0,$
(2.11)

and that, for all $c\in(0,c_{j+1})$,

$\sup_{\delta<|x|<ct}|R_{k_{j+1}}(t,x)-\rho_{j+1}|\underset{\delta,t\to+\infty}{\longrightarrow}0.$
(2.12)

_Step 1.Proof of ( 2.11)._
Owing to Proposition 2.1, for $\varepsilon>0$ small enough, there is $B>0$
such that

$R_{k_{j+1}}(t,x)\leq
Be^{-\frac{(c_{k_{j+1}}(S_{j})+\varepsilon)}{2d_{k_{j+1}}}\big{(}|x|-(c_{k_{j+1}}(S_{j})+\varepsilon)t\big{)}},\quad\text{for
all }\ t>0,\ x\in\mathbb{R}.$

Then, for any $c>c_{j+1}+\varepsilon=c_{k_{j+1}}(S_{j})+\varepsilon$, we have

$\sup_{|x|>ct}R_{k_{j+1}}(t,x)\leq
Be^{-\frac{(c_{j+1}+\varepsilon)}{2d_{k}}(c-c_{j+1}-\varepsilon)t}\underset{t\to+\infty}{\longrightarrow}0.$

Then, because we can take $\varepsilon$ as close to zero as we want, (2.11)
holds true.

_Step 2.Upper bound._
The remaining steps are dedicated to prove (2.12). This particular step is
dedicated to prove that, for any $\eta>0$, there are $M,\varepsilon>0$ such
that

$R_{k_{j+1}}(t,x)\leq\rho_{j+1}+\eta+Me^{-\varepsilon|x|}.$ (2.13)

This will imply in particular that, for any $c>0$,

$\limsup_{\begin{subarray}{c}t\to+\infty\\\
\delta\to+\infty\end{subarray}}\sup_{\delta<|x|<ct}\left(R_{k_{j+1}}(t,x)-\rho_{j+1}\right)\leq
0.$

The remaining step below will prove that this is also true with interverting
$\rho_{j+1}$ and $R_{k_{j+1}}$ (but then we will have to restrict to
$c<c_{j+1}$).

First, let us take $c\in(c_{j+1},c_{j})$. Observe that, owing to (2.10) from
Lemma 2.6, we have, for any $\theta>1$, for $|x|<ct$,

$\partial_{t}R_{k_{j+1}}\leq d_{k_{j+1}}\Delta
R_{k_{j+1}}+\mu_{k_{j+1}}I_{k_{j+1}}^{0}+Ce^{-q|x|}+f_{k_{j+1}}(R_{k_{j+1}},\theta
S_{j}),$

for some $C,q>0$.

We denote $\rho(\theta)$ the unique positive solution of the equation
$f_{k_{j+1}}(\rho(\theta),\theta S_{j})=0$ (this solution exists because
$z\mapsto f_{k_{j+1}}(z,\theta S_{j})$ is concave and vanishes at $z=0$ where
it has a strictly positive derivative, and goes to $-\infty$ as
$z\to+\infty$). We have $\rho(\theta)\to\rho_{j+1}$ as $\theta\to 1$.

Let us define the function

$u(t,x)=\rho(\theta)+Me^{-\varepsilon x}$

Because the function $f_{k_{j+1}}(\cdot,\theta S_{j})$ is concave, we have
$f_{k_{j+1}}(u,\theta S_{j})\leq f_{k_{j+1}}^{\prime}(\rho(\theta),\theta
S_{j})Me^{-\varepsilon x}$. Therefore,

$\partial_{t}u-d_{k_{j+1}}\Delta u-f_{k_{j+1}}(u,\theta S_{j})\geq
Me^{-\varepsilon
x}(-d_{k_{j+1}}\varepsilon^{2}-f_{k_{j+1}}^{\prime}(\rho(\theta),\theta
S_{j})).$

Up to taking $M>0$ large enough and $\varepsilon>0$ small enough, we can
guarantee that

$M(-d_{k_{j+1}}\varepsilon^{2}-f_{k_{j+1}}^{\prime}(\rho(\theta),\theta
S_{j}))\geq(\mu_{k_{j+1}}I_{k_{j+1}}^{0}+Ce^{-q|x|})e^{\varepsilon x},$

indeed, $f_{k_{j+1}}^{\prime}(\rho(\theta),\theta S_{j})<0$ and
$I_{k_{j+1}}^{0}$ is compactly supported.

Therefore, $u$ and $R_{k_{j+1}}$ are respectively supersolution and
subsolution of the same parabolic equation on the domain $|x|<ct$ ($u$ is
actually supersolution everywhere in space and time). We have $u(0,\cdot)\geq
R_{k_{j+1}}(0,\cdot)=0$. To apply the parabolic comparison principle, we need
to have $R_{k_{j+1}}(t,\pm ct)\leq u(t,ct)$, but this comes again from the
exponential estimates of Proposition 2.1, up to taking $M$ large enough (and
because we took $c>c_{j+1}$).

Doing the same reasoning with the supersolution $\rho(\theta)+Me^{\varepsilon
x}$, we finally end up with finding, thanks to the parabolic comparison
principle, that

$R_{k_{j+1}}(t,x)\leq\rho(\theta)+Me^{-\varepsilon|x|},\quad t>0,\ |x|<ct.$

Taking $\theta$ close enough to $1$ yields that (2.13) is verified at least
for $|x|<ct$, but because $R_{k_{j+1}}$ is non-decreasing with respect to $t$,
this is actually true for all $t>0,x\in\mathbb{R}$.

_Step 3. Lower bound._
It now remains to prove that, for $c\in(0,c_{j+1})$, we have

$\limsup_{\begin{subarray}{c}\delta\to+\infty\\\
t\to+\infty\end{subarray}}\sup_{\delta<|x|<ct}\left(\rho_{j+1}-R_{k_{j+1}}(t,x)\right)\leq
0,$

or, equivalently, that

$\liminf_{\begin{subarray}{c}\delta\to+\infty\\\
t\to+\infty\end{subarray}}\inf_{\delta<|x|<ct}R_{k_{j+1}}(t,x)\geq\rho_{j+1}.$
(2.14)

To do so, we use (2.9) from Lemma 2.6. Let us take
$\underline{c},\overline{c}\in(\max\\{\tilde{c},c\\},c_{j+1})$ (where
$\tilde{c}$ is from Lemma 2.6), such that $\underline{c}<\overline{c}$.

Let us take $\eta>0$ small enough and $\omega\in(0,1)$ close enough to $1$
such that

$2\sqrt{d_{k_{j+1}}(\omega\alpha_{k_{j+1}}S_{j}-\mu_{k_{j+1}}-\eta)}>\overline{c},$

which is possible because
$2\sqrt{d_{k_{j+1}}(\alpha_{k_{j+1}}S_{j}-\mu_{k_{j+1}})}=c_{j+1}>\overline{c}$.
The equation (2.9) from Lemma 2.6 then implies that there is $T>0$ such that

$\partial_{t}R_{k_{j+1}}\geq d_{k_{j+1}}\Delta
R_{k_{j+1}}+f_{k_{j+1}}(R_{k_{j+1}},\omega S_{j})-\eta
R_{k_{j+1}},\quad\text{for}\ t>T,\ |x|>\underline{c}t.$

Let us now define $u(t,x)$ as the solution of the equation

$\partial_{t}u=d_{k_{j+1}}\Delta u+f_{k_{j+1}}(u,\omega S_{j})-\eta
u,\quad\text{for}\ t>T,\ |x|>\underline{c}t,$

with (moving) Dirichlet boundary condition $u(t,\pm\underline{c}t)=0$ for
$t>T$, and with initial condition at time $t=T$ given by
$u(T,\cdot)=R_{k_{j+1}}(T,\cdot)$.

Then, applying the parabolic comparison principle on the domain $E:=\\{(t,x)\
:\ t>T,\ |x|>\underline{c}t\\}$, we find that $R_{k_{j+1}}(t,x)\geq u(t,x)$
for $(t,x)\in E$.

Now, because $R_{k_{j+1}}(t,x)$ is non-decreasing with respect to $t$, and by
continuity, we have that there is $x^{\delta}_{t}$ such that
$\delta<|x^{\delta}_{t}|<ct$ that satisfies

$\inf_{\delta<|x|<ct}R_{k_{j+1}}(t,x)=R_{k_{j+1}}(t,x^{\delta}_{t})\geq
R_{k_{j+1}}\left(\frac{|x^{\delta}_{t}|}{\overline{c}},x^{\delta}_{t}\right).$

Observe that we used the fact that $\overline{c}>c$ here.

Now, for $\delta,t$ large enough, we have
$\left(\frac{|x^{\delta}_{t}|}{\overline{c}},x^{\delta}_{t}\right)\in E$.
Hence, for $\delta,t$ large, we have

$\inf_{\delta<|x|<ct}R_{k_{j+1}}(t,x)\geq
u\left(\frac{|x^{\delta}_{t}|}{\overline{c}},x^{\delta}_{t}\right).$

Now, owing to classical results from reaction-diffusion equations theory, we
have

$u(t,\pm\overline{c}t)\to\rho_{j+1}(\omega,\eta),$

where $\rho_{j+1}(\omega,\eta)$ is the unique positive solution of

$f_{k_{j+1}}(\rho_{j+1}(\omega,\eta),\omega
S_{j})-\eta\rho_{j+1}(\omega,\eta)=0.$

This solution exists and is unique provided $\eta,\omega$ are small enough and
close enough to $1$ respectively. Clearly, we have
$\rho_{j+1}(\omega,\eta)\to\rho_{j+1}$ as $\eta$ goes to zero and $\omega$
goes to $1$, which yields (2.14), and concludes the proof. ∎

To conclude the proof of Theorem 1.3, we need the following:

######  Proposition 2.7.

Let $(S,I_{1},\ldots,I_{N},R_{1},\ldots,R_{N})$ be the solution of (1.3)
arising from the initial datum
$(S_{0},I_{1}^{0},\ldots,I_{N}^{0},0,\ldots,0)$.

Assume that $(H_{p})$ holds true. Then, for every $k\neq k_{1},\ldots,k_{p}$,
the trait $k$ vanishes.

###### Proof.

Let $k\neq k_{1},\ldots,k_{p}$ be chosen. Using (2.10) from Lemma 2.6, we know
that, for $\theta>1$, for $c<c_{p}$, there are $C,q,T>0$ such that

$\partial_{t}R_{k}\leq d_{k}\Delta
R_{k}+\mu_{k}I_{k}^{0}+Ce^{-q|x|}+f_{k}(R_{k},\theta S_{p}),\quad\text{for}\
t>T,\ |x|<ct.$

By definition of the propagation sequence, up to taking $\theta$ close enough
to $1$ and thanks to hypothesis (1.7), we can guarantee that
$-\eta:=\alpha_{k}\theta S_{p}-\mu_{k}<0$. Hence, by concavity of $z\mapsto
f_{k}(z,\theta S_{p})$, and because $f^{\prime}_{k}(0,\theta S_{p})=-\eta$, we
have

$\partial_{t}R_{k}\leq d_{k}\Delta R_{k}+\mu_{k}I_{k}^{0}+Ce^{-q|x|}-\eta
R_{k},\quad\text{for}\ t>T,\ |x|<ct.$

Now, owing to the estimates from Proposition 2.1, we know that, for every
$\varepsilon>0$, we have (remember that $c_{k}(S_{p})=0$)

$R_{k}(t,x)\leq
B^{\varepsilon}_{k,p}e^{-\frac{\varepsilon}{2d_{k}}\big{(}|x|-\varepsilon
t\big{)}},\quad\text{for all }\ t>0,\ x\in\mathbb{R}.$

Let now $M,\lambda>0$ be fixed, and define $w(t,x):=Me^{-\lambda x}$. We have

$\partial_{t}w-d_{k}\Delta w+\eta w=(-d_{k}\lambda^{2}+\eta)Me^{-\lambda
x}\geq\mu_{k}I_{k}^{0}+Ce^{-q|x|},$

provided $\lambda$ is small enough and $M$ is large enough. Moreover, up to
decreasing $\lambda$ if needed, we have

$Me^{-\lambda ct}\geq
B^{\varepsilon}_{k,p}e^{-\frac{\varepsilon}{2d_{k}}\big{(}c-\varepsilon\big{)}t}.$

Therefore, $R_{k}(t,\pm ct)\leq w(t,\pm ct)$. Hence, we can apply the
parabolic comparison principle to get that

$R_{k}(t,x)\leq Me^{-\lambda x},\quad\text{for }\ t>T,\ |x|<ct.$

Because $R_{k}(t,x)$ is non-decreasing with respect to $t$, then this is
actually true for all $t>0,x\in\mathbb{R}$. Therefore, the $k$-th trait does
not propagate, hence the result. ∎

##  3 Qualitative properties

Now that we have proved Theorem 1.3, we use it to study some qualitative
properties of our model (1.3). More precisely, Theorem 1.3 tells us that the
dynamic of the system is given by the propagation sequences from Definition
1.2. Hence, understanding how the propagation sequences depend on the
parameters of the model will yield qualitative properties.

As we already mentioned, this task may be rather intricate, because the
propagation sequences depend on the parameters in a complex way. Hence, we do
not plan on giving a general picture here but rather we want to emphasize the
three properties mentioned at the end of Section 1.3. These three properties
are stated as the three following results, that are corollaries of Theorem
1.3.

First, we show that the basic reproduction numbers, introduced in Section 1.2
as the ratio $\frac{\alpha S_{0}}{\mu}$, are not good indicators of the
dynamic of the system.

######  Corollary 3.1.

Consider the system (1.3) with $N=2$. Take $d_{1}=d_{2}=1$. Then, one can
choose $\alpha_{1},\alpha_{2},\mu_{1},\mu_{2}$ and $S_{0}$ such that

$1<\frac{\alpha_{1}S_{0}}{\mu_{1}}<\frac{\alpha_{2}S_{0}}{\mu_{2}},$

but where the trait $1$ propagates and the trait $2$ vanishes.

This corollary gives the existence of situations where the trait with the
lower basic reproduction number propagates, and where the trait with the
larger basic reproduction number disappears. We mention it can be adapted to
the case where $N\geq 2$.

This is in contradiction with the intuition one could have built from the
model with one single trait.

The second property we prove concerns the contagiousness of each successive
trait. More precisely, we show that each trait that propagates is less
contagious, and has a lower recovery rate, than the traits that have
propagated before. This is not completely direct from Theorem 1.3, which only
tells us that each trait that propagates has a smaller speed than the trait
that propagates before.

######  Corollary 3.2.

Consider the system (1.3) with all diffusion equals, i.e.,
$d_{1}=\ldots=d_{N}$. Let $(k_{i})_{i\in\llbracket 1,p\rrbracket}$ be the
sequence of the indices that propagate, given in Definition 1.2.

Assume that the coefficients satisfy the hypotheses of Theorem 1.3 (in
particular (1.6) and (1.7)).

Then, for all $i\in\llbracket 1,p-1\rrbracket$, one has

$\alpha_{k_{i+1}}<\alpha_{k_{i}}\quad\text{ and
}\quad\mu_{k_{i+1}}<\mu_{k_{i}}.$

To conclude, we present a result concerning the total number of casualties,
or, what is equivalent, the final number of susceptible individuals.

We denote

$S_{\infty}:=\lim_{|x|\to+\infty}\lim_{t\to+\infty}S(t,x).$

As we already mention in Section 1.3, when we can apply Theorem 1.3, we have
$S_{\infty}=S_{p}$, where $S_{p}$ is given by the propagation sequences.

When $N=1$, and when $\frac{\alpha S_{0}}{\mu}\leq 1$, the single trait does
not propagate, and $S_{\infty}=S_{0}$, whereas when $\frac{\alpha
S_{0}}{\mu}>1$, the trait propagates, and
$S_{\infty}=S_{0}e^{-\frac{\alpha}{\mu}\rho}$, were $\rho$ solves
$S_{0}(1-e^{-\frac{\alpha}{\mu}\rho})=\rho$.

We can rewrite this by saying that, in this case $N=1$, $S_{\infty}$ is the
smallest solution of

$S_{0}-\frac{\mu}{\alpha}\ln(S_{0})=S_{\infty}-\frac{\mu}{\alpha}\ln(S_{\infty}).$

This comes easily from the result we just mentioned. The function
$f(z)=z-\frac{\mu}{\alpha}\ln(z)$ is strictly convex, goes to $+\infty$ as $z$
goes to $0$ or to $+\infty$, and has a unique minimum, reached at
$z=\frac{\mu}{\alpha}$.

Observe that we necessarily have
$S_{0}e^{-\frac{\alpha}{\mu}\rho}<\frac{\mu}{\alpha}$ when
$S_{0}>\frac{\mu}{\alpha}$ (this is natural, when a disease propagates, the
number of susceptible left untouched is too small for disease to propagate
again). We shall use this fact several time in the following proofs.

We can plot $S_{\infty}$ as a function of $S_{0}$. One would obtain the
following graph:

Our next corollary shows how the final number of susceptible $S_{\infty}$
depends on the initial number of susceptible $S_{0}$ when $N=2$.

######  Corollary 3.3.

Consider the system (1.3) with $N=2$. We denote
$S_{\infty}:=\lim_{|x|\to+\infty}\lim_{t\to+\infty}S(t,x)$.

Without loss of generality, assume that

$\frac{\mu_{1}}{\alpha_{1}}<\frac{\mu_{2}}{\alpha_{2}}.$

Then, there are $0<\underline{r}\leq\overline{r}$ such that

  * •

If

$\frac{d_{2}}{d_{1}}<\underline{r},$

the situation is the same as if there were only the trait $1$ in the model:

    * –

When $S_{0}<\frac{\mu_{1}}{\alpha_{1}}$, we have $S_{\infty}=S_{0}$.

    * –

When $S_{0}>\frac{\mu_{1}}{\alpha_{1}}$, only the trait $1$ propagates, and
$S_{\infty}$ is the smallest solution of

$S_{\infty}-\frac{\mu_{1}}{\alpha_{1}}S_{\infty}=S_{0}-\frac{\mu_{1}}{\alpha_{1}}\ln(S_{0}).$

  * •

If

$\frac{d_{2}}{d_{1}}>\overline{r}$

there are $\underline{S},\overline{S}$ that verify
$\frac{\mu_{1}}{\alpha_{1}}<\frac{\mu_{2}}{\alpha_{2}}<\underline{S}<\overline{S}$
and $\varepsilon\geq 0$ such that

    * –

When $S_{0}<\frac{\mu_{1}}{\alpha_{1}}$, no trait propagate, and
$S_{\infty}=S_{0}$.

    * –

When $S_{0}\in(\frac{\mu_{1}}{\alpha_{1}},\underline{S})$, only the trait $1$
propagates, and then
$S_{\infty}=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{0})}$.

    * –

When $S_{0}\in(\underline{S},\underline{S}+\varepsilon)$, we can not apply
Theorem 1.3.

    * –

When $S_{0}\in(\underline{S}+\varepsilon,\overline{S})$, then the traits $2$
and $1$ propagate, and
$S_{\infty}=S_{1}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{1})}$, where
$S_{1}=S_{0}e^{-\frac{\alpha_{2}}{\mu_{2}}\rho_{2}(S_{0})}$.

    * –

When $S_{0}>\overline{S}$, then only the trait $2$ propagates, and
$S_{\infty}=S_{0}e^{-\frac{\alpha_{2}}{\mu_{2}}\rho_{2}(S_{0})}$.

In addition, we have that $\underline{S}\to\frac{\mu_{2}}{\alpha_{2}}$ and
$\varepsilon\to 0$ as $\frac{d_{2}}{d_{1}}\to+\infty$, and $\overline{S}$ does
not depend on $d_{1},d_{2}$.

Let us give here some remarks concerning this lemma. Of course, the most
interesting situation is when $d_{1}<<d_{2}$. In this case, plotting
$S_{\infty}$ as a function of $S_{0}$ would give

The dashed zone, for $S_{0}\in(\underline{S},\underline{S}+\varepsilon)$,
corresponds to the zone where we can not apply our theorem ((1.6) is not
verified there). However, observe that the zone can be made as small as we
want up to increasing $d_{2}$ or decreasing $d_{1}$.

We now turn to the proofs of these results.

###### Proof of Corollary 3.1.

We take $d_{1}=d_{2}=1$ fixed without loss of generality.

Let $\alpha_{1},\mu_{1},S_{0}$ fixed in such a way that

$\frac{\alpha_{1}S_{0}}{\mu_{1}}>1.$

Now, let $S_{1}=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}}$ where $\rho_{1}$
is the unique positive solution of $f_{1}(\rho_{1},S_{0})=0$. We have

$\frac{\alpha_{1}S_{1}}{\mu_{1}}<1.$

Take $\lambda\in(\frac{\alpha_{1}}{\mu_{1}},\frac{1}{S_{1}})$ and define
$\alpha_{2}=\varepsilon\lambda$ and $\mu_{2}=\varepsilon$, for
$\varepsilon>0$. Then

$\frac{\alpha_{1}S_{0}}{\mu_{1}}<\lambda
S_{0}=\frac{\alpha_{2}S_{0}}{\mu_{1}}.$

Moreover, up to taking $\varepsilon>0$ small enough, we ensure that

$\alpha_{2}S_{0}-\mu_{2}=\varepsilon(\lambda
S_{0}-1)<\alpha_{1}S_{0}-\mu_{1}.$

We also have

$\alpha_{2}S_{1}-\mu_{2}=\varepsilon(\lambda S_{1}-1)<0.$

We can compute the propagation sequence associated with these parameters: we
find $p=1$ and $k_{1}=1$. Moreover, the hypotheses of Theorem 1.3 are
verified, then only the trait $1$ propagates and the trait $2$ goes extinct. ∎

We now turn to the proof of the second qualitative property.

###### Proof of Corollary 3.2.

Assume that the coefficients satisfy the hypotheses. Then, by definition of
the propagation sequences, we know that, for all $i\in\llbracket
1,p-1\rrbracket$,

$c_{k_{i+1}}(S_{i-1})<c_{k_{i}}(S_{i-1}),$

that is (recall that we assume that the diffusions are equal),

$\alpha_{k_{i+1}}S_{i-1}-\mu_{k_{i+1}}<\alpha_{k_{i}}S_{i-1}-\mu_{k_{i}},$

hence

$(\alpha_{k_{i+1}}-\alpha_{k_{i}})S_{i-1}<\mu_{k_{i+1}}-\mu_{k_{i}}.$ (3.15)

In addition, because the trait $k_{i+1}$ propagates, we necessarily have

$\alpha_{k_{i+1}}S_{i}-\mu_{k_{i+1}}>0.$ (3.16)

Moreover, by definition of $S_{i}$, we have

$\alpha_{k_{i}}S_{i}-\mu_{k_{i}}<0.$ (3.17)

Combining (3.15), (3.16) and (3.17) yields

$(\alpha_{k_{i+1}}-\alpha_{k_{i}})S_{i-1}<(\alpha_{k_{i+1}}-\alpha_{k_{i}})S_{i}.$

Now, because $S_{i-1}>S_{i}$, we have

$\alpha_{k_{i+1}}<\alpha_{k_{i}}.$

Combining this again with (3.16) and (3.17) yields and this implies, thanks to
(3.15)

$\mu_{k_{i+1}}<\mu_{k_{i}},$

and this concludes the proof. ∎

We now prove our third qualitative property.

###### Proof of Corollary 3.3.

We define

$\underline{r}:=\frac{\alpha_{1}}{\alpha_{2}}.$

Then, when $\frac{d_{2}}{d_{1}}\leq\underline{r}$, we have
$d_{2}\alpha_{2}\leq d_{1}\alpha_{1}$ and, for all $S_{0}>0$,

$c_{1}(S_{0})\geq c_{2}(S_{0}).$

If $S_{0}<\frac{\mu_{2}}{\alpha_{2}}$, then
$\alpha_{2}S_{0}-\mu_{2}<\alpha_{1}S_{0}-\mu_{1}<0$. We can compute the
propagation sequences: we find $p=0$, the sequences are empty. We can apply
Theorem 1.3 to get that no trait propagate, and then $S_{\infty}=S_{0}$.

If $S_{0}>\frac{\mu_{2}}{\alpha_{2}}$, then $c_{1}(S_{0})>0$. Computing the
propagation sequence, we find that $k_{1}=1$,
$S_{1}=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{0})}<\frac{\mu_{1}}{\alpha_{1}}$.
This implies that $S_{1}<\frac{\mu_{2}}{\alpha_{2}}$, then
$\alpha_{2}S_{1}-\mu_{2}<0$, hence $p=1$. We can apply Theorem 1.3 to find
that only the trait $1$ propagates, and $S_{\infty}=S_{1}$.

Now, we let $\overline{r}>0$ be a real number to be taken sufficiently large
after (unlike $\underline{r}$, it will not be explicit). We assume for now
that $\overline{r}>\frac{\alpha_{1}}{\alpha_{2}}$.

If $S_{0}<\frac{\mu_{1}}{\alpha_{1}}$, the situation is the same as above, no
trait propagate and $S_{\infty}=S_{0}$.

Now, let

$\underline{S}:=\frac{d_{2}\mu_{2}-d_{1}\mu_{1}}{d_{2}\alpha_{2}-d_{1}\alpha_{1}}.$

Because we assumed that $\overline{r}>\frac{\alpha_{1}}{\alpha_{2}}$, an easy
computation shows that $\underline{S}>\frac{\mu_{2}}{\alpha_{2}}$, and, for
$S_{0}\in(\frac{\mu_{1}}{\alpha_{1}},\underline{S})$, we have
$c_{1}(S_{0})>c_{2}(S_{0})$.

Hence, computing the propagation sequence, we find that $k_{1}=1$ and
$S_{1}=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{0})}$. We have
$S_{1}<\frac{\mu_{1}}{\alpha_{1}}<\frac{\mu_{2}}{\alpha_{2}}$, then
$d_{2}(\alpha_{2}S_{1}-\mu_{2})<0$, and we find $p=1$.

We can apply Theorem 1.3, we find that only the trait $1$ propagates and
$S_{\infty}=S_{1}=S_{0}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{0})}$.

Now, let $\overline{S}$ be the largest solution of

$\overline{S}-\frac{\mu_{2}}{\alpha_{2}}\ln(\overline{S})=\frac{\mu_{1}}{\alpha_{1}}-\frac{\mu_{2}}{\alpha_{2}}\ln(\frac{\mu_{1}}{\alpha_{1}}).$

On the one hand, $\overline{S}$ does not depend on $d_{1},d_{2}$. On the other
hand, it is readily seen that $\underline{S}$ is a decreasing function of
$\frac{d_{2}}{d_{1}}$ and that, when $\frac{d_{2}}{d_{1}}\to+\infty$, we have
$\underline{S}\to\frac{\mu_{2}}{\alpha_{2}}$. We assume from now on that
$\overline{r}$ is large enough so that, for
$\frac{d_{2}}{d_{1}}>\overline{r}$, we have $\overline{S}>\underline{S}$.

We find that, for $S_{0}>\underline{S}$, we have

$c_{1}(S_{0})<c_{2}(S_{0}).$

Computing the propagation sequence, we find $k_{1}=2$, and
$S_{1}=S_{0}e^{-\frac{\alpha_{2}}{\mu_{2}}\rho_{2}(S_{0})}$. If
$S_{0}\in(\underline{S},\overline{S})$, owing to our choice of $\overline{S}$,
we have $S_{1}>\frac{\mu_{1}}{\alpha_{1}}$, and then

$c_{1}(S_{1})>0.$

Then, we find $p=2$, and
$S_{\infty}=S_{2}=S_{1}e^{-\frac{\alpha_{1}}{\mu_{1}}\rho_{1}(S_{1})}$.

Now, to apply Theorem 1.3, we need to verify that (1.6) holds true, i.e., we
can apply the theorem only when $S_{0}$ is such that

$c_{1}(S_{0})+c_{1}(S_{1})<c_{2}(S_{0}).$

Unfortunately, this can not be everywhere verified (indeed,
$c_{2}(S_{0})-c_{1}(S_{0})\to 0$ when $S_{0}\to\underline{S}$). Finding the
minimal $S_{0}$ that satisfies this is rather intricate ($S_{1}$ is a function
of $S_{0}$), but observe that

$c_{1}(S_{0})+c_{1}(S_{1})<2c_{1}(S_{0})=4\sqrt{d_{1}(\alpha_{1}S_{0}-\mu_{1})}.$

Up to increasing $\overline{r}$ so that
$\overline{r}>4\frac{\alpha_{1}}{\alpha_{2}}$, it is easy to see that

$4\sqrt{d_{1}(\alpha_{1}S_{0}-\mu_{1})}<2\sqrt{d_{2}(\alpha_{2}S_{0}-\mu_{2})}\iff
S_{0}>\underline{S}+\varepsilon,$

where
$\varepsilon=\frac{3d_{2}d_{1}(\mu_{2}\alpha_{1}-\mu_{1}\alpha_{2})}{(d_{2}\alpha_{2}-4d_{1}\alpha_{1})(d_{2}\alpha_{2}-d_{1}\alpha_{1})}$
indeed goes to zero when $\frac{d_{2}}{d_{1}}\to+\infty$.

Now, when $S_{0}>\overline{S}$, we still have $c_{2}(S_{0})>c_{1}(S_{0})$, but
now $\alpha_{1}S_{1}-\mu_{1}<0$. Hence, the propagation sequence contains one
element, $p=1$, and we can apply Theorem 1.3 to find that only the trait $2$
propagates, and
$S_{\infty}=S_{1}=S_{0}e^{-\frac{\alpha_{2}}{\mu_{2}}\rho_{2}(S_{0})}$. ∎

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