Classical SIR compartment models are cornerstone models of epidemiology which allow one to study the evolution of an infected population at a given spatial scale (e.g. whole countries, regions, counties or cities). Such models date back to the pioneer work of Kermack and McKendrick [15] and describe the evolution of susceptible (S) and infected (I) populations which eventually become removed (R) via systems of ordinary differential equations which typically take the form where τ > 0 is a contact rate between susceptible and infected populations, and 1/η > 0 is the average infectious period; see [12] for a review on SIR models. These models have been used in the past to reproduce data of epidemic outbreaks such as the bubonic plague [15] , malaria [19] , SARS influenza [6, 18] and most recently COVID-19 [16, 20, 21] ; see also [20] for other applications.
In classical SIR models such as (1.1), interactions among the infected population are oversimplified, and when taken into account they typically involve transfer matrices of populations of infected between various uniform patches [17, 18, 26] . Our interest lies in the understanding of the intricate interplay between spatial effects and heterogeneous interactions among infected populations. Schematically, we propose a model composed of cities linked by a given transportation network (roads, railroads or rivers), see Figure 1 for an illustration in the case of France. It will turn out that the appropriate theoretical framework will be graph theory where each vertices of the graph will be thought as the cities and the edges the lines of transportation. In a first approximation, we will assume that infected populations are only subject to spatial diffusion along the lines, as it is traditionally assumed in classical spatial SIR models [1, 3, 9, 22] . As a consequence, in our model, the dynamics of the epidemic only takes place in the cities. Interactions are then modeled by flux exchanges between cities and lines where we assume that some fraction of infected individuals can either leave a city to be on a line, or leave a line and stop in a city, or pass from one line to another through a city. The typical question that we address here can easily be stated as follows. Given a connected graph of cities linked by roads and an initial configuration of infected individuals, how does the epidemic spread into the network and what is the eventual final configuration of the infected population? Our aim here is to gain insight into this spreading aspect at the fundamental mathematical level of a SIR type model that incorporates the possibility of infected individuals to travel along a specific given transportation network.
Our framework is at the crossroad of models that take into account lines of transportation such as recent reaction-diffusion models that study propagation of epidemics along lines with fast diffusion [3] and models that incorporate networks with more sophisticated interactions dynamics [2, 4, 5, 25] . On a formal level, our proposed model can be thought of as being a one-dimensional version of the planar reaction-diffusion system of [3] with a line of fast diffusion in the case of one city and one line of transportation. Actually, the graph structure of the transportation network provides a natural embedding into a planar spatial domain. From a mathematical point of view, our model shares also some similarities with the PDE-ODE model of [8] which studies the spread of airborne diseases where the movement of pathogens in the air is assumed to follow a linear diffusion.
Throughout, we denote by G = (V, E) a compact metric graph, i.e. a collection of vertices V and edges E and further assume that G is finite and connected. Each edge e ∈ E is identified with a segment Ω e = [0, e ] with e ∈ (0, ∞), where e is the finite length of the edge. A real valued function u : G −→ R is a collection of one dimensional maps defined for each edge e ∈ E: u e : Ω e −→ R. 
Given a graph G, we let X v (t) := (S v (t), I v (t), R v (t)) ∈ R 3 , for each v ∈ V, where S v (t) represents the population of susceptible individuals, I v (t) the population of infected individuals and R v (t) the population of susceptible individuals at vertex v ∈ V and time t > 0. We assume that X v evolves according to a SIR model of the form
where τ v , η v > 0 are the intrinsic parameters of the epidemic which may depend on the vertex v. The contribution −λ v I v (t) in the right-hand side of the equation for the infected population traduces the fact that infected individuals can leave the vertex v to incident edges whereas e∼v α v e u e (t, v) reflects the contribution of incoming infected individuals from incident edges. Here, e ∼ v denotes the edges incident to the vertex v and
such that λ v e I v (t) infected individuals leave vertex v to edge e. We have assumed that only the infected population is subject to movement, and we think of S v being an ambiant population whose movement does not affect its distribution. We recover the standard SIR model (1.1) by considering the trivial graph G = ({v}, ∅). Throughout the manuscript, we will assume the following standing hypothesis on the coefficients α v e and λ v e in (2.1).
Hypothesis 2.1. For each (e, v) ∈ E × V we assume that
α v e ∈ (0, 1) and λ v e ∈ (0, 1), together with e∼v λ v e ∈ (0, 1) and e∼v α v e ∈ (0, 1).
Next, for each e ∈ E, we let d e > 0 and we assume that u e evolves according to
x u e (t, x), t > 0, x ∈
• Ω e .
Assuming that infected individuals have local diffusion along the edges of the graph is a first approximation, and this can be viewed as a limiting Brownian movement of individuals. We shall come back to this modeling hypothesis later in the manuscript, but possible extensions could be to incorporate nonlocal diffusion or transport terms.
It now remains to model the exchanges of infected individuals at the vertices. Fo each v ∈ V, we associate an integer δ v ≥ 1 which we refer to as its degree (i.e. number of edges incident to the vertex v). We define u v (t) ∈ R δv as the column vector function
where we recall that e ∼ v denotes the edges incident to the vertex v, and thus u e (t, v) is the corresponding limit value of u e at x = v. Define also ∂ n u v (t) ∈ R δv as the column vector function
where ∂ n u e (t, v) is the outwardly normal derivative of u e at the vertex v. Our boundary conditions at the vertex v that link (2.1) and (2.2) are described by
and K v ∈ M δv (R) whose structure will be specified below. Formally, (2.3) traduces the balance of fluxes of infected individuals at the vertex v, and we will demonstrate this heuristic rigorously by showing in the forthcoming Subsection 2.4 the conservation of total population.
We now precise the form of the matrix K v entering in the boundary condition (2.3). Essentially, K v gathers two contributions. One contribution comes from the exchanges between infected individuals at the vertex with the incoming infected individuals for the incident edges. The second contribution traduces exchanges between edges. Indeed we allow infected individuals to pass from one edge to another one. More precisely, we have that K v splits into two parts
is such that the sum of each column is zero. More precisely, if we label by e ∼ v = (e 1 , · · · , e δv ) the edges incident to the vertex v, we have that for all i = 1, · · · , δ v
In the case δ v = 3, we get
see Figure 2 for an illustration in that case.
Furthermore, for the diagonal term we will use the shorthand notation
The fact that N v ∈ M δv (R) is such that the sum of each column is zero precisely traduces the fact that there is the conservation of infected individuals through exchanges between incident edges at each vertex. And, we remark that it implies that the matrix K v has a strict column diagonal dominance in the sense that for each i = 1,
because of this specific structure of N v . From now on we also assume that K v has a diagonal dominance with respect to its lines. This property will be crucial later on in the proof of existence of solutions. As a consequence, we impose the following running assumptions on the matrices K v .
Hypothesis 2.2. For each v ∈ V and (e, e ) ∈ E × E, we assume that ν v e,e ∈ [0, 1).
• 
Finally, we complement our coupled PDE-ODE (2.1)-(2.2)-(2.3) with some initial conditions. We assume that at t = 0, we have
On the other hand, for the ODE system (2.1), we suppose that
We further assume that (2.3) is satisfied at t = 0
with obvious notations u 0 v := (u 0 e (v)) e∼v and I 0 v := (I 0 v , · · · , I 0 v ) t . Last, we impose that the initial total population of infected individuals is strictly positive,
and that susceptible individuals are initially present at each vertex of the graph
This in turn implies that the total population is initially
Assuming that there is a solution to (u, (X v ) v∈V ) to (2.1)-(2.2)-(2.3), we have that the total mass of the system M (t) defined as
is a conserved quantity and thus independent of t.
To see that, we first remark that
and ·, · is the standard Euclidean inner product on R δv . On the other hand let us define
and assume that u is a classical solution of (2.2), which we will prove in the next section, and compute
The fact that
is a direct consequence on the specific structure of each matrix N v and the fact that the sum of each column is zero. We therefore conclude that M (t) = 0 and e∈E Ωe
Biological interpretation. Our model is thus consistent with the conservation of the total population as it is traditionally the case for SIR model in the case of zero natality/mortality rate. The exchanges between the vertices and the edges exactly compensate each other as is natural.
We now present our main results regarding our model (2.1)-(2.2)-(2.3). At this stage of the presentation, we remain formal and refer to the following sections for precise statements and assumptions.
Main result 1: Existence and uniqueness of classical solutions. We prove in Theorem 1 below that for each well prepared initial condition our model (2.1)-(2.2)-(2.3) admits a unique positive classical solution which is global in time. We remark that the system (2.1)-(2.2)-(2.3) is not standard as it couples a system of PDEs to ODEs at each vertices through inhomogeneous Robin boundary conditions. As a consequence, the existence and uniqueness of classical solutions has to be proved. This analysis is conducted in Section 3.
Main result 2: Long time behavior of the solutions. We fully characterize the long time behavior of the unique solution of our model. More precisely, we prove that the final total population of infected individuals at each vertex, denoted by I ∞ v , is a well defined quantity: 0 < I ∞ v < ∞ for v ∈ V and (I ∞ v ) v∈V are solutions of a system of c V + 1 implicit equations, where c V stands for the cardinal of V, which belong to the parametrized submanifold
where M 0 is the initial total mass. We refer to Theorem 2 for a precise statement. We also present further qualitative results on the final total configuration (I ∞ v ) v∈V in the fully symmetric case where we obtain closed form formula (see Lemma 4.3) and in the case of two vertices where we manage to obtain sharp bounds on the final total populations of infected individuals (see Lemma 4.4). In each case, we manage to relate these quantities to standard basic and effective reproductive number for classical SIR model. The aforementioned results are proved in Section 4.
Main result 3: A mass preserving semi-implicit numerical scheme. We propose and analyze a semi-implicit in time numerical scheme based on finite differences in space which has the property to preserve a discrete total mass associated to the discretization. We prove that if the time discretization constant is smaller than a universal constant depending only on the parameters of the system (and not on the space discretization constant) and if N v is symmetric for each v ∈ V, then our mass preserving semi-implicit numerical scheme is well-posed and preserves the positivity of the solutions. We refer to Section 5 for a presentation of the numerical scheme and Theorem 3 for a precise statement of our main result.
Main result 4: Numerical results for various types of graphs. We illustrate our theoretical findings with selection of numerical simulations for various types of graphs in Section 6. We respectively study the case of 2 vertices and 1 edge, 3 vertices and 3 edges (closed graph), 4 vertices and 3 edges (star-shape graph) and N + 1 vertices and N edges with N being arbitrarily large (lattice graph). Most notably, in the last case, we show the propagation of the epidemics across the vertices of the graph in the form of a traveling wave.
This section is devoted to the proof of the following main theorem which guarantees that our model is well-posed.
The proof of Theorem 1 is divided into two parts. We first prove the existence of positive global classical solutions and then show that such constructed solutions are unique. We look for solutions that satisfy (2.1)-(2.2)-(2.3) in the classical sense, and we always assume that (S 0 v , I 0 v ) ≥ 0 with S 0 v > 0, v∈V I 0 v > 0 and u 0 ∈ BC(G, R + ), that is for all e ∈ E, u 0 e ≥ 0 is bounded continuous on Ω e . We further assume that the initial conditions satisfy the boundary condition (2.3). We remark that the system (2.1)-(2.2)-(2.3) is not standard as it couples a system PDEs to ODEs at each vertices through inhomogeneous Robin boundary conditions. As a consequence, the well-posedness of the Cauchy problem has to be proved.
Remark 3.1. Our existence and uniqueness result extends trivially in the case that parameters τ v > 0, α v e ∈ (0, 1), λ v e ∈ (0, 1) and ν v e,e ∈ [0, 1) are continuous functions of time satisfying τ v (t) > 0, α v e (t) ∈ (0, 1), λ v e (t) ∈ (0, 1) and ν v e,e (t) ∈ [0, 1) together with Hypotheses 2.1-2.2 verified at all times t > 0.
In this section, we construct a classical solution to (2.1)-(2.2)-(2.3) through a limiting argument. We will obtain a solution (u, (X v ) v∈V ) has the limit of a subsequence of solution ((u n , (X n v ) v∈V )) n≥0 of the following problems
and ∂ t u n e (t, x) = d e ∂ 2 x u n e (t, x), t > 0, x ∈
• Ω e , ∀e ∈ E. starting from u 0 ∈ BC(G, R + ) and (X 0 v ) v∈V . Note that (3.1)-(3.2)(3.3) is supplemented by the same initial condition (u 0 , (X 0 v ) v∈V ) at each step. We proceed along three main steps.
Step #1: solvability of (3. ) v∈V ) such that for each t → u n−1 e (t, v) is continuous, then we get the existence of a unique solution of (3.1) which is C 1 in time. Next we solve the system of PDEs (3.3)-(3.2) whose coupling comes from the boundary conditions and owing that now the right-hand side of (3.2) can be seen as given inhomogeneous term of class C 1 in time. As both D v and K v are invertible matrices, we get the existence of a classical solution u n ∈ C 1,2 which then ensures that t → u n e (t, v) is continuous.
Step #2: a priori estimates. Let 0 < T < 1 be fixed. We first show by a recursive argument that 0 < S n v , 0 ≤ I n v , 0 ≤ R n v for each v ∈ V and 0 ≤ u n e for each e ∈ E. It is trivial at n = 0. Let assume that is it true at n − 1. We start from (3.1) and a direct integration gives
Now owing that 0 ≤ I n v for each v ∈ V, the maximum principle implies that u n e ≥ 0 for each e ∈ E. Assume by contradiction that e * ∈ E is the component which reaches a negative minimum, namely u n e * (t * , x * ) = −δ < 0 with u n e * (t, x) > −δ for t < t * and x ∈ Ω e * and for each e = e * we have u n e (t, x) > −δ for t ≤ t * and x ∈ Ω e . We know that x * ∈ ∂Ω e * and let denote v * = x * ∈ V the vertex where this occurs. The Hopf lemma implies that ∂ n u n e * (t * , v * ) < 0. Inspecting the boundary condition (3.2) at v * , we obtain that
and leads to a contradiction. Here we have used the fact that
from Hypothesis 2.2 on the matrices (K v ) v∈V .
Next, from the positivity of solutions, we obtain some uniform L ∞ bounds. More precisely, we claim that there exists a constant K > 0 depending only on (T,
First, using (3.1) we obtain that
which gives that
which in turn implies that
for C,C > 0 only depend on the initial condition (u 0 , (X 0 v ) v∈V ) and the parameters of the system. We now claim that by induction, we have for all 0 < t ≤ T ,
with a n = 2 + 2a n−1 for n ≥ 2 and a 1 = 1 for someĈ > 0 only depending on (u 0 , (X 0 v ) v∈V ). As 0 < T < 1, we get that
Step #3: existence of a solution. Parabolic Schauder estimates give that the time derivative and the space derivatives up to order 2 of u n are uniformly Hölder continuous in compact sets. As a consequence, we can use the Arzela-Ascoli theorem to show that (u n , (X n v ) v∈V ) converges (up to sequences) toward (u,
As a by product of the proof we get that for the just constructed solution (u, (X v ) v∈V ) we have the uniform bounds:
The fact that I v (t) ≥ 0 implies thanks to the strong maximum principle that in fact
which in turn gives that
Finally, we use the conservation of mass which tells us that e∈E Ωe
such that both I v (t) and R v (t) are uniformly bounded in time, together with their derivatives. This also implies that there exists a constant M > 0, depending only (u 0 ,
Using again parabolic regularity, we obtain the solution (u, (X v ) v∈V ) is global in time and satisfies (2.1)-(2.2)-(2.3) in the classical sense.
Let assume that (u, (X v ) v∈V ) and (ũ,
By linearity, we get that for e ∈ E
On the other, one computes thatX v satisfies for each v ∈ V
We define the energy
and note that E (0) = 0 by definition. Next, differentiating E (t), we obtain
On the one hand, we have
as N v is symmetric positive. On the other hand, we compute
where C > 0 is some large positive constant. Next, we see that
such that we obtain
As a consequence, we get
for someC > 0 and we conclude that E (t) = 0 for all time which then implies thatû = 0 and X v = 0. 
is strictly decreasing, it asymptotically converges towards a limit that we denote
Furthermore, as R v (t) is strictly increasing and uniformly bounded, it asymptotically converges towards a limit that is denoted
But as for each t > 0
this implies that
which in turn proves that
If one recall the notation m(t) for the total population on the edges then we have
The above computations shows that m(t) has a limit as t −→ +∞, that we denote m ∞ and which satisfies
We shall also keep in mind that
And so if we introduce the function Ψ v (x) := x − ηv τv ln x, then the above conservation of mass can be written as
On the other, one can compute that
Now, as m(t) and each I v (t) are convergent we deduce that all
which proves that
And the boundary conditions imply that
Next, we define the sequence of functions u n e (t, x) = u e (t + n, x) for each e ∈ E and X n v (t) = X v (t + n) for each v ∈ V which are uniformly bounded such that one can extract a convergent subsequence. On the one hand we have that lim
with the boundary conditions
This then shows that u ∞ e (t, x) = 0, t > 0 and x ∈
• Ω e for each e ∈ E. As there is unicity of the limit, we deduce that lim
From which we also get that m ∞ = 0 and that
Finally, we use the fact that
to obtain that
As a consequence, the final total populations of infected individuals at each vertices satisfy the following scalar differential equation
Passing to the limit as t → +∞, we get
To summarize, we have proved the following result.
that satisfy the boundary condition (2.3), the long time behavior of the unique corresponding solution (u, (X v ) v∈V ) is given by
where the final total populations of infected individuals
(4.5)
As a consequence, (I ∞ v ) v∈V belongs to the parametrized submanifold given by
and (R ∞ v ) v∈V belongs to the parametrized submanifold given by
The equations (4.6), (4.7), and (4.8) also read v∈V (S ∞ v + R ∞ v ) = M 0 , which is nothing but (4.1) since we have proved that m ∞ = 0.
. Then, equations (4.6), (4.7), and (4.8) are respectively equivalent to
The common right hand side features τ η M 0 that is nothing but the traditional basic reproductive number R 0 .
The aim of this section is to present further qualitative results on the final total configuration (I ∞ v ) v∈V in the fully symmetric case where one can obtain closed form formula and in the case of two vertices where we manage to obtain sharp bounds on the final total populations of infected individuals. In each case, we manage to relate these quantities to standard basic and effective reproductive number for classical SIR model [10] .
Fully symmetric case. We assume that the length e of every edge e ∈ E is equal to a reference length . For every e ∈ E, the diffusion coefficient d e is equal to d. We moreover suppose that
We also assume that τ = τ v > 0 and η = η v > 0 are independent of v ∈ V. In the same spirit, λ v e = λ and α v e = α for every e ∈ E and v ∈ V. We also assume ν v e i ,e j = ν for every edges incident to the vertex v. Finally, the components u 0 e of initial condition on each edges e ∈ E are supposed to be even with respect to the center of the interval Ω e = [0, ]. Thanks to all these assumptions, I ∞ v does not depend on the vertex v ∈ V and we set I ∞ v = I ∞ . Let us recall the notation c V for the cardinal of the set V. The parametrized submanifold given by (4.6) becomes
We can transform this relation as
(4.9)
Let I = −τ I ∞ . We have to solve
The solutions are given in terms of Lambert W function that is the multivalued inverse relation of the function f (w) = we w for w ∈ C [7] . Let us recall how to compute the real solutions of the equation αe x + βx + γ = 0 for (α, β, γ) ∈ R * × R * × R. Let ∆ = α/β exp(−γ/β) be the discriminant. If ∆ ≥ 0 or ∆ = − exp(−1), the solution is unique and
In our symmetric case, the discriminant writes
Since ∆ < 0, there exist solutions to (4.9) if ∆ ≥ − exp(−1), which is equivalent to
(4.10)
We recall that when we consider the standard SIR model (meaning in the context of this paper that we consider an isolated vertex), we can define the effective reproductive number R e and the basic reproductive number R 0 respectively given by R e := S 0 τ η , and R 0 := M 0 τ η , (4.11) see [10, 12, 26] for further properties of effective and basic reproductive numbers. If we denote R 0 = M 0 τ /η, the equation (4.10) reads
This inequality is satisfied as long as S 0 ≤ M 0 , which is always true
However, we can show that I ∞ 0 > 0 and I ∞ −1 < 0. Thus, the only possibility is
We also have access to S ∞ and R ∞ thanks to (4.4). Since exp(−W 0 (x)) = W 0 (x)/x, we obtain
We can summarize these results in the following lemma. . Assume that our model is fully symmetric, then the final total population of infected individuals as given by Theorem 2 is independent on the vertex that is
and I ∞ has the following closed form formula
where R e and R 0 are respectively the effective and basic reproductive number defined in (4.11) and c V the cardinal of V. See Figure 3 for an illustration. 
, in the (I v1 , I v2 ) plane, resp. in the (S v1 , S v2 )-plane, in the fully symmetric case. The asymptotic value I ∞ , resp. S ∞ , lies at the intersection of the diagonal I v1 = I v2 , resp. S v1 = S v2 , and the implicit curve given by (4.6), resp. (4.7).
Case of two vertices. In this simple case, it is possible to build explicit formulas to deal with the implicit submanifold equations (4.6), (4.7), and (4.8).
be respectively the local to vertex v k basic and effective reproductive number. Then,
where the Lambert W function W can be either W 0 or W −1 . Indeed, the argument of W being negative, two solutions have to be considered. We obviously also have
Due to the definition of the domain of the Lambert W function, the argument has to be greater than − exp(−1). So, the following inequality must be satisfied for
Solving the equality part of this inequality, we find that
This equation has to be verified both for W 0 and W −1 . Let Σ v 2 0,−1 be defined by
Then, the domain of
Thus,
We present on Figure 4 (left) the functions W 0 and W −1 defining S ∞ v 2 as a function of S ∞ v 1 and the domain Ω S for a given set of the parameters and initial conditions. We refer to Section 5 for details regarding the numerical integration of the model and Section 6 for further numerical results on the case of two vertices.
Actually, we can reduce the domain of validity of (4.12)-(4.13) for S ∞ v 1 and S ∞ v 2 . Indeed, we know that S v k , k = 1, 2, decay with respect to time, so
The domain ω S is drawn on Figure 4 (right).
Concerning I ∞ v 1 and I ∞ v 2 , we can perform the same analysis. Let
We obtain for k = 1, 2,
still with W equal to W −1 and W 0 . Let ι 1 −1,0 and ι 2 −1,0 be defined by and
with A v 1 and A v 2 given by (4.14) and (4.15). Then,
We can show that min ι v k −1 , ι v k 0 < 0 for k = 1, 2. So, we can reduce this domain since I ∞ v k > 0. So, we define the domain ω I
As a consequence, we have proved the following lemma.
Lemma 4.4. Case of two vertices. Assume that |V| = 2 and |E| = 1. The final total population of infected individuals at each vertex I ∞ v k , k = 1, 2 can be expressed as 
with ι v k −1,0 , k = 1, 2 defined in (4.16)-(4.17). See Figure 5 for an illustration.
Remark 4.5. As the solutions R ∞ v k , k = 1, 2, are simply given by
We represent on Figure 5 the domain ω R .
In this section, we propose a semi-implicit in time numerical scheme based on finite differences in space which has the property to preserve the discrete total mass.
For each e ∈ E, we denote δx e > 0 the space discretization of each edge, and J e ∈ N the number of points of the corresponding discretization. For each i = 1, · · · , J e , the space grid on each edge is given by x i = (i − 1)δx e with e = (J e − 1)δx e . And we let J := e∈E J e ∈ N. Let δt > 0 be the time discretization and denote t m = mδt for m ≥ 0.
For a given function u ∈ C 1,2 (R + × G, R + ), its space-time discretization is given by some sequence of vectors
For each e ∈ E, there exists an integer j e ∈ N such that u e (t m , x i ) ∼ U m je+i , i = 1, · · · , J e , m ≥ 0.
We approximate the laplacian on each edge via finite differences. That is, for each e ∈ E,
where we have only considered the interior points of the discretized domain. Let us now precise how we approximate the laplacian at a given vertex v ∈ V of the graph. So let v ∈ V such that there are δ v edges incident to the vertex. We locally label e ∼ v = (e 1 , · · · , e δv ) all these incident edges. For each v ∈ V, we introduce the map σ v : {e 1 , · · · , e δv } → {1, · · · , J} such that σ v (e k ) corresponds to the global index of the grid discretization associated to the vertex v on edge e k . Finally, we denote by n(σ v (e k )) the global index of the nearest neighbor on edge e k to the vertex v. Note that either n(σ v (e k )) = σ v (e k ) − 1 or n(σ v (e k )) = σ v (e k ) + 1. To approximate the laplacian at a given vertex v ∈ V on edge e k , we use the following formula
The unknown U * ,m σv(e k ) can be expressed by discretization of the boundary condition as follows. For each v ∈ V with e ∼ v = (e 1 , · · · , e δv ), we approximate the normal derivative ∂ n u e k (t m , v) as
2δx e k , k = 1, · · · , δ v , m ≥ 1.
Using (2.3), and denoting I m v the time approximation of I v (t m ), we obtain the following expression for U * ,m
As a consequence, we obtain that for each k = 1, · · · , δ v and m ≥ 1
We introduce the following scheme for each m ≥ 0
One can find similar semi-implicit discretization for the SIR part of the model in [23] .
Well-posedness and positivity. We prove that the numerical scheme defined through (5.1) is well defined and preserves positivity under some condition on δt. Indeed, we first remark that the equation for S m+1 v and I m+1 v in (5.1) can be used to obtain that
can be expressed only in terms of elements of U m+1 as
.
As a consequence, there exists a matrix A ∈ M J (R) such that
Lemma 5.1. There exists a constant C 0 > 0, which only depends on the parameters of the system, such that if 0 < δt < C 0 then we have
Proof. Let U ∈ R J = 0 be such that (I J + A)U = 0. Without loss of generality, assume that U j 0 = max j=1,··· ,J U j > 0. If there exists e ∈ E such that j 0 = j e + i 0 for some i 0 ∈ {2, · · · , J e − 1}, then we have
which is a contradiction by definition of U j 0 . Next if j 0 is such that there is v ∈ V and k ∈ {1, · · · , δ v } such that j 0 = σ v (e k ), then we have
The left-hand side of the above equality is strictly positive and we claim that the right-hand side is negative. We use the fact that (
As a consequence, we deduce that
The last two terms are positive by definition of U j 0 = U σv(e k ) = max j=1,··· ,J U j > 0. Now using Hypothesis 2.2, we have that
ν v e l ,e k > 0, such that the term in bracket is positive provided that
ν v e l ,e k .
As a consequence, we impose that
where it is understood that when the positive part is zero there is no condition on δt. And we have reached a contradiction since
This shows that I J + A is invertible.
Next let U ∈ R J be the unique solution of (I J + A)U = V with V ≥ 0. We denote by U − ∈ R J the vector with components given by
Our aim is to evaluate (I J + A)U, U − J where ·, · J is the following scalar product on R J :
We divide (I J + A)U, U − J into three parts:
The first and second terms are handled as follows
For the third term Q 3 , if we further assume that N v is symmetric, then the matrix K v = A v + N v is symmetric positive definite, and thus for each v ∈ V there exists some β v > 0 such that
And thus, we get an estimate for Q 3 of the form
which is positive provided that δt is small enough. As a consequence, we have proved that
which implies that U − = 0 and thus U ≥ 0.
The previous lemma demonstrates the well-posedness of our numerical scheme (5.1). It also ensures that if we start with positive initial conditions U 0 ≥ 0 and S 0 v > 0, I 0 v ≥ 0 with v∈V I 0 v > 0 and R 0 v ≥ 0, then for all m ≥ 1 we also have that U m ≥ 0, S m v > 0, I m v ≥ 0 and R m v ≥ 0, provided δt > 0 is small enough and N v is symmetric for each v ∈ V.
Preservation of total discrete mass. For any U ∈ R J , we define the following quantity
The expression trap J (U ) is simply the trapezoidal rule applied to the elements of U adapted to our graph G. From (5.1), we get that
Upon denoting Z 1,m v,k the following quantity U m+1 n(σv(e k )) − U m+1 σv(e k ) .
Next, we observe that
where the cancellation comes from the specific structure of the discretized laplacian through finite differences. As a consequence, we have that
We also have that
as the sum over the lines of N v vanishes. And thus we get
On the other hand, from (5.1) we also have
As a conclusion, we have proved the following result.
a solution of (5.1), then we have for each m ≥ 0
This is the discrete conter part of conservation of mass for the continuous model. Now, combining Lemma 5.1-5.2, we have proved the following theorem.
Theorem 3. There exists a constant C 0 > 0, which only depends on the parameters of the system, such that if 0 < δt < C 0 , then the numerical scheme (5.1) defines a unique sequence
If we further assume that N v is symmetric for each v ∈ V, then the numerical scheme (5.1) preserves the positivity of the initial condition. Finally, for each solution of (5.1), the total discrete mass is preserved, namely for each m ≥ 0, we have
In the present section, we illustrate our theoretical results with a collection of numerical simulations for various types of graphs. Throughout this section the time discretization is set to δt = 0.01 while the space discretization to δ xe = 0.01 for each e ∈ E.
6.1 Case of 2 vertices and 1 edge
We first consider the case where c V = 2 and c E = 1, where c E denotes the cardinal of E. In this setting, we recall that our model reads as follows
where v 1 = 0 and v 2 = . This system is complemented by some initial condition (u 0 , S 0 i , I 0 i , R 0 i ) with S i > 0, I 0 1 + I 0 2 > 0, R 0 i = 0 and u 0 ≥ 0 such that the boundary condition is satisfied initially. Finally, we normalize the total mass as follows
For the numerical simulations, we have fixed initial conditions to be of the form
where S 0 and I 0 may vary. In Figures 6-7-8 , S 0 and I 0 are fixed to (S 0 , I 0 ) = (1/2, 10 −6 ), while in Figure 10 , S 0 is allowed to vary and I 0 is fixed to I 0 = 10 −6 . Figure 6 : Profiles of the solutions (S i (t), I i (t)) together with the total population on the edge 0 u(t, x)dx and the total mass of the system M (t) as the parameter λ 1 is varied from 0.05 to 0.95. All other parameters are fixed and set to d = = 1, λ 2 = 1/10, α 1 = α 2 = 1/4, and τ 1 = τ 2 = 1 with η 1 = η 2 = 1/3. For the initial condition we have (S 0 , I 0 ) = (1/2, 10 −6 ).
In Figure 6 , we report the profiles of the solutions (S i (t), I i (t)) together with the total population on the edge 0 u(t, x)dx and the total mass of the system M (t) as the parameter λ 1 is varied from 0.05 to 0.95, while all other parameters are being kept fixed. We observe that the dynamics of the epidemic at the second vertex is almost independent of the parameter λ 1 while it has a significant impact on the dynamics at the first vertex. Indeed, as λ 1 is increased, the maximum of infected individuals max t≥0 I 1 (t) is decreased. In the last panel of the figure, we also illustrate the conservation of total population where the fluctuations around M 0 = 1 is of order 10 −12 .
In the top panel of Figure 7 , we present the final total populations of infected individuals and corresponding final population of susceptible individuals as λ 1 is varied. The blue curve is the location of (I ∞ 1 , I ∞ 2 ) respectively (S ∞ 1 , S ∞ 2 ) while the dark red circles indicate the numerically computed values. We recover the fact that λ 1 has a more significant impact on the final total populations at the first vertex than it has at the second vertex. The get a better understanding of the intricate dynamics between the epidemic at the two vertices, we also present the relative distance ∆T := T 2 − T 1 between time of maximal infection T j in each population as λ 1 is varied. We observe that ∆T is not monotone in λ 1 , as it first decreases and then increases. But we also note that ∆T < 0 for λ 1 ≥ 0.1 traducing the fact that the pick of the epidemic occurs at the second vertex before it does at the first vertex, although initially I at the second vertex decreases as λ 2 increases while at the first vertex it varies less significantly. As α 1 increases, the final total population of infected individuals at the first vertex increases while it decreases at the second vertex. This time the relative distance ∆T := T 2 − T 1 between time of maximal infection is monotonically increasing with α 1 . We get the opposite monotonicity properties as α 2 is varied.
In Figure 8 , we investigate the joint effect of the diffusion coefficient d and the length of the edge on the dynamics of the epidemic at the vertices. Here, we focus on the delay between time of maximal infection T j in each infected population I j (t). As expected, when the diffusion coefficient is really small while the length is being kept at order one, ∆T takes large value: ∆T ∼ 10 4 when d = 10 −3 and = 1. Biologically, this means that when the diffusion coefficient is really small it takes more time for infected individuals from vertex one to reach the second vertex and start an epidemic. We also note that at fixed , ∆T monotonically decreases as d increases, while at fixed d, ∆T monotonically increases as increases.
In Figures 9-10 , we vary respectively the initial population of susceptible individuals S 0 and infected individuals I 0 . We visualize the final total populations of infected individuals and corresponding final population of susceptible individuals on the parameterized surfaces (I ∞ 1 , I ∞ 2 , S 0 ) and (S ∞ 1 , S ∞ 2 , S 0 ), respectively (I ∞ 1 , I ∞ 2 , I 0 ) and (S ∞ 1 , S ∞ 2 , I 0 ), where the level sets of the parameterized surface are given by the conservation of total mass (4.6). We note that (I ∞ 1 , I ∞ 2 ) and (S ∞ 1 , S ∞ 2 ) are almost independent of I 0 when I 0 ≤ 10 −3 with sensible variations only occurring for larger values of I 0 . On the other hand, we observe that as S 0 is increased the final total population of infected individuals increases at the first vertex while it decreases at the second one. The dependence of (S ∞ 1 , S ∞ 2 ) as a function of S 0 is more subtile and is presented in Figure 11 . In the same figure, we also show the location of max I j (t) and its amplitude. We observe a strong nonlinear dependence with respect to S 0 . As S 0 increases, we first see that the time at which I 1 (t) is maximal increases and then decreases, while max I 1 (t) is monotonically increasing. The converse is observed 
for each value of S 0 , while the dark red circles indicate the numerically computed values. Each dark blue curve is a level set of the parameterized surface given by the conservation of total mass (4.6). All other parameters are fixed to d = = 1, λ 1 = λ 2 = 1/10, α 1 = α 2 = 1/4, and τ 1 = τ 2 = 1 with η 1 = η 2 = 1/3. 
for each value of S 0 , while the dark red circles indicate the numerically computed values. Each dark blue curve is a level set of the parameterized surface given by the conservation of total mass (4.6). All other parameters are fixed to d = = 1, λ 1 = λ 2 = 1/10, α 1 = α 2 = 1/4, and τ 1 = τ 2 = 1 with η 1 = η 2 = 1/3. at the second vertex. Figure 10 . The initial condition is of the form (S 0 , 1 − S 0 ) with S 0 ∈ [0, 1] and to each initial configuration is associated a color code from blue to red. The curve in the top right panel is a projection on the (S ∞ 1 , S ∞ 2 )-plane of the parametrized a curve from Figure 10 , right panel.
Next, we consider the case of 3 vertices and 3 edges arranged in a triangular configuration. For the numerical simulations presented in Figure 12 , we have assumed full symmetry in the parameters that is
Regarding the initial condition, we have chosen
for a given (S 0 , I 0 ), while for each e ∈ E we have set u 0 e (x) = 0 on Ω e . Note that, we have initially a boundary layer as our initial condition does not satisfy (2.3) for small times. We remark that the final total populations of infected individuals and corresponding final population of susceptible individuals belong to a surface as provided by (4.6)-(4.7) from Theorem 2.
In Figure 13 , we tested a different configuration. Upon labeling by A the edge between vertices v 1 and v 2 , B the edge between vertices v 2 and v 3 and C the edge between vertices v 1 and v 3 , we have set the parameters to
. The length of each edge is fixed e = = 1 and (τ v , η v ) = (1, 1/7) at each vertex v ∈ V. Finally, we have set different coefficients on each edge, namely d A = 1, d B = 10 −2 and d C = 10 −3 . Initially, we assume that infected individuals are only present at vertex v 1 and each vertex has the same number of susceptible individuals fixed to 1/3. Finally, for each e ∈ E we have set u 0 e (x) = 0 on Ω e . We see in Figure 13 that such a configuration can generate a second wave of infection at the first and second vertices showing that transient dynamics can be complex with multiple bumps of infection. Next, we consider a star-shape graph with 4 vertices and 3 edges where one vertex is connected to the three others. In this configuration, we assume that our parameters may vary with respect to time, modeling locked down strategies for example [11, 16] . More precisely, we will assume that there exists T lock and µ lock such that the transmission rates can be written as
, t > T lock , for each v ∈ V and for a given 0 < τ lock < τ . We will assume that the four vertices are at equal distance such that e = for each e ∈ E and that the coefficient diffusion are equal on each edge, d e = d, e ∈ E. We further assume that at the central vertex v 2 exchanges are no longer allowed. That is, we impose that
while α j e = α for j = 2 and e ∈ E, together with
while λ j e = λ for j = 2 and e ∈ E, and also
Finally, we set η v = η for all v ∈ V. Regarding the initial condition, we work with (S 0 1 , I 0 1 , S 0 2 , I 0 2 , S 0 3 , I 0 3 , S 0 4 , I 0 4 ) = (S 0 − I 0 , I 0 , S 0 , 0, S 0 − , 0, S 0 + , 0) , for given (S 0 , I 0 , ), while for each e ∈ E we have set u 0 e (x) = 0 on Ω e . In Figure 14 , we report the location of the time of maximal infection T max for each vertex together with the corresponding amplitude I j max = max t≥0 I j (t) as a function of τ lock . We observe that below a critical value of τ lock , the time of maximal infection always occurs at t = T max = T lock traducing the fact that the locked down strategy has no effect on the dynamics of the epidemic. At each vertex, we observe the same pattern: as τ lock is decreased the corresponding I j max is decreasing while T max is increasing up to some value of τ lock where we observe a sudden turning point (see the right panel of Figure 14 ). We observe that τ tp lock,v k , the value of the turning point, is well approximated (actually always bounded by below) by the value at which the effective reproduction number of each vertex is equal to 1. We also point out that when τ lock is below the turning point τ tp lock,v k , the corresponding value of I j max is below 10 −3 . On the other hand, in Figure 15 , we present similar results but this time τ lock is fixed and T lock varies. Above some critical value of T lock , I j max saturates to a fixed value independent of T lock traducing the fact that the locked down strategy has no effect on the dynamics of the epidemic if it occurs to late in time. Depending on the initial configuration of susceptible populations at each vertex, we observe intricate nonlinear relationships on the location of the time of maximal infection T max .
In our final example, we have considered a network of N + 1 vertices and N edges arranged in a lattice, in the sense that vertex v j is only connected to vertices v j−1 and v j+1 via two different edges. Figure 16 shows the time evolution of the infected population I v j (t) and susceptible populations S v j (t) at each vertex for several different initial conditions when the length and diffusion coefficient of each edge are equal. In the first case (top panel), we assume that I 0 v 1 > 0 while I 0 v j = 0 for all other vertices, and observe a propagation of burst of activity among infected and susceptible populations. In the second case (middle panel), we assume that I 0 v N/2 > 0 while I 0 v j = 0 for all other vertices, and we see the propagation of two bursts of activity among infected and susceptible populations going leftwards and rightwards. In the last case (bottom panel), we assume that I 0 v 1 = I 0 v N +1 > 0 while I 0 v j = 0 for all other vertices, and we note the propagation of two waves activity which collide at the middle vertex v N/2 . For very small values of the diffusion coefficient d, this burst of epidemic activity seems to travel coherently and forms a coherent traveling wave, as can be seen in Figure 17 where we represent the location of max t>0 I v j (t) at each vertex. Such a traveling wave of epidemic activity share similarities with traveling waves in excitable media such as the propagation of electrical activity along a nerve cell [13, 14] or calcium waves [24] . When d = 10 −3 , they are all aligned on the same line, where for smaller values d ∈ 10 −1 , 10 −2 the location is a nonlinear curve. We also demonstrate that larger diffusion coefficient leads to a faster propagation of epidemic burst across vertices. Finally, we also remark that if I d max,1 denotes the maximum as a function of d at the first vertex, we have I d 1 max,1 ≤ I d 2 max,1 for d 1 ≤ d 2 while for larger vertices j ≥ 6 we have the reverse ordering I d 1 max,j ≥ I d 2 max,j for d 1 ≤ d 2 . For the numerical simulations presented in Figures 16-17 , we have assumed full symmetry in the parameters that is Regarding the initial condition on the edge, we have set u 0 e (x) = 0 on Ω e for each e ∈ E. Figure 17 : Left: Location of the maxima of the infected population I j (t) at each vertex j ∈ {1, · · · , 25} for several different diffusion coefficients d ∈ 10 −3 , 10 −2 , 10 −1 . Right: Zoom of the left figure for small times. We observe that for very small d the location of the maxima is a long a line while it is curved for larger values of d. We also remark that if I d max,1 denotes the maximum as a function of d at the first vertex, we have I d1 max,1 ≤ I d2 max,1 for d 1 ≤ d 2 while for larger vertices j ≥ 6 we have the reverse ordering I d1 max,j ≥ I d2 max,j for d 1 ≤ d 2 . Other parameters were set to = 1, (τ, η) = (1, 1/75), and (α, λ, ν) = (1/8, 1/10, 1/20), while the initial condition is (S 0 , I 0 ) = (1/25, 10 −6 ).