key: cord-0190571-ruk1o8kw
authors: Ottaviano, Stefania; Sensi, Mattia; Sottile, Sara
title: Global stability of multi-group SAIRS epidemic models
date: 2022-02-07
journal: nan
DOI: nan
sha: df866153ad7738df8782cdbb6f0873d961668dba
doc_id: 190571
cord_uid: ruk1o8kw

We study a multi-group SAIRS-type epidemic model with vaccination. The role of asymptomatic and symptomatic infectious individuals is explicitly considered in the transmission pattern of the disease among the groups in which the population is divided. This is a natural extension of the homogeneous mixing SAIRS model with vaccination studied in Ottaviano et. al (2021). We provide a global stability analysis for the model. We determine the value of the basic reproduction number $mathcal{R}_0$ and prove that the disease-free equilibrium is globally asymptotically stable if $mathcal{R}_0<1$. In the case of the SAIRS model without vaccination, we prove the global asymptotic stability of the disease-free equilibrium also when $mathcal{R}_0=1$. Moreover, if $mathcal{R}_0>1$, the disease-free equilibrium is unstable and a unique endemic equilibrium exists. First, we investigate the local asymptotic stability of the endemic equilibrium and subsequently its global stability, for two variations of the original model. Last, we provide numerical simulations to compare the epidemic spreading on different networks topologies.

One of the most common assumptions in classic population models is the homogeneity of interactions between individuals, which then happen completely at random. While such an assumption significantly simplifies the analysis of the models, it can be beneficial to renounce it and to formulate models with more realistic interactions. Heterogeneity in the interactions among the population can depend on many factors. The most common division regards the geographical distinction and the membership to different communities, cities or countries, in which the same infectious disease can have a different behaviour based on the group under study.

The division in groups can also depend on the specific disease under study. For example, individuals can be divided into age groups to study children's diseases, such as measles, mumps or rubella, or can be differentiated by the number of sexual partners for sexually transmitted infections.

Multi-group models can also be useful to study disease transmitted via vectors or multiple hosts, such as Malaria or West-Nile virus.

The concept of equitable partitions has been used to study networks partitioned into local communities with some regularities in their structure, in the case of SIS and SIRS models [1, 27, 26] , by means of the N-Intertwined Mean-Field approximation [40] . In the aforementioned works, the macroscopic structure of hierarchical networks is described by a quotient graph and the stability of the endemic equilibrium can be investigated by a lower-dimensional system with respect to the starting one.

Several authors proposed multi-group models to describe the transmission behaviour between different communities or cities, see for example [14, 10, 42] . In this paper, we assume that each individual interacts within a network of relationships, due e.g. to different social or spatial patterns; individuals are hence divided into groups, which are not isolated from one another.

As in the homogeneous mixing case, the stability analysis of the equilibrium points of the system under investigation allows to understand its long-term behaviour and, hence, to obtain some insight into how the prevalence of an endemic disease depends on the parameters of the model [37] and, in this case, on the network topology. However, the problem of existence and global stability, especially for the endemic equilibrium, is often mathematical challenging; unfortunately, for many complex multi-group models it remains an open question, or requires cumbersome conditions [20] . In this framework, Guo et al. [7, 6] and Li and Shuai [17] developed a graph-theoretic method to find Lyapunov functions for some multi-group epidemic models which has recently allowed to obtain various results on the global dynamics of SIRS-type models [21, 22] and SEIRS-type models [4] .

In this paper, we present a multi-group model, as extension of the SAIRS-type model proposed in [28] , where the role of asymptomatic and symptomatic infectious individuals in the disease transmission has been explicitly considered. Asymptomatic cases often remain unidentified and possibly have more contacts than symptomatic individuals, allowing the virus to circulate widely in the population [3, 24, 25, 11] . The so-called "silent spreaders" are playing a significant role even in the current Covid-19 pandemic and numerous recent papers have considered their contribution in the virus transmission (see, e.g., [2, 30, 29, 18, 35] ). However, this contribution has proved relevant also for other communicable diseases, such as influenza, cholera, and shigella [12, 23, 36, 31] .

Although models incorporating asymptomatic individuals already exist in the literature, they have not been analytically investigated as thoroughly as more famous compartmental models. Since these types of models have been receiving much more attention lately, we believe they deserve a deeper understanding from a theoretical point of view. Thus, we aim to partially fill this gap and provide a stability analysis of the multi-group system under investigation.

In our model, we denote with S i , A i , I i and R i , i = 1, . . . , n, the fraction of Susceptible, Asymptomatic infected, symptomatic Infected and Recovered individuals, respectively, in the i−th group, such that S i + A i + I i + R i = 1. We remark that, from here on, we will use the terms community and group interchangeably.

The disease can be transmitted by individuals in the classes A i and I i , within their group, to the susceptible S i , with transmission rate β A ii and β I ii , respectively, but also between different groups: e.g., individuals A j and I j , belonging to the j-th community, may infect susceptible individuals S i of group i with transmission rate β A ij and β I ij , respectively. From the asymptomatic compartment, an individual can either progress to the class of symptomatic infectious or recover without ever developing symptoms. We assume that the average time of the symptoms developing, denoted by 1/α, and the recovery rates from both the infectious compartments, δ A and δ I , do not depend on the community of origin, i.e. these parameters depend only on the disease. Furthermore, the average time to return to the susceptible state, 1/γ, only depends on the specific disease under study, and not on the community to which an individual belongs. The remaining parameters of the model depend on the community's membership. First, the proportion of susceptible individuals who receive the vaccine might be different for each group; we denote with ν i , i = 1, . . . , n, the proportion of susceptible in the i−th group who receive a vaccine-induced temporary immunity. Moreover, µ i , i = 1, . . . , n represent both the birth rates and the natural death rates in community i. Finally, individuals of different communities may have contacts each other, by direct transport, but they never permanently move to another community. Therefore, the total population in each group may only change through births and natural deaths; we do not distinguish between natural deaths and disease-related deaths.

The paper is organised as follows. In Sec. 2, we present the system of equations for the multi-group SAIRS model with vaccination, providing its positive invariant set. In Sec. 3, we determine the basic reproduction number R 0 and prove that the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) if R 0 < 1 and unstable if R 0 > 1. Moreover, we prove the GAS of the DFE also in the case R 0 = 1, for the model in which no vaccination is administered to the susceptible individuals. In Sec. 4, we prove the existence and uniqueness of an endemic equilibrium (EE) by a fixed point argument, as in [37] , since there is no explicit expression for R 0 . In Sec. 4.1, we provide sufficient conditions for the local asymptotic stability of the EE. In Sec. 5, we discuss the uniform persistence of the disease and we investigate the global asymptotic stability of the EE for two variations of the original model under study. Precisely, in Thm. 14, we study the global stability of the SAIR model (i.e. γ = 0) and we prove that the EE is GAS if R 0 > 1. In Sec. 5.2, we establish sufficient conditions for the GAS of the EE for the SAIRS model (i.e., γ = 0) with vaccination, under the restriction that asymptomatic and symptomatic individuals have the same average recovery period, i.e. δ A = δ I . The problem of the global stability of the endemic equilibrium in the most general case, i.e. δ A = δ I , remains open. In Sec. 6, we provide some numerical simulations in which we simulate the evolution of the epidemics in four different structures of community networks.

The system of ODEs which describes the evolution of the disease in the i-th community is the following:

with initial condition (S 1 (0), A 1 (0), I 1 (0), R 1 (0), . . . , S n (0), A n (0), I n (0), R n (0)) belonging to the set Γ = {(S 1 , A 1 , I 1 , R 1 , . . . , S n , A n , I n , R n ) ∈ R 4n

where R 4n + indicates the non-negative orthant of R 4n . The flow diagram representing the interaction among two groups of system (1), as well as their internal dynamics, is given in Figure 1 . Assuming initial conditions inΓ, S i (t) + A i (t) + I i (t) + R i (t) = 1, for all t ≥ 0 and i = 1, . . . , n; hence, system (1) is equivalent to the following 3n-dimensional dynamical system:

with initial condition (S 1 (0), A 1 (0), I 1 (0), . . . , S n (0), A n (0), I n (0)) belonging to the set

. . , n}. System (3) can be written in vector notation as

where x(t) = (S 1 (t), A 1 (t), I 1 (t), . . . , S n (t), A n (t), I n (t)) and f (x(t)) = (f 1 (x(t)), f 2 (x(t)), . . . , f 3n (x(t))) is defined according to (3) .

We make the following assumptions:

Assumption 1.

• The matrices [β A ij ] i,j=1,...,n and [β I ij ] i,j=1,...,n are irreducible. This means that every pair of communities is connected by a path.

• β A ii = 0, β I ii = 0, i = 1, . . . , n. This means that infection can spread within each community.

Theorem 1. Γ is positively invariant for system (3) . That is, for all initial values x(0) ∈ Γ, the solution x(t) of (3) will remain in Γ for all t > 0.

Proof. Let us consider the boundary ∂Γ, as in [28, Th. 1] . It consists of the following hyperplanes: Let x ∈ H k,1 , k = 1, . . . , 4, and consider the following cases:

Case 3:

Case 4:

The proof for the hyperplanes H k,i , k = 1, . . . , 4 and i = 2, . . . , n is analogous.

System (3) always admits a disease-free equilibrium, whose expression is:

x 0 = (S 0,1 , A 0,1 , I 0,1 , . . . , S 0,n , A 0,n , I 0,n ) ,

Note that, in general,

Lemma 2. Consider the matrix

The basic reproduction number R 0 of (3) is

where ρ(M 1 ) is the spectral radius of the matrix M 1 .

Proof. We shall use the next generation matrix method [38] to find R 0 . System (3) has 2n disease compartments, namely A i and I i , i = 1, . . . , n. Rearranging the order of the equations such that the disease compartments can be written as x = (A 1 , . . . , A n , I 1 , . . . , I n ) T , we can rewrite the corresponding ODEs as

Thus, we obtain

which can be written in matrix notation

. . , µ n ), and 0 and I are the zero matrix and the identity matrix of order n, respectively. Since V is a block lower triangular matrix, its inverse is the 2n × 2n block matrix:

The next generation matrix is defined as M := F V −1 . By direct calculation, we obtain

The basic reproduction number R 0 is defined as the spectral radius of M , denoted by

As a direct consequence of the Perron Frobenius Theorem, ρ(M 1 ) > 0. This proves our claim.

In the following, we present some results to prove the global asymptotic stability of the DFE x 0 . Recall that a matrix M is called non-negative if each entry is non-negative; we simply write M ≥ 0 to indicate this. We use the following results from [39] : Lemma 2] ). If F is non-negative and V is a non-singular M-matrix, then R 0 = ρ(F V −1 ) < 1 if and only if all eigenvalues of (F − V ) have negative real parts.

Note that the matrices F and V defined in Lemma 2 satisfy the hypotheses of Lemma 3, thus the following result holds:

. . , I n (t)) be the solutions of system (3) with initial condition x(0) ∈ Γ, in which we have rearranged the order of the equations. In view of Theorem 4, it is sufficient to prove that for all i = 1, . . . , n

whit S 0,i as in (5) . From the first n equations of (3), it follows that

Thus, S 0,i is a global asymptotically stable equilibrium for the comparison equation

Then, for any ε > 0, there existst i > 0, such that for all t ≥t i , it holds

Lett = max{t 1 , . . . , t n }, then for all t ≥t, from (11) and the remaining 2n equations of (3) it follows that

Let us now consider the comparison system

Let w = (v 1 , . . . , v n , u 1 , . . . , u n ) T , then one can rewrite this system as

where F ε and V ε are the matrices defined in (7) and (8), respectively, evaluated in x 0 (ε) whose components are S 0,i + ε for i = 1, . . . , n and 0 in the remaining 2n components. Notice that we can choose ε > 0 sufficient small such that ρ(F ε V −1 ε ) < 1 and then, from Lemma 3, all the eigenvalues of matrix (F ε − V ε ) have negative real parts. It follows that lim t→∞ w i (t) = 0 from any initial conditions in Γ, from which lim

Thus, for any ε > 0, there existst 1 > 0 such that, for all t ≥t 1 , we have

From that and the first n equations of system (3), we get that for all i = 1, . . . , n and for t ≥t 1

The comparison system

has a globally asymptotically stable equilibrium

.

Thus, we get that for any ζ > 0, there existst 2 > 0 such that for all t ≥t 2 ,

This implies that for all ε > 0 lim inf

Letting ε go to 0, we have lim inf t→∞ S i (t) ≥ S 0,i for all i = 1, . . . , n, which combined with (12) gives us

Let us consider the SAIRS model without vaccination, that is (3) with ν i = 0, i = 1, . . . , n. From (6), the expression of the basic-reproduction number is

and the components of the DFE (5) become S 0,i = 1, A 0,i = I 0,i = 0, for all i = 1, . . . , n.

In Theorem 4 and 5 we proved that the DFE is globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1. In the following theorem, which describe the case when we do not have any vaccination, we are able to prove that the DFE is globally asymptotically stable also when R 0 = 1.

Proof. To prove the statement, we use the method presented in [34] .

Rearranging the order of the equations such that the disease compartments can be written as x = (A 1 , . . . , A n , I 1 , . . . , I n ), system (3), restricted to these compartments, can be rewritten as:

Let ω T be the left eigenvector of M corresponding to the eigenvalue R 0 . Note that in our case the irreducibility assumption for M in [34, Thm 2.2] fails. However, we can show that ω > 0. Indeed, let ω T = (ω 1 , ω 2 ), where ω 1 and ω 2 are both vectors with n components. It is easy to see that ω 1 is the lefteigenvector of the non-negative matrix M 1 (10) corresponding to its spectral radius ρ(M 1 ) = R 0 . Since M 1 is irreducible and non-negative, it follows by the Perron-Frobenius theorem that ω 1 > 0. Moreover, from (9), let

then, we have ω 1 M 2 = R 0 ω 2 ; since ω 1 M 2 > 0 it follows that ω 2 > 0. Hence, ω > 0. Now, consider the following Lyapunov function

By differentiating Q along the solution of (3), we obtain

Since 

To prove the existence and uniqueness of an endemic equilibrium point, we recall the following definition and theorem from [8] .

Definition 7. A function F (x) : R n + → R n + is called strictly sublinear if for fixed x ∈ (0, ∞) n and fixed r ∈ (0, 1) n , there exists an ε > 0 such that F (rx) ≥ (1 + ε)rF (x), where ≥ denotes the pointwise ordering in R n . By using the above result, we can prove the following theorem. Proof. An equilibrium point is a solution of the non linear equations obtained by setting the right-hand side of equations (3) equal to zero. Then, the following must hold:

for i = 1, 2, . . . , n. By excluding as solution the DFE (5), we assume A * i > 0, for some 1 ≤ i ≤ n. From (16), we immediately obtain

for all i = 1, 2, . . . , n. Substituting (17) in (15), we obtain

By our assumption on x * , the denominator of (18) is strictly positive. Lastly, substituting (17) and (18) into (14), we obtain

which can be rearranged to give

We can collect (µ i + ν i + γ)(α + δ A + µ i ) and (µ i + γ) in both the numerator and denominator, to obtain

with M 1 as in (6). Define a function H = (h 1 , . . . , h n ) : R n + → R n + , in the following way:

Then, since ∂h j ∂y i > 0, for all i, j = 1, 2, . . . , n, H is monotonically increasing in all its components. Moreover, J H (0) = M 1 that is a non-negative and irreducible matrix and the function H(x) is bounded and strictly sublinear with

Thus, by Theorem 8, we have that system (3) has an unique endemic equilibrium inΓ. Remark 1. From Eq. (17) we can note that since I * i < 1, we have that A * i < δI +µi α .

In this section, we prove local asymptotic stability of the endemic equilbrium, under additional assumptions.

Theorem 10. Assume that fixed j, β I ij = h j β A ij for all i = 1, . . . , n,. Moreover, let us assume that

for i = 1, . . . , n. Then, the endemic equilibrium x * := (S * 1 , A * 1 , I * 1 , . . . , S * n , A * n , I * n ) is local asymptotically stable.

Proof. Usually, the asymptotic local stability of the endemic equilibrium point is studied by linearizing system (4) around that point. However, it is know that the endemic equilibrium is asymptotically stable if the linearized system dy dt = J f (x * )y has no solution of the form [8, 37] . To prove our statement with this strategy, we consider the following system, equivalent to (4):

where x(t) = (A 1 (t), I 1 (t), R 1 (t), . . . , A n (t), I n (t), R n (t)) and f (x(t)) = (f 1 (x(t)), f 2 (x(t)), . . . , f 3n (x(t))), with

Now, to prove the asymptotic local stability of x * , we consider the following equations:

We proceed by assuming that Re z ≥ 0 and showing that this assumption leads to at a contradiction.

From the second and third equation of (20), we have respectively

and

Now, considering the assumption that fixed j, β I ij = h j β A ij for all i = 1, . . . , n, and replacing (21) and (22) in the first equation of (20), we obtain

and consider the following transformation:

Then, we get

Now, let us note that if Re z ≥ 0, then

Hence, we can rewrite (24) in the following form:

where C = (c ij ) with

From (25), we have that Reη i (z) ≥ 0. Moreover, the following claim, whose proof is given in Appendix A, holds:

Claim 11. If Re z ≥ 0, then Re η i (z) > 0.

Now, let us note that C is a non-negative matrix and that A * = CA * , where A * = (A * 1 , . . . , A * n ). Let η(z) = inf{Re η i (z), i = 1, . . . , n}, and |Ũ | = (|Ũ 1 |, . . . , |Ũ n |),

and taking the absolute values in (26), we get

It is easy to verify that if Re z ≥ 0, then Re η i (z) > 0 for all i, hence η(z) > 0. Now, we define ǫ to be the minimum value for which |Ũ | ≤ ǫA * . Since the components of A * belong to (0, 1), ǫ < ∞. Hence, by (27), (1 + η(z))|Ũ | ≤ C|Ũ | ≤ ǫCA * = ǫA * . This inequality contradicts the minimality of ǫ because η(z) > 0 if Re z ≥ 0, thus we can conclude that Re z < 0 and the equilibrium is stable.

As in [28] , we conjecture that some, if not all, these technical assumptions could be relaxed, as our numerical simulations suggest. However, the techniques we use in this paper require such assumptions on the parameters in order to reach a result, and multigroup models often require cumbersome hypotheses [9, 16, 32] .

In this section, we first discuss the persistence of the disease, then we investigate the global stability property of the endemic equilibrium for some variations of the original model (1).

Theorem 13. If R 0 > 1, system (3) is uniformly persistent.

Proof. From Theorem 9 we know that DFE x 0 is the unique equilibrium of (3) on ∂Γ, i.e., the largest invariant set on ∂Γ is the singleton {x 0 }, which is isolated. If R 0 > 1, we know from Theorem 4 that x 0 is unstable. Then, by using [5, Thm 4.3] , and similar arguments in [15, Prop. 3.3] , we can assert that the instability of x 0 implies the uniform persistence of (3).

In this section, we study the global asymptotic stability of the endemic equilibrium of the SAIR model, which describes the dynamic of a disease which confers permanent immunity (i.e. γ = 0). The dynamic of an SAIR model of this type is described by the following system of equations:

The basic reproduction number is derived by substituting γ with 0 in (6):

If R 0 > 1, system (29) has a unique equilibrium inΓ, which satisfies

Theorem 14. The endemic equilibrium x * is globally asymptotically stable inΓ if R 0 > 1.

Proof. In order to prove the statement, we use a graph-theoretic approach as in [34] to establish the existence of a Lyapunov function. Let us definẽ

Substituting (30), (31) , and (32) in (29), we obtain

For i = 1, . . . , n, differentiating V i along the solutions of (29) and using (33), we have

Thus, from (34) and (35), we obtain

Using the fact that 1 − x ≤ − ln(x), we can write

Thus, we obtain Figure 2 : The weighted digraph (G,B) constructed for system (29) .

Moreover, for all i = 1, . . . , n

and again, using the fact that 1 − x ≤ − ln(x), we have

We can construct a weighted digraph G, associated with the weight matrixB = (β ij ) i,j=1,...,2n , with β ij > 0 as defined above and zero otherwise; see Figure 2 . Let us note that, from Assumption 1, the digraph (G,B) is strongly connected. Since G i,n+j + G n+j,j = −ã i + ln(ã i ) +ã j − ln(ã j ) = G i,j , i, j = 1, . . . , n, it can be verified that each directed cycle C of (G,B) has (s,r)∈E(C) G rs = 0, where E(C) denotes the arc set of the directed cycle C. Thus, the assumptions of [34, Theorem 3.5] hold, hence the function

for constants c i > 0 defined as in [34, Prop. 3.1] , satisfies dV dt ≤ 0, meaning that V is a Lyapunov function for system (29) . It can be verified that the largest compact invariant set in which dV dt = 0 is the singleton {x * }. Hence, our claim follows by LaSalle's Invariance Principle [13] .

Remark 2. We observe that the proof of Theorem 14 also holds for the case δ A = 0 in system (3) . That is to say, for a model with two stages of infection I 1 and I 2 , in which from the first class of infection one passes to the second at the rate α and one cannot directly pass into the compartment of recovered individuals. Then, from the second stage of infection, one can recover at the rate δ I2 . It is known that, if α = δ I2 , the length of the infectious period follows a gamma distribution; otherwise, the resulting distribution is not a standard one. Moreover, we remark that Theorem 14 only requires R 0 > 1, and no additional conditions on the parameters, despite the complexity of the model under study. Models with multiple infected compartments have been studied, e.g., in [19, 33, 41] .

In the δ A = δ I =: δ case, from (6) we have

If R 0 > 1, system (1) with δ A = δ I =: δ has a unique equilibrium inΓ, which satisfies

Theorem 15. Assume that (µ i + ν i )S * i ≥ γR * i and δ > ν i , for each i = 1, . . . , n. Then, the endemic equilibrium x * is globally asymptotically stable inΓ if R 0 > 1.

Proof. Lets i ,ã i ,ĩ i , V i , and V n+i as in Theorem 14. Let us definer i = Ri

, i = 1, . . . , n.

By using equations (39) , and differentiating along the solution of (1) with δ A = δ I =: δ, we obtain

and the derivatives dVi,2 dt and dVn+i dt as in (35) and (37), respectively. Moreover,

by assumption δ > ν i , thus

Let us consider the weighted digraph G, the weight matrixB, and the functions G i,j , for i, j = 1, . . . , 2n defined as in Theorem 14. Consider the following function:

where the constant c i > 0 are defined as in [34, Prop. 3.1] . Then, by following similar steps as in Theorem 14 and from (42), we obtain

Now, since it can be verified that over each directed cycle C of (G,B), (s,r)∈E(C) G rs = 0, by following the same arguments in the proof of [34, Thm 3.5] , we have that 2n i=1 2n j=1 c iβij G i,j = 0. Moreover, by assumption (µ i + ν i )S * i ≥ γR * i , for each i = 1, . . . , n, hence

Thus, we have dV dt ≤ 0. Since the largest compact invariant set in which dV dt = 0 is the singleton {x * }, by LaSalle invariance principle our claim follows.

Remark 3. Note that if ν i = 0 for all i, we obtain the same sufficient conditions for the GAS of the EE found for the SIRS model in [21] .

In this section, we explore the role of the network structures in the evolution of the epidemics. The primary criterion for parameter selection is the clarity of the resulting plot. Hence, the simulations were carried out with a set of parameters considered in [28] . These parameters, summarized in Table 1 , ensure that R 0 > 1 in all the networks we consider, whose shapes are represented in Figure 3 . In particular, we remark on how sensitive R 0 is on the topology of the network, which is reflected in its adjacency matrix. Indeed, let us consider (6) and let

Let us defineĀ = A + I n , where A is the adjacency matrix and n the number of nodes of the network we are considering, respectively. Then, as a consequence of the Perron-Frobenius theorem, the following lower and upper bounds for R 0 hold:

In the case of the cycle-tree network in Figure 3 (a), we have ρ(A) = 3.2877, for the star network in Figure  3 (b), ρ(A) = 3.8284, in the case of the ring network in Figure 3 (c), ρ(A) = 3, and for the line network in Figure 3 (d) we have ρ(A) = 2.9021. Consequently, in the star network, we found the largest R 0 = 4.91, for the cycle-tree network we have R 0 = 4.37. In the other two networks, i.e. the ring and the line, we find R 0 = 4.07 and R 0 = 3.97, respectively; we can see that the presence of one additional link in the ring increases the spectral radius of the transmission matrices and thus facilitates the spread of the disease. We provide numerical simulations of the evolution of an epidemics for the different 9-communities networks considered, see Figures 4, 5, 6 and 7. In each simulations, the epidemics starts in community 1, with a small asymptomatic fraction of the population, and no symptomatic individuals. We obtain a delay in the start of the epidemics, directly proportional to the path distance of any community from community 1: this is particularly visible in Figure 7 . We observe a delay in the time of the peak, as well, although this is often less pronounced; this is clear in in Figure 6 , in which communities with the same distance (path length) from Community 1 reach the peak at the same time. We can see that in the star network the peak of the non-central communities happens exactly at the same time and has the same magnitude, as one would expect, see Figure 5 .

For ease of interpretation, we plot the total number of Asymptomatic infected individuals and symptomatic Infected individuals in all four cases, see Figure 8 . The qualitative behaviour of all simulations is the same: after a first spike, the dynamics converges towards the endemic equilibrium, through quickly damping oscillations. In all our simulations, the endemic equilibrium values of I are greater than the ones of A, as we expected from (17) and our choice of the parameters involved in the formula.

Notice the significantly lower peaks in Figure 8d , when compared to 8c, even though the corresponding networks only differ for one edge, connecting Community 9 to Community 1, in which the epidemics start. Table 1 for the values of the parameters. Figure 5 : Evolution of the epidemic in each community of the star network, see Figure 3b . The title of each subplot indicates the community it represents, as well as the peak time of infected individuals. In this setting, from (6) we obtain R 0 = 4.91. Refer to Table 1 for the values of the parameters. Figure 6 : Evolution of the epidemic in each community of the ring network, see Figure 3c . The title of each subplot indicates the community it represents, as well as the peak time of infected individuals. In this setting, from (6) we obtain R 0 = 4.07. Refer to Table 1 for the values of the parameters. Figure 7 : Evolution of the epidemic in each community of the line network, see Figure 3d . The title of each subplot indicates the community it represents, as well as the peak time of infected individuals. In this setting, from (6) we obtain R 0 = 3.97. Refer to Table 1 for the values of the parameters. (c) (d) Figure 8 : Total amount of Asymptomatic infected ( A(t)) and symptomatic Infected ( I(t)) in the four networks we simulate. Respectively: (a) cycle-tree network, see Figure 3a ; (b) star network, see Figure 3b ; (c) ring network, see Figure 3c ; and (d) line network, see Figure 3d . The qualitative behaviour is the same, i.e. convergence towards the endemic equilibrium through damped oscillations. Refer to Table 1 for the values of the parameters.

We analysed a multi-group SAIRS-type epidemic model with vaccination. In this model, susceptible individuals can be infected by both asymptomatic and symptomatic infectious individuals, belonging to their communities as well as to other adjacent communities. We derived the expression of the basic reproduction number R 0 , which depends on the matrices which encode the transmission rates between and within communities. We showed that if R 0 < 1, the disease-free equilibrium is globally asymptotically stable, i.e. the disease will be eliminated in the long-run, whereas if R 0 > 1 it is unstable. Moreover, in the SAIRS model without vaccination (ν i = 0, for all i = 1, . . . , n), we were able to generalize the result on the global asymptotic stability of the disease-free equilibrium also in the case R 0 = 1. Weproved the existence of a unique endemic equilibrium if R 0 > 1. We gave sufficient conditions for the local asymptotic stability of the endemic equilibrium; then, we investigated the global asymptotic stability of the endemic equilibrium in two cases. The first one regards the SAIR model (i.e. γ = 0), and does not requires any further conditions on the parameters besides R 0 > 1.

The second is the case of the SAIRS model, with the restriction that asymptomatic and symptomatic individuals have the same mean recovery period, i.e. δ A = δ I . In this case, we provided sufficient conditions for the GAS of the endemic equilibrium.

We leave as open problem the study of the global asymptotic stability of the endemic equilibrium for the SAIRS model with vaccination, in the case β A = β I and δ A = δ I . Lastly, we conjecture that the conditions we derived to prove the asymptotic behaviour of the model are sufficient but not necessary conditions, as our numerical exploration of various settings seems to indicate.

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The last inequality holds since by hypothesis (δ I − ν i )α ≤ 2(µ i + 2ν i + γ + δ I ) (µ i + ν i + γ)(δ I + ν i ) + (µ i + 2ν i + γ + δ I )

The authors would like to thank Prof. Andrea Pugliese for the fruitful discussions, suggestions and careful reading of the paper draft.The research of Stefania Ottaviano was supported by the University of Trento in the frame "SBI-COVID -Squashing the business interruption curve while flattening pandemic curve (grant 40900013)".

Proof of Claim 11. We recall thatIt is easy to see that if Re(z) ≥ 0, then Re(K 1 i (z)) > 0. Now, we show that if Re(z) ≥ 0, thenFor ease of notation, we define:Now, let us show thatWe have thatwhere we have introduced the notation P = (a + h 1 )(a + h 2 ) and S = (2a + h 1 + h 2 ).Since we assume δ I ≥ ν i , we can see that the minimum of g(b) is equal to−(δ I − ν i )α 2(µ i + 2ν i + γ + δ I ) ((µ i + ν i + γ)(δ I + ν i )) + (µ i + 2ν i + γ + δ I ) 2 ≥ −1.