key: cord-0017879-5hppz7vw
authors: Zhou, Jinling; Ma, Xinsheng; Yang, Yu; Zhang, Tonghua
title: A Diffusive Sveir Epidemic Model with Time Delay and General Incidence
date: 2021-06-01
journal: Acta Math Sci
DOI: 10.1007/s10473-021-0421-9
sha: 8d4133a8c66b98dcdbfa1b71da68b000460124a9
doc_id: 17879
cord_uid: 5hppz7vw

In this paper, we consider a delayed diffusive SVEIR model with general incidence. We first establish the threshold dynamics of this model. Using a Nonstandard Finite Difference (NSFD) scheme, we then give the discretization of the continuous model. Applying Lyapunov functions, global stability of the equilibria are established. Numerical simulations are presented to validate the obtained results. The prolonged time delay can lead to the elimination of the infectiousness. ELECTRONIC SUPPLEMENTARY MATERIAL: Supplementary material is available in the online version of this article at 10.1007/s10473-021-0421-9.

Vaccination is an effective way of controlling the transmission of infectious diseases such as tuberculosis and tetanus etc.. Thus, many countries provide routine vaccination against all of these diseases. However, vaccine-induced immunity may wane as time goes on. To better understand this phenomenon, mathematical models have been developed. Kribs-Zaleta and Velasco-Hernández [1] considered an SIS disease model with vaccination. Arino et al. [2] investigated an SIRS model with vaccination. Li et al. [3] indicated that vaccine effectiveness plays a key role in disease prevention and control. To describe vaccination strategy, Liu et al. [4] considered SVIR epidemic models. Let S, V, I and R be the susceptible, vaccinated, infectious and recovered individuals, respectively. Furthermore, Li and Yang [5] proposed the following model for t > 0:

dV dt = µqA + αS − σβV I − (µ + η)V, dI dt = βI(S + σV ) − (γ + µ + δ)I, dR dt = γI − µR.

(1.1)

Here µ and A represent the death rate and the birth rate, respectively. q < 1 denotes the fraction of the vaccinated newborns, p is the unvaccinated newborns, 0 < σ < 1 represents that the vaccine is not completely effective, β is the transmission coefficient of the susceptible, γ is the recovery rate, δ is the per capita disease-induced death rate. The susceptible population is vaccinated at a constant rate α and the vaccine-induced immunity wanes at rate η. Li and Yang discussed the global dynamics of system (1.1) by applying Lyapunov functions.

As seen from the existing models, incidence rates play a very important role in determining model dynamics; for example, the bilinear incidence rate is applicable to Hand-Foot-and-Mouth disease [6] , H5N1 [7] and SARS [8] , but not to sexually transmitted diseases [9] . To model the effect of behavioural changes, Liu et al. [10] proposed an incidence rate βSI p 1+αI q . To model the cholera epidemics in Bari, Capasso and Serio [11] considered the incidence rate p = q = 1. Due to a diseases latency, or factors of immunity, infection processes are not instantaneous. Hence, time delay is important in studying infectious disease dynamics. Hattaf et al. [12] studied a delayed SIR model with general incidence. Wang et al. [13] proposed a delayed SVEIR model with nonlinear incidence. Recently, Hattaf [14] proposed a generalized viral infection model with multi-delays and humoral immunity. For more works on delayed epidemic models with vaccination, we refer readers to [15] [16] [17] [18] [19] .

All of the above mentioned works are location independent, but location-dependent phenomenon are not uncommon in mathematical biology (see [20] [21] [22] ). Webby [23] pointed out that infectious cases can first be found at one location and can then spread to other areas. Therefore, it is interesting to study epidemic models with spatial diffusion. Xu and Ai [24] considered an influenza disease model with spatial diffusion and vaccination. Abdelmalek and Bendoukha [25] proposed a diffusive SVIR epidemic model allowing continuous immigration of all classes of individuals. Xu et al. [26] discussed a vaccination model with spatial diffusion and nonlinear incidence.

Let Ω be a bounded domain in R n with smooth boundary ∂Ω. Let D i (i = 1, 2, 3, 4, 5) be the diffusion rate and ∆ be the Laplace operator. Then, motivated by the aforementioned works, particularly [13, 23] , we study the delayed SVEIR model with spatial diffusion as follows:

Here, τ represents the latent period of the disease. The other parameters are as described for system (1.1) . Denote by n the outward unit normal vector of ∂Ω as in [20, 21] . We further consider model (1.2) with initial condition

where ϕ i (i = 1, 2, 3, 4, 5) are uniformly continuous and bounded. Functions g and f satisfy g(0) = f (0) = 0 and (H1) for I > 0, g(I) > 0 and f (I) > 0; (H2) for I ≥ 0, g ′ (I) > 0 and f ′ (I) > 0, g ′′ (I) ≤ 0 and f ′′ (I) ≤ 0. Epidemiologically, (H1) means that individuals are positive. (H2) implies that the incidences of Sf (I) and V g(I) become faster with an increase in the number of the infectious individuals. However, the per capita infection rate will slow down because of a certain inhibition effect, since g ′′ (I), f ′′ (I) ≤ 0 imply that ( f (I) I ) ′ , ( g(I) I ) ′ < 0. For example, the commonly used nonlinear incidence function x 1+αx (α > 0) satisfies both (H1) and (H2). In this study, in addition to model (1.2), we will also investigate the discrete analogue, due to the fact that epidemiological data is usually collected daily, monthly, or even yearly, but not continuously. Hence, it is more reasonable to use a discrete model to study the transmission mechanism of infectious disease. Furthermore, it is an interesting problem as to whether or not a selected difference scheme can preserve the positivity, boundedness and global stability for the corresponding continuous model. In this regard, some researchers have applied the NSFD scheme proposed by Mickens [27] to discuss the dynamical behaviors of different epidemic models ( [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] ).

The rest of the paper is organized as follows: in Section 2, we establish the global dynamics of the continuous model (1.2). In Section 3, we derive the discretization of (1.2) by the NSFD scheme and establish the positivity and boundedness of the solution. By using discrete Lyapunov functionals, we discuss the global stability of the equilibria of the discretised model in Section 4. This is then followed by numerical simulations in Section 5 to illustrate the obtained results.

From system (1.2), we only discuss the following system:

In this section we assume that D 1 = D 2 = D 4 = D. Let X := C(Ω, R 3 ) be a Banach space with the supremum form || · || X and τ > 0. Let C τ := C([−τ, 0], X) with the form ||φ|| := max θ∈[−τ,0] ||φ(θ)|| X , ∀φ ∈ C τ . Define X + := C(Ω, R 3 + ) and C + τ := C([−τ, 0], X + ). Then (X, X + ) and (C τ , C + τ ) are strongly ordered spaces. For any given functionφ(t) :

whereΓ is the Green function associated with D∆. According to [35, Corollary 4] ,T j (t) are strongly positive and compact (j = 1, 2, 3, ∀t > 0).

Then, system (2.1) can be rewritten as

Theorem 2.1 System (2.1)-(2.2) admits a unique solutionT (·, t,φ) on [0, ∞) satisfying Z(·, 0,φ) =φ for allφ ∈ X + . Given ∀x ∈ Ω and t ≥ 0, the semiflow Θ(t)φ = (S(., t,φ), V (., t,φ), I(., t,φ)) is point dissipative.

Proof Taking ∀φ ∈ X + and k > 0 (sufficiently small), we havẽ

The above inequality implies that

According to [38, Corollary 4] , one can derive that system (2.1) has a unique mild solutioñ Z(·, t,φ) ∈ X + for t ∈ [0, τφ).

. Then, we can get

where d 1 = min{µ + α, µ + β}. Then, S(x, t) and V (x, t) are bounded on [0, τφ) by using a comparison principle. DefineG(

where d 2 = min{µ, µ + β, µ + δ + γ}. Thus,G(x, t) are bounded on [0, τφ), by the comparison principle. This implies that I(x, t) are also bounded on [0, τφ). The remaining proofs are similar to Theorem 2.1 of Zhou et al. [34] , which we omit here.

3)

The endemic equilibrium should satisfy

Theorem 2.2 If R 0 < 1, then system (2.1) has a unique disease-free equilibrium E 0 (S 0 , V 0 , 0); if R 0 > 1, then system (2.1) has a unique endemic equilibrium E * (S * , V * , I * ) with S * = µpA µ+α+f (I * ) and V * = µqA(µ+α+f (I * ))+αµpA (µ+β+g(I * ))(µ+α+f (I * )) , except for E 0 . Proof When R 0 < 1, the result is obvious. According to the first two equations of (2.4), one can get

Then, we have

Obviously, h(+∞) = −∞ and h(0) = 0. It follows from h ′ (0) > 0 that h(I) = 0 has at least one positive solution denoted by I * , where

This is equivalent to R 0 > 1. Thus, (2.4) has at least one positive solution with

.

We now prove that the endemic equilibrium is unique. Note that

By (H2), we know that h ′′ (I) < 0 for I > 0. If there exists more than one positive equilibrium, then there must exist a point E * (S * , V * , I * ) such that h ′′ (I * ) = 0. We obtain a contradiction.

Let 0 = µ 0 < µ i < µ i+1 be the eigenvalues of −∆ on Ω, and E(µ i ) be the space of eigenfunctions with µ i (i = 1, 2 . . .). Then, we define the orthonormal basis of E(µ i )(i = 1, 2 . . .) by {φ ij : j = 1, 2, . . . , dimE(µ i )} as follows:

Here, X ij = {cφ ij : c ∈ R 3 }. In a fashion similar to [20, Theorem 3.1], one gets the following result:

If R 0 < 1, then we have

Thus, (2.6) has no positive real root. Assume that (2.6) has a complex root λ = ω 1 + iω 2 with ω 1 ≥ 0; substituting it into (2.6), one has

Squaring and adding these equations together, we obtain

Using ω 1 ≥ 0 and µ i ≥ 0, we have

when R 0 < 1. This is a contradiction. Therefore, (2.6) has no complex root with a non-negative real part. Considering i = 0 and the space X 0 corresponding to µ 0 = 0, we get

when R 0 > 1. Therefore, there exists a constant λ 0 > 0 such that λ 3 (λ 0 , 0) = 0, yielding that (2.6) has at least one positive root.

It follows from (H2) that g ′ (I) is nonincreasing, so one can obtain g(I) = g(I) − g(0) = g ′ (η)(I − 0) ≤ g ′ (0)I, where η is between 0 and I. Similarly, one has f (I) ≤ f ′ (0)I.

According to ln x ≤ x − 1 and

Thus, we get

Since

Clearly, the largest invariant subset of dL(t) dt = 0 is {E 0 }. The conclusion is correct.

Theorem 2.5 If R 0 > 1, then E * of system (2.1) is globally asymptotically stable.

Proof Define

Clearly, H ≥ 0 with the equality holds if and only if S = S * , V = V * and I = I * .

It follows from (2.4) that

Notice that

and

By Assumption (H2) and ln x ≤ x − 1, we can get ln IG(I * )

In a manner similar to the proof of Theorem 2.4, the conclusion is proved.

Define Ω = [a, b], △x = (b − a)/M and m = [τ /△t]. ∆t is the time stepsize. The mesh points are (x n , t k ), where x n = a+n△x and t k = k∆t with n ∈ {0, 1, . . . , M } and k ∈ N. Denote S(x n , t k ), V (x n , t k ) and I(x n , t k ) by S k n , V k n and I k n , respectively. We use a (M + 1)-dimensional vector

to denote the approximation S, V and I at time t k . () T is the transposition of a vector. According to the NSFD scheme, the discretization of system (2.1) is

with initial condition S k n = φ k n ≥ 0, V k n = ψ k n ≥ 0 and I 0 n = ϕ k n ≥ 0, (3.2) where k ∈ {−m, −m + 1, . . . , 0} and n ∈ {0, 1, . . . , M } and the boundary condition is

The equilibria of system (3.1) is the same as for (2.1). Applying M-matrix theory [39] , we have the following result: 

Proof According to (3.1), we get 

Similarly, we also have CI k+1 = e −µτ ∆tT k+1 + I k .

Here 

with c 1 = 1 + D 4 ∆t/(∆x) 2 + ∆t(µ + δ + γ), c 2 = −D 4 ∆t/(∆x) 2 and c 3 = 1 + 2D 4 ∆t/(∆x) 2 + ∆t(µ + δ + γ). Since C is a M-matrix, one has

Thus, the solution of system (3.1) remains nonnegative. Define

By (3.1), we get

Thus,

It can be concluded by induction that lim sup k→+∞ G k ≤ (M + 1)A. Therefore, for all k ∈ N,

In this section, we discuss the global stability of equilibria for system (3.1).

Theorem 4.1 For any ∆x > 0 and ∆t > 0 , if R 0 ≤ 1, then E 0 of system (3.1) is globally asymptotically stable.

Clearly, L k ≥ 0 with equality holds if and only if S k n = 0, V k n = 0 and I k n = 0 for all k ∈ N and n ∈ {1, 2, . . . , M }.

Applying µpA = (µ + α)S 0 , µqA + αS 0 = (µ + β)V 0 , we can get 

Then,

Thus,

It is clear that L k+1 − L k ≤ 0, when R 0 ≤ 1. Then {L k } is a non-increasing sequence. There must existL > 0 such that lim k→+∞ L k =L, meaning that lim

Furthermore, we can get lim k→+∞ I k n = 0. Theorem 4.2 For any ∆x > 0 and ∆t > 0, if R 0 > 1, then E * of system (3.1) is globally asymptotically stable.

We conclude that H k ≥ 0 if and if only S k n = S * , V k n = V * and I k n = I * for all k ∈ N and n ∈ {0, 1, . . . , M }.

By (2.4), we obtain Hence,

Thus,

.

Applying Assumption (H2) and that ln x ≤ x − 1, we can get where G = {f, g}. Therefore,

Clearly, H k is a non-increasing sequence. There exists H > 0 such that lim 

From (3.1), set f (I) = β1I 1+I , g(I) = β2I 1+I , ∆x = 0.2 and ∆t = 0.1. Referring to [13, 26] , we take δ = 0, p = 1, q = 0, D 1 = D 2 = D 4 = D = 1 and the other parameters as follows: By (2.3) and simple calculations, we have that R 0 = 0.9278 < 1 and that E 0 = (77.4336, 7743.3628, 0). Using Theorem 4.1, E 0 is globally asymptotically stable. One gets that the disease is extinct (see Figure 1 ). is globally asymptotically stable when R0 = 0.9278 < 1 Case 2 Choose α = 0.9, τ = 20 and initial condition S(n, k) = 700 sin n + 700, V (n, k) = 8000(1 + cos n), I(n, k) = sin(0.5n) + 2, n ∈ {0, 1, . . . , 200}, k ∈ {−m, −m + 1, . . . , 0}.

We obtain that R 0 = 2.8999 > 1 and that E * = (744.4733, 7444.0831, 1.8991), respectively. Thus, E * is globally asymptotically stable, by Theorem 4.2. Hence, the disease will eventually become endemic (see Figure 2 ). is globally asymptotically stable when R0 = 2.8999 > 1 Case 3 Effect of time delay. Choose τ = 5, 10, 15, 20 with α = 0.9 and an initial condition as in Case (2) . We obtain that R 0 = 5.2840, 4.3262, 3.5420, 2.8999 and that I * = 4.2821, 3.3247, 2.5408, 1.8991, respectively. Here, we give the simulations of solutions of the infectious I at x = 10 with different values of τ . We observe that the number of those who are infectious decreases with an increase of τ (see Figure 3 ). Biologically, this delay can play an important role in eliminating the number of people who are infectious. By increasing the delay, we can decrease the number of people who are infectious. 

In this paper, we proposed a diffusive SVEIR epidemic model with time delay and general incidence. For this model, we first considered the global dynamics of the continuous case. Then, by using the NSFD scheme, we derived the discretization of the model. It has been shown that the global stability of the equilibria is completely determined by the basic reproduction number R 0 : if R 0 ≤ 1, then the disease-free equilibrium E 0 is globally asymptotically stable; if R 0 > 1, then the endemic equilibrium E * is globally asymptotically stable. One sees that the NSFD scheme can preserve the global properties of solutions for an original continuous model, such as the positivity and ultimate boundedness of solutions, and global stability of the equilibria. It is our intention to use this method to study other delayed diffusive epidemic models.

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