THE SEIR MODEL WITH PULSE AND DIFFUSION OF VIRUS IN THE ENVIRONMENT

This paper addresses a reaction-diﬀusion problem featuring impulsive eﬀects under Neumann boundary conditions. The model simulates the periodic eradication of viruses in an environment. Initially, we establish the well-posedness of the reaction-diﬀusion model. We deﬁne the basic reproduction number R 0 for the problem in the absence of pulsing and compute the principal eigenvalue of the corresponding elliptic eigenvalue problem. Utilizing Lyapunov functionals and Green’s ﬁrst identity, we derive the global threshold dynamics of the system. Speciﬁcally, when R 0 < 1, the disease-free equilibrium is globally asymptotically stable; conversely, if R 0 > 1, the system exhibits uniform persistence, and the endemic equilibrium is globally asymptotically stable. Additionally, we consider the generalized principal eigenvalues for the problem with pulsing and provide suﬃcient conditions for the stability of both the disease-free equilibrium and the positive periodic solution. Finally, we corroborate our theoretical ﬁndings through numerical simulations, particularly discussing the impacts of periodic environmental cleaning.


Introduction
It is widely recognized that certain viruses, including the human immunodeficiency virus (HIV) and hepatitis B and C viruses (HBV, HCV), possess the ability to survive and propagate for extended periods in the absence of hosts [4,5,8,9,29].This capability amplifies their potential for causing further infections, complicating efforts to prevent and control viral spread, and posing significant risks to public health.Given these challenges, there is considerable importance in investigating the influence of environmental transmission of free-living viruses on the effectiveness of viral control measures.
It has been posited that the likelihood of viral infection is often related to spatial location [10,28].The diffusive process essentially represents the random spatial movement of the virus, which exhibits no preferential direction as indicated in [31].As a result, reaction-diffusion models serve as reliable mathematical frameworks for exploring the influence of diffusion and spatial heterogeneity on disease transmission.For further details and discussions, please refer to [17,18,24,26,[32][33][34] and the references cited therein.
Numerous theoretical studies have been conducted on epidemic models incorporating diffusion.For example, Allen et al. introduced a classical SIS reactiondiffusion system with spatial heterogeneity in 2008 [2].This work was later extended in 2014, where a HIV viral infection model that accounted for both virus diffusion and spatial heterogeneity was considered [23].More recently, Pang and Xiao examined a SIS-W model aimed at controlling hospital infections and explored the effects of direct transmission via free-living viruses (W ) on disease progression [15].
Considering individuals in the "exposed"(E) category-who do not exhibit symptoms immediately after coming into contact with the virus-research has been conducted on SEIR models both without diffusion [12,35] and with diffusion [1,7].Moreover, transient perturbations in virus levels can be triggered by various factors, such as climatic changes or human interventions like periodic environmental disinfection, which result in a rapid decline in environmental viral concentrations over short periods.The primary objective of this article is to investigate and elucidate the dynamics of virus transmission at the population level, specifically in the context of hospital-acquired infections.In this study, we assume that only the virus undergoes diffusion, while the populations themselves do not.This leads to a set of hybrid differential equations: the first three equations are ordinary differential equations representing the dynamics of the susceptible, exposed, and infected populations, while the final equation is a partial differential equation modeling virus diffusion.Consequently, we propose the following SEIW epidemic model, which incorporates pulse effects as well as diffusion of the virus within the environment.x ∈ Ω, n = 0, 1, 2, ..., V (x, (nT ) + ) = V (x, nT ), x ∈ Ω, V = S, E, I, ∂W ∂η = 0, x ∈ ∂Ω, t > 0, (S(x, 0), E(x, 0), I(x, 0), W (x, 0)) where t ∈ ((nT ) + , (n + 1)T ] means the equations hold for t ∈ (nT, (n + 1)T ], while the initial value of the unknown takes its right-hand limit at t = nT .S(x, t), E(x, t), I(x, t) and W (x, t) represent the density of susceptible patients, exposed patients, infected patients, free viruses in the environment, respectively.α(x) is the locationdependent growth rate of the susceptible patients, µ 1 (x) means the transmission rate between the susceptible and the environmental virus.µ 2 (x) expresses the transmission rate between the susceptible and the infectious individuals, γ 1 (x) is the death rate of the susceptible patients.δ represents the proportion of individuals who progress from susceptible to exposed, and a(x) is the rate of exposed turn into infectious individuals.γ 2 (x) and γ 3 (x) are the death rate of the exposed patients and the infected patients, respectively.d expresses the diffusion coefficient of the virus, b(x) is the rate of virus production and 1/γ 4 (x) is the average survival time of free viruses without hosts.Owing to regular cleaning at each time t = nT , some of free viruses remain in the environment, we assume that its ratio is c(0 < c ≤ 1), while all patients do not change significantly in a short time.Apparently, c = 1 means no cleaning and c = 0 implies that there is no virus existing in the environment after thorough cleaning.We assume all location-dependent parameters are continuous and strictly positive.This paper is arranged as follows.Section 2 deals with the well-posedness of the problem.In Section 3, we first establish the basic reproduction number and the threshold dynamics for a corresponding linearized system when c = 1, and then the generalized eigenvalue is defined to obtain the sufficient conditions for the stability of the disease-free equilibrium and the positive periodic solution when 0 < c < 1. Numerical simulations in Section 4 are devoted to discussing the impacts of spatial heterogeneity and diffusion rate d on the basic reproduction number, and the role of the cleaning ratio c on the control of virus.A brief discussion is finally presented in Section 5.

Analysis of the model
Some basic properties of the solution to problem (1.1) is firstly analysed in this section.
Let X = C( Ω, R 4 ) be a Banach space with the supremum norm || • || X .Define + ) is a positive cone of X, then (X, X + ) is an ordered Banach Space.For any initial function φ = (S 0 (x), E 0 (x), I 0 (x), W 0 (x)) T ∈ X + , we next define where G is a Green function related to d∆−γ 4 (x) subject to the Neumann boundary condition.It follows from Corollary 7.2.3 in [21] that T 4 (t) : C( Ω, R) → C( Ω, R) is compact and strongly positive for any t > 0.

Threshold dynamics
It is easy to see that problem (1.1) admits a disease-free equilibrium E 0 (S * (x),0,0,0), where S * (x) = α(x) γ1(x) .We first linearize problem (1.1) in E 0 and consider the following problem If c = 1, then there is no impulse in problem (3.1).Let E(x, t) = e λt ϕ 2 (x), I(x, t) = e λt ϕ 3 (x) and W (x, t) = e λt ϕ 4 (x), we obtain the following eigenvalue problem Let R(t) be the solution semiflows on C( Ω, R 2 ) corresponding to linear system (3.1).Then R(t) is a positive C 0 -semigroup with generator It follows from Theorem 3.12 in [11] that A is a closed and resolvent positive operator and A = F + V , where Let R(t) be the solution semigroup generated by the operator V .We describe the distribution of initial infections by φ(x) = ( φ2 (x), φ3 (x), φ4 (x)).Thus, R(t) φ is the distribution of infected individuals that affected by mobility, mortality, recovery or transform.The distribution of new infections at time t becomes F (x) R(t) φ(x).The total distribution of new infections can be described by L as the following The basic reproduction number R 0 can be defined by the spectral radius of L as Let η 0 be the principal eigenvalue of the following eigenvalue problem we have R 0 = 1 η0 .It is easy to see that the expression of R 0 is difficult to give, so we here take the following auxiliary eigenvalue problem (3.4) into account.Let τ 0 be the principal eigenvalue of eigenvalue problem (3.4) We will prove that R 0 − 1 and 1 τ0 − 1 have the same sign.Since the principal eigenvalue of A and A T are same, so τ 0 is also the principal eigenvalue of problem (3.5) which can be written by It is clear that and hold.Substituting (3.7) and (3.8) into the first equation of (3.6) yields where It follows form Theorem 2.4 in [6] that elliptic eigenvalue problem (3.9) admits a unique principal eigenvalue τ 0 and its corresponding positive eigenfunction is x) can be obtained by equations (3.7) and (3.8), respectively.The following results can be obtained by Theorem 3.2 in [26].
Theorem 3.1.The eigenvalue problem (3.5) admits a unique positive principal eigenvalue τ 0 , and its corresponding positive eigenfunction is . In addition, It is easy to see from the expression of 1 τ0 that τ 0 is an nondecreasing function with respect to d.
The following lemma can be obtained by Theorem 3.1 in [26].
It follows from Lemma 3.1 and 3.2 that R 0 − 1 and 1 τ0 − 1 have the same sign.
When 0 < c < 1 in problem (3.1), it means harvesting pulse occurs in this system and 1 − c is the harvesting rate.Let E(x, t) = e −λt φ 2 (x, t), I(x, t) = e −λt φ 3 (x, t) and W (x, t) = e −λt φ 4 (x, t) in problem (3.1), we obtain the corresponding eigenvalue problem Problem (3.15) is the periodic and degenerate eigenvalue problem with pulse, as we know, there is no results for the existence of the principal eigenvalue for this kind of problem, so now we consider the generalized principal eigenvalues.
In what follows, we construct by the definition of the generalized eigenvalue λ * .
Similarly, the third inequality in problem (3.24) can be written as and then integrating both sides of inequality aforementioned over (0 + , T ) to get λ ≥ −k 5 − 1 T ln c.It can be derived through problems (3.24) and (3.27) that h 2 (t) ≤ e − ln c T t , t ∈ (0 + , T ] and h 2 (0) ≤ 1 c .We still construct which is substituted into the first two inequalities in (3.24), assert λ ≥ −k 1 − ck 2 and λ ≥ −k 3 − ck 4 .Therefore, we obtain by the definition of the generalized eigenvalue λ * .To sum up, we have the following estimates of the generalized principal eigenvalues.
Similar as Lemma 2.3 in [15], we have the following positivity of the solution by using the strong maximum principle and the Hopf boundary lemma.Lemma 3.3.Assume that u(x, t) is the solution to problem (1.1)with nontrivial initial value φ ∈ X + .Then S(x, t) > 0, E(x, t) > 0, I(x, t) > 0 and W (x, t) > 0 for any x ∈ Ω, t > 0.
Next, we give the long time behavior of the solution according to the principal eigenvalue (for c = 1) or the generalized principal eigenvalues (for 0 < c < 1).Theorem 3.3.When c = 1, i.e. there is no pulse in problem (1.1), if R 0 < 1, then the disease-free equilibrium E 0 (S * (x), 0, 0, 0) of problem (1.1) is globally asymptotically stable.
Proof.We first define a Lyapunov function through positive eigenfunction pair Recall problem (3.5), we get It follows from the Green's first formula and Neumann boundary condition that Therefore, we obtain Recalling that R 0 −1 and 1 τ0 −1 have the same sign, R 0 < 1 means that 1−τ 0 < 0, we then have dW dt ≤ 0, and E 0 (S * (x), 0, 0, 0) is globally asymptotically stable by the similar argument as in [16].
According to Lemma 2.8 in [15] and Theorem 4.2 in [30], we have the following results.
The first three equations of problem (3.31) indicates that I(x) satisfies In view of I(x) > 0, we get , where If γ 1 (x)γ 3 (x)(a(x)+γ 2 (x))−α(x)µ 2 (x)[a(x)+(1−δ)γ 2 (x)] > 0, then the equation (3.32) can be rewritten as . After careful calculation, we derive that the derivative of I(x) W (x) with respect to W is less than zero when W is nonincreasing with respect to W .The uniqueness is now completed.
Next we consider global asymptotic property of the endemic equilibrium of the system (1.1) with no pulse.Theorem 3.5.Assume that c = 1, which means no pulse occurs.If R 0 > 1, then the endemic equilibrium E 1 (S(x), E(x), I(x), W (x)) of problem (1.1) is globally asymptotically stable.
By calculating the derivative of H along the positive solution of system (1.1), we have Since E 1 is the steady state solution to problem (1.1), substituting (3.31) into (3.33)yields It then follows from Green's first formula and Neumann boundary condition that Hence, dH dt ≤ 0, and the equality sign holds if and only if (S(x, t), E(x, t), I(x, t), W (x, t)) = S(x), E(x), I(x), W (x) .
We finally obtain that E 1 is globally asymptotically stable.
It is quite difficult to analyze the stability of the endemic equilibrium of problem (1.1) with harvesting pulse (i.e.0 < c < 1), however, the following numerical approximations (see Fig. 6) show that if λ * < 0, the solution to problem (1.1) tends to a positive periodic solution.

Numerical simulations
In this section, numerical approximations are carried out to verify the correctness of the theoretical results and explore the impact of virus reproduction rate and diffusion coefficient on the basic reproduction number.
We secondly fix d = 0.01, c = 1, b 0 = 0.12, then 1 τ0 ≈ 1.9537 > 1 and R 0 > 1. Problem (3.1) admits an endemic equilibrium E 1 , and if R 0 > 1, E 1 is globally asymptotically stable.Fig. 2 indicates that I(x, t) and W (x, t) ultimately stabilize to a positive steady state.We still fix d = 0.01, c = 1, and then choose b 0 = 0.06, 0.07, 0.08, respectively.It is worth noting in Fig. 3 that 1 τ0 is a nondecreasing function with respect to k. Owing to the same sign of R 0 − 1 and 1 τ0 − 1, we can see that the increase of the virus reproduction rate in heterogeneous environment will lead to an increase in R 0 , which increases the risk of disease infection and bring about more infections.In the following, the stability of the solution is illustrated by numerical simulation.Firstly, fix parameters b(x) = 0.6 1 + 0.5 sin( 9πx 10 ) , d = 0.137, c = 0.8, and choose γ 4 = 0.5, then λ * ≥ 0.1029 > 0. From Fig. 5 we can see that the density of viruses in the environment eventually goes to zero.Secondly, Choose γ 4 = 0.1, then λ * ≤ −0.2495 < 0. Numerical approximation in Fig. 6 indicates the virus in the environment stabilizes to a positive periodic steady state.
This indicates that the smaller γ 4 (x) is, the longer the average survival time of free viruses without hosts has, the more likely viruses are to persist in the environment.Therefore, it can be seen that, the average survival time of free viruses without hosts which is 1/γ 4 (x) plays an important role in the persistence or extinction of viruses in the environment.We also note that a short average survival time of free viruses is beneficial for the extinction of the viruses when the pulse takes place.

Discussion
This paper focuses on a reaction-diffusion problem featuring impulsive effects under Neumann boundary conditions.Initially, we formulate an infectious disease model that incorporates both pulsing and environmental virus diffusion, and prove the well-posedness of problem (1.1).Subsequently, we utilize the principal eigenvalue τ 0 of the corresponding elliptic eigenvalue problem to establish a symbolic relationship with R 0 in problem (1.1) when c = 1 (no pulse).We also estimate the generalized eigenvalues in problem (1.1) for 0 < c < 1 (harvesting pulse).In the case where c = 1, we examine the global stability of both the disease-free and endemic equilibria in heterogeneous environments by constructing a Lyapunov functional.Specifically, we find that if R 0 < 1, the disease-free equilibrium is globally asymptotically stable.Conversely, if R 0 > 1, problem (1.1) exhibits uniform persistence and admits a unique, globally asymptotically stable, positive steady-state solution.Furthermore, in the case where 0 < c < 1, we establish that if λ * > 0, the disease-free equilibrium is globally asymptotically stable.However, if λ * < 0, the solution to problem (1.1) converges to a positive periodic solution.
Our numerical simulations demonstrate that an elevated rate of viral reproduction contributes to an increase in new infections in spatially heterogeneous environments.In the other hand, extensive viral diffusion coupled with regular environmental cleaning results in a decrease in new infections.These results suggest strategy for infection control.Specifically, frequent ventilation can facilitate the diffusion of the virus, thereby reducing the number of new infections.Additionally, regular environmental disinfection is effective in eliminating viruses originating from the environment.

Remark 3 . 2 .
Considering the impact of the virus reproduction rate in the heterogeneous environment, we take b(x) := b 0 1 + k sin 9πx 10 in Example 4.2 in Section 4. It can be seen from simulations of 1 τ0 that if b 0 is fixed, 1 τ0 is a nondecreasing function with respect to k. Lemma 3.1.Assume that λ 0 be the principal eigenvalue of the following eigenvalue problem 15), where the equal signs of the first three equations are replaced by ≥},λ * = inf{λ ∈ R : λ satisfies (3.15),where the equal signs of the first three equations are replaced by ≤}.
into the third inequality in problem (3.23) to derive that λ ≤ −c 4 − 1 T ln c − c3 c e ln c T t .Then we have λ ≤ −c 4 − 1 T ln c − c3 c , and