Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental e ff ects

: In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasi-ble equilibria with or without antibody response are established. The basic reproduction number R b 0 is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if R b 0 < 1. Two endemic equilibria exist if R b 0 < 1, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number R 0 is calculated, from which the disease-free equilibrium is globally asymptotically stable if R 0 ≤ 1, and the unique endemic equilibrium is globally asymptotically stable if R 0 > 1. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.


Introduction
Infectious diseases have serious impacts on human health, social stability, economic development, happiness of families, and even national security. In particular, humans beings are facing serious threats from various kinds of viruses, e.g., AIDS, influenza, hepatitis B, dengue, cholera, MERS, SARS, Ebola and COVID-19 [1][2][3][4][5][6]. Therefore, it is vitally important to study the dynamic properties uals will make the control of disease spread more difficult [23,25]. Based on the above consideration, in this paper, we propose an SEIR epidemic model coupling virus spread within the body and between hosts, incubation, antibody response and environmental effects. The paper is organized as follows: In Section 2, the virus infection model with humoral immunity in the host, and the SEIR epidemic model of in vitro transmission between hosts are proposed with incubation at fast and slow time scales respectively, and both models are coupled by the environmental concentration of the virus. In Section 3, some properties and important results of the virus infection model with humoral immunity are given. In Section 4, the positivity and boundedness of solutions for the SEIR model without antibody responses are discussed, criteria on the existence and local stability of equilibria are established, and the uniform persistence of the model is obtained. In Section 5, we further study the SIER model with an antibody response. The basic reproduction number is calculated, and the existence and global stability of equilibria are studied. In Section 6, the main conclusions are illustrated by numerical examples and discussion. In Section 7, we give a brief conclusion.

Model description
In this section, we provide a detailed description of the SEIR epidemic model coupling virus transmission in the body and between hosts, incubation and environmental effects. We first proposed the following assumptions.
(A 1 ) The transmission of diseases is mainly caused by susceptible individuals being exposed to a virus in the environment. This kind of infection is commonly called indirect infection. The environmental contamination rate is related to the quantity of infected individuals and the virus load V of hosts, with the form θV I.
(A 2 ) Susceptible individuals have an incubation period after being infected by the virus. Usually, different lurks have different incubation periods. To facilitate discussion, we take an average incubation period provided that the infected individuals will not transmit disease in the incubation period.
(A 3 ) The infected individuals usually will either die or be cured as a remover after being infected. To facilitate discussion, we assume that the removed individuals will not become susceptible again or that the disease will not recur.
(A 4 ) The Viruses in the environment primarily comes from unique releasers, i.e., infected individuals. The amount of virus released by different infected individuals is related to the total load of virus within the bodies of hosts. To facilitate discussion, we take the average number of viruses within the infected individuals.
Based on the above assumptions, we can first establish the following SEIR epidemic model of viral infection: dU(t) dt = θV(s)I(t)(1 − U(t)) − (ξ + γ)U(t), where S (t), E(t), I(t) and R(t) indicate the numbers of susceptible, latent, infected and removed indi-viduals at time t, U(t) is the contamination rate of the virus in the environment at time t, β indicates the probability of susceptible individuals being exposed to environmental viruses, Λ indicates the recruitment rate of susceptible individuals, µ indicates the natural mortality rate of the entire population (including susceptible, latent, infected, and removed individuals), α indicates the conversion rate of lurks into infected individuals, σ indicates the cure rate of infected individuals, ζ indicates the diseaserelated mortality rate of infected individuals, θ indicates the emission rate of virus to the environment released by each infected individual, θV I(t)(1 − U(t)) indicates the increase in virus concentration in the environment per unit time as a whole, ξ indicates the decay rate of virus in the environment, and γ is the per capita virus clearance rate in the environment. In the above model, V represents the amount of virus carried in infected individuals. To obtain the specific form of V, we need to further study the dynamic process of virus infection in host. Dynamic models of virus infection in different types of hosts have been investigated [9, 11-13, 20, 21]. In this article, we further propose a virus infection model with humoral immunity in the host as follows: In this article, the influence of environmental viruses is also introduced. The number of viruses in the body of a host will increase due to contact with environmental viruses, breathing and diet. Therefore, we improve model (2.2) and change it into the following form: where function g(U(t)) represents the increase in the number of viruses in the host caused by environmental viruses invading into the host per unit time. Virus invasion usually indicates that the virus invades the host through breathing, eating, or contact with air, water, food or other objects contaminated by environmental viruses. Obviously, as the number of environmental viruses increases, the number of viruses that invade the host will also increase. Thus, we finally establish the SEIR model coupling virus transmission both in the body and between hosts, incubation and environmental effects as follows: There are two time scales in model (2.4). One is the dynamic evolution time s of susceptible cells, infected cells, viruses and B cells within the host. The other is the dynamic evolution time t of susceptible, latent, infectious, and recovered individuals and the level of environmental in vitro contamination of hosts. Time s is a fast time scale within hosts, and time t is a slow time scale between hosts. Usually, s is faster than t; then, we can assume that t = ωs, where ω is a very small positive number. However, there is a variable g(U(t)) in subsystem (2.3) within hosts and a variable V(s) in subsystem (2.1) between hosts. In this case, model (2.4) can be seen as a coupling models (2.1) and (2.3) in the body and between hosts.

Virus infection model with humoral immunity in a host
For the virus infection model (2.3), since the virus infection change process in the host is much faster than the disease transmission process between hosts, we can assume that the environmental contamination rate U(t) in model (2.3) remains constant U (0 ≤ U ≤ 1). Thus, model (2.3) becomes an isolated virus infection dynamics model.
We further suppose that environmental contamination is mainly related to the density of viruses living in polluted environment, and the host is infected through the ingestion of contaminated food. Therefore, function g(U) is assumed to satisfy the following condition: There are some special forms of g(U) satisfy condition (H 1 ). For example, the linear function g(U) = aU used in [9,11,12], means that the amount of environmental virus entering the host linearly increases with U. The nonlinear function g(U) = a(U) 1+bU is used in [12], where a > 0, b > 0, which means that the amount of environmental virus entering the host will reach saturation.
From the biological significance of model (2.3), any solution (T (s), T * (s), V(s), B(s)) is assumed to satisfy the initial conditions: Based on the positivity and boundedness of solutions, the conclusion is as follows. .
(2) If R w > 1, then the antibody response infection equilibrium A 4 (U) is globally asymptotically stable.
Proof. For equilibrium A 3 (U), we define the Lyapunov function L 3 as follows: For equilibrium A 4 (U), we define the Lyapunov function L 4 as follows: Computing the derivatives of L 3 and L 4 along the solution of model (2.3), we can obtain When R w ≤ 1, V 3 ≤ ω h . Hence, for any (T, T * , V, B) ∈ R 4 + , dL 3 dt ≤ 0. Moreover, dH 3 dt = 0 implies that T = T 3 , T * = T * 3 and V = V 3 . Furthermore, B = 0 when V = V 3 . Thus, from the LaSalle invariance principle [33], equilibrium A 3 (U) is globally asymptotically stable.
and V = V 4 . In addition, B = B 4 when V = V 4 . Thus, from the LaSalle invariance principle [33], equilibrium A 4 (U) is globally asymptotically stable. This completes the proof. □

Positivity and boundedness
Here, we discuss the nonnegativity and boundedness of solutions for model (2.4) in the general case, so we discuss neither the antibody response, nor the antibody dose response. Since the viral infection change process in the host is much faster than the disease transmission process between the hosts, we can assume that the state of virus infection in hosts has reached its equilibrium while the state of disease transmission between the host has not changed. Therefore, we can assume that the fast time variable V(s) =V(U) in model (2.4); furthermore we have the following form of the coupled model: According to Theorem 4, we haveV(U) with the following expression, For the convenience of statements, we denote R 5 Due to the biological significance of model (4.1), any solution (S (t), E(t), I(t), U(t)) of model (4.1) satisfies the following initial condition: where (S 0 , E 0 , I 0 , R 0 , U 0 ) ∈ R 5 + and S > 0. Regarding the nonnegativity and boundedness of solutions for model (4.1), we have the following conclusions.
Next, we prove the boundedness of the solutions. Let However, from the fifth equation of model (4.1), we have dU(t 0 ) dt = −(ξ+γ), which causes a contradiction. Therefore, U(t) < 1 for any t ∈ [0, τ ∞ ). Furthermore, assume that U(0) ≤ 1; then, by the continuity of solutions with respect to initial values, we can obtain U(t) ≤ 1 for any t ∈ [0, τ ∞ ). Thus, we finally determine that solution (S (t), This shows that solution u(t) is also ultimately bounded. This completes the proof. □

Basic reproduction number and equilibria
Now, we consider the SEIR model without an antibody response. Because the removed R does not emerge in the other four equations of model (4.1), we can only investigate the following sub-model For the convenience of discussion, we always assume that R w0 > 1 and R w ≤ 1 in this section. Thus, from (4.2), we can determine that the fast time variableV(U) in model (4.5) has the formV(U) k . Now, we define the basic reproduction number for model (4.5) as follows: . Therefore, the basic regeneration number R b0 denotes the number of new cases in which in the initial stage of infection, an infected individual releases viruses into the environment during the infection period. These viruses infect susceptible individuals during the survival period and make susceptible individuals become latent individuals, and last these latent individuals become infected individuals in the incubation period.
Define the functions be the positive equilibrium of model (4.5) that satisfies the following equations: Combining the basic reproduction number R b0 , assuming the existence of equilibria, we can obtain the following conclusion. Proof. With calculation we obtain The second order derivative of H(U) is given as follows: In addition, a 2 > 0, where a 1 and a 2 are defined in (3.5). This implies that T ′ > 0. Therefore, we further obtain H ′′ (U) < 0 for any U ≥ 0. H(U) is an upper convex function that has at most two zeros.

Stability of equilibria
We first consider the stability of the disease-free equilibriumW 0 = (S 0 , 0, 0, 0) of the model (4.5). The following conclusion is established.

Uniform persistence
Now, we discuss the uniform persistence of positive solutions for model (4.5). We can establish the following conclusion.
If E(t) ≡ 0, then from the third and fourth equations of model (4.5), we know that I(t) ≡ 0 and U(t) ≡ 0. Thus, model (4.5) is reduced to the following equation: From this, we can obtain lim t→∞ S (t) = S 0 , which implies that ω(x 0 ) = {W 0 }. If I(t) ≡ 0, then from the second and fourth equations of model (4.5), we know that I(t) ≡ 0 and U(t) ≡ 0. Thus, model (4.5) is also reduced to Eq (4.11), which implies that ω(x 0 ) = {W 0 }. If U(t) ≡ 0, then similar to the above discussions, we obtain ω(x 0 ) = {W 0 }. Therefore, we finally obtain M 0 = ∪ x 0 ∈M ∂ ω(x 0 ). Moreover, M 0 is isolated and noncyclic in ∂X. Now we prove that K s (W 0 ) ∩ X = ∅, where K s (W 0 ) is the stable set of W 0 . By contradiction, we assume that there is a x 0 ∈ X such that lim t→∞ u(t) = W 0 . Since R b0 > 1, we can choose a sufficiently small constant ε > 0 such that Thus, there is a t * > 0 such that S (t) ≥ S 0 − ε, U(t) < ε, E(t) < ε and I(t) < ε for all t ≥ t * . Furthermore, from lim U→0V (U) =V(0), we can also obtainV(U(t)) >V(0) − ε for all t ≥ t * . The following function is defined: Then lim t→∞ L(t) = 0. When t ≥ t * , Clearly, dL(t) dt > 0 for all t ≥ t * . Therefore, L(t) is an increasing function of t ≥ t * . lim t→∞ L(t) 0, which leads to a contradiction. Then, K s (W 0 ) ∩ X = ∅. According to theory of persistence for dynamic systems, there is a constant ε such that for any x 0 ∈ X, Therefore, the uniform persistence of model (4.5) is obtained. This completes the proof. □

SEIR model with antibody response
In this section, the SEIR model with an antibody response is considered. We always assume that R w0 > 1 and R w > 1 in this section. Hence, from (4.2), the fast time variableV(U) = ω h in model (4.5) is a constant that does not depend on U. Thus, model (4.5) takes the following form: The positivity and boundedness of model (5.1) has been established in Theorem 6 in Section 4.1. The basic reproduction number for model (5.1) is defined as With the existence of equilibrium of model (5.1), the conclusion is given below.
The proof of Lemma (2) is simple. Hence, we omit it here. Based on the global stability of equilibria W 0 andŴ for model (5.1), the conclusion is given below.
For conclusion (ii), we define the Lyapunov function L 1 as follows: The derivative of L 1 (t) along with any solutionŴ of model (5.1) is given by .
When R 0 = 1, we obtain the critical value of ψ as follows: Thus, according to Theorem 12, we obtain the following conclusions.
Corollary 1. If the number ψ of B cells in the host satisfies ψ > ψ 0 , then the disease between the hosts will be extinct. In contrast, if ψ < ψ 0 , then the disease between the hosts will be persistent.
Remark 6. Corollary 1 shows that the increasing number of B cells in infected and susceptible individuals can effectively control the spread of diseases between the hosts. However, the main way to increase the number of B cells in susceptible and infected individuals is to carry out active and effective treatment and vaccination. Therefore, when a certain infectious disease appears in an area, actively carrying out large-scale effective vaccination, timely treatment of patients who have been infected with the disease, and effective treatment are important means of preventing the disease in a timely and effective manner. By calculation, R w0 = 1.0849 > 1 and R b0 = 0.7208 < 1. Then, from Figure 1, the disease-free equilibrium W 0 = (68.1818, 0, 0, 0) of model (4.5) is locally asymptotically stable, which means that Theorem 7 is true.    Figure 2 shows that the unique endemic equilibriumW = (S ,Ẽ,Ĩ,Ũ) is locally asymptotically stable, which means that the Theorem 8 is true.

Conclusions
In this article, we investigate an SEIR epidemic model (2.4) coupling virus transmission in the body and vitro of hosts, incubation, humoral immunity and environmental effect, which are characterized and linked by two subsystems, i.e., (2.1) and (2.3).
With respect to the virus transmission process with humoral immunity in the body of hosts, we assume that the environmental contamination rate U(t) in model (2.3) remains constant at U (0 ≤ U ≤ 1) since the spread of virus infection within hosts is much faster than that in vitro of hosts. The basic reproduction number R w with an antibody response is defined, by which antibody-free infection equilibrium A 3 (U) is globally asymptotically stable if R w ≤ 1, while infection equilibrium A 4 (U) is globally asymptotically stable if R w > 1.
As is the SEIR model without an antibody response, we assume that the virus load in the body of hosts will tend to reach equilibrium, i.e., the fast time variable V(s) in model (2.4) satisfies V(s) = V(U) = 1 c (g(U) + pT * 3 (U)) in Theorems 4. The basic reproduction number R b0 is defined, from which we find that disease-free equilibriumW 0 is locally asymptotically stable if R b0 < 1, while the unique positive equilibriumW is locally asymptotically stable if R b0 > 1 (or R b0 = 1 and H M > 0) and F ′ (Ũ) ≤ 0. When R b0 < 1 and H M > 0, the system has two different positive equilibriaW 1 andW 2 with U 1 < U 2 , which means that system (4.5) experiences backward bifurcation at R b0 = 1. Meanwhile,W 2 is locally asymptotically stable if F ′ (Ũ 2 ) ≤ 0 andW 1 are unstable.
Furthermore, for the SEIR model with an antibody response, we assume that the fast time variable V(s) in model (2.4) satisfies V(s) =V(U) = ω h in Theorems 4. The basic reproduction number R 0 is defined, by which we find the disease-free equilibrium W 0 is globally asymptotically stable when R 0 ≤ 1, while the unique endemic equilibriumŴ is globally asymptotically stable if R 0 > 1.
From the numerical examples we know that F ′ (Ũ) ≤ 0 and F ′ (Ũ 2 ) ≤ 0 are pure mathematical conditions, and can only be used to prove the local stability of endemic equilibriaW andW 2 . Generally, we hope that the local stability of model (4.5) can only be determined by the basic reproduction number R b0 , but we used some additional conditions to do that. So we have summarized several open problems. The first is whether condition F ′ (Ũ) ≤ 0 and F ′ (Ũ 2 ) ≤ 0 can be taken off in the proof of Theorems 8 and 10. The second is whether we can get that (4.9) directly holds. The third one is whether we can further establish an appropriate Lyapunov function to obtain the global stability ofW. We will continue to investigate these questions in the future.
The results obtained in this paper show that the strength of antibodies in hosts has a great effect on the spread of diseases between hosts. When the antibodies in hosts do not work or are weak, the results obtained in Section 4 show that backward bifurcation could occur. Even if the basic reproduction number R b0 is less than 1, the disease will continue to spread, which will results in the control of the spread of disease between hosts being very difficult, making it difficult to effectively treat the disease.
When hosts have an extensive antibody response, the results obtained in Section 5 show that the spread of disease between hosts will be easy to control. Under antibody action, the basic reproduction number R 0 could be less than 1, and the diseases between hosts could be effectively controlled if we could decrease the incubation period of latent individuals, increase the production rate of B cells in hosts, and reduce the virus concentration (or load) both in the environment or in the body of hosts. Therefore, based on the above discussion with the explanation of the basic reproduction number R 0 , we can take the following prevention and control measures.
1. Vaccination and antibody immunotherapy. After vaccination, susceptible individuals will widely produce antibodies , which could effectively prevent the virus from invading. Even if a susceptible individual is infected by the virus, due to the effect of the antibody response, the amount of virus in hosts can only reachV(U) = ω h , so most of the infected individuals will only have some minor symptoms, and will not experience long-term infection. Furthermore, if antibody immunotherapy is performed on susceptible individuals, it will increase the number of antibody cells ψ in infected hosts, and eventually make the basic reproduction number R 0 less than 1, then the spread of disease will be effectively controlled until the disease eliminated.
2. Timely strict isolation and treatment of infected individuals, isolation control of close human contacts. These measures could effectively prevent close contact between infected and susceptible individuals, and reduce the release of virus into the environment by infected individuals, further reducing the infection rate of susceptible individuals and the virus emission rate of infected individuals, and finally making the basic reproduction number R 0 less than 1, this could effectively control the disease transmission between hosts until its disappearance.
3. Legitimately expand the range of management, supervision and publicity, e.g., wearing masks (surgical masks, goggles, etc.), practicing safe social distancing in public or crowded places, frequent hand-washing, exercising of the body, maintain a healthy diet, and obtaining sufficient rest. These measures could effectively reduce the infection rate of susceptible individuals, so that the basic reproduction number R 0 will be less than 1, effectively controlling the spread of disease. 4. Manually eliminating the virus in the environment. These measures could not only effectively improve the clearance rate of environmental viruses but also reduce the average survival period of environmental viruses. Therefore, the basic reproduction number R 0 would be less than 1, the spread of disease would be effectively controlled, and the disease would eventually become extinct.
In this paper we proposed the SEIR model (2.4) coupling virus transmission both in body and vitro of hosts. Since the virus infection changes process in host is much faster than the disease transmission process between hosts, to simplify discussions, we assumed that the environmental contamination rate U(t) in the fast time model (2.3) remains constant at U, and then, the fast time variable V(s) in the slow time model (2.1) reaches its equilibriumV(U) while the state of disease transmission between the host is unchanged. Thus, we established the coupled model (4.1). Obviously, model (4.1) is a special limit state coupled model. In this paper, we mainly investigated the dynamical behavior of model (4.1). However, it is more realistic to directly investigate the coupling model (2.4) with both fast time s and slow time t. Particularly, in model (2.4) we can assume that fast and slow times satisfy t = ωs is sufficiently, where ω is an enough small positive number. Therefore, an interesting open problem is to investigate coupling model (2.4) with assumption t = ωs. We will discuss this problem in the future.