Stability and stabilization of ecosystem for epidemic virus transmission under Neumann boundary value via impulse control

In this paper, by using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction-diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point. Besides, employing impulse control technique derives the globally exponential stability criterion of delayed feedback ecosystem. And numerical examples illuminate the effectiveness of impulse control, which has a certain enlightening effect on the actual epidemic prevention work . That is, in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation. particularly, the newly-obtained theorems quantifies this feasible step.


Introduction
Infectious diseases have existed for thousands of years in the history of human development and brought great harm to human beings. In the 6th century, the prevalence of smallpox in the Middle East reduced the population by nearly one tenth. From 1347 to 1352, the spread of plague in Europe killed nearly one third of the European population. AIDS is due to the infection of human immunodeficiency virus and make the human immune system destroyed, loss of immune capacity. In 1927, Kermack and McKendrick established the famous Sir chamber model in order to analyze the spreading law of London black death from 1665 to 1666 and Bombay plague in 1906 ( [3]).
In 1932, Kermack and McKendrick established the famous SIS model ( [4,5]). The idea of compartment model is to divide the population into susceptible individuals, infected individuals and recovering individuals, and ignore the birth and death of the population, that is, the total population remains unchanged. Such infectious diseases as measles and chickenpox, patients with lifelong immunity after recovery, recovery individuals will not enter the susceptible individuals. On this basis, numerous scholars have done a lot of work ( [6][7][8][9][10]). Stability analysis is an important part of infectious disease dynamics. The stability of equilibrium point is studied by qualitative theory of delay differential equations and uniform persistence theory. The stability of equilibrium includes local stability, uniform persistence and global stability. Local stability means that the initial value is near the equilibrium point, and the value at any time is near the equilibrium point [6]. Uniform persistence means that the system eventually has a positive lower bound, which reveals that infectious diseases eventually spread in the population [7]. Global stability means that the equilibrium point is stable. For any initial value, when the time is sufficiently large, the solution of the system will converge to the equilibrium point [7,8], The disease is extinct or epidemic. The research on the stability and consistent persistence of the dynamic model of infectious diseases is helpful for people to find the epidemic law of infectious diseases, and is of great significance for the prevention and control of infectious diseases. Time delay effect exists widely in the objective world. The epidemic law of infectious diseases not only depends on the current state, but also related to the previous state. In the process of infectious disease transmission, for some infectious diseases, such as tuberculosis, AIDS, etc., the susceptible become infected after infection, but they do not necessarily have the ability to infect immediately. It takes a period of time for the susceptible to become infected to have the ability to infect, so people need to take the infection delay into account in the infectious disease model. For influenza and other diseases, since the infected person can still reinfect the disease after recovery, it is necessary to consider the recovery delay. The recovery delay is the time from the recovery stage of the convalescent person to becoming a susceptible person again.
In order to study gonorrhea, Cooke and Yorke established an epidemic dynamics model with time delay ( [9]). There are a lot of classical works on the time-delay epidemic model (see [10 -13]). In the virus dynamics model, since it takes a period of time from the virus infecting the host cell to the host cell producing virus particles, it is necessary to take the infection delay into account in the virus dynamics model. Because there is a period of time between the immune system receiving antigen stimulation and producing immune cells, it is necessary to take the immune delay into account in the virus dynamics model. In this paper, a dynamic model of infectious disease patients and susceptible patients with diffusion is studied, and a global stability result is obtained. Furthermore, since the artificial epidemic prevention can be regarded as an impulsive effect, the impulsive delay dynamical system is also considered, and the global exponential asymptotic stability criterion is obtained.
In this paper, the innovative points are listed as follows, ⋄ By using the variational method, a sufficient condition for the unique existence of the stationary solution of the reaction-diffusion ecosystem is obtained, which directly leads to the global asymptotic stability of the unique equilibrium point.

System descriptions
Consider the following ecosystem under Neumann boundary value: where u, v represent population density of people infected with virus and that of virus susceptible people or animal, respectively. d > 0 represents the birth rate of the virus susceptible, a 1 uv represents the increasing degree on the infected with virus due to the cross infection, and −a 2 uv represents the decreasing degree on the susceptible due to the cross infection. b 1 uv 2 represents decreasing degree on the infected due to epidemic preventions while b 2 u 2 v is the increasing degree on the susceptible after the patients are cured and became susceptible to infection. c 1 u represents the decreasing degree on the patients due to the death of some patients. c 2 u represents the decreasing degree on the virus susceptible due to increasing patients.
Throughout this paper, the following assumption is considered, (H1) There are two positive constants M i (i = 1, 2) such that Remark 1. Due to the limited natural resources, the boundedness hypothesis (H1) on population density is reasonable.

Remark 2. For any given positive numbers
But the condition (H2) can not warrant that (u * , v * ) T must be the unique stationary solution of the system (2.1). In fact, each (u(x), v(x)) T satisfying the following elliptic equations (2.6) must be a stationary solution of the ecosystem (2.6) The following lemma may be necessary to some extent (see, e.g. [1,Lemma 4] by Ω. v(x) is a real-valued function belonging to H 1 0 (Ω) and ∂v(x) ∂ν | ∂Ω = 0. Then which λ 1 is the smallest positive eigenvalue of the Neumann boundary problem (2.7)

Unique existence of the globally asymptotical stationary solution
Due to the viewpoint of Remark 2, the author has to prove that (u * , v * ) T defined in (2.5) is the unique equilibrium point of the system (2.1). The following uniqueness proof technique imitates that of the author's another work [14].
Proof. It is easy to verify that (u * , v * ) defined in (2.5) is a solution of the following equations which implies that (u * , v * ) T is a positive equilibrium point of the system (2.1).
Let (u(x), v(x)) T be any stationary solution of the system (2.1). Set Based on the boundedness assumptions (H1) on u, v, one can get and where α i , β i are defined in (3.2).
Similarly, other parameters are also simplified to some extent.

Impulse control and global stabilization on delayed feedback system
Consider the following delayed feedback system with impulse: Obviously, the null solution of (4.1) is corresponding to the equilibrium point (u * , v * ) T of the following system: where each B k > 0 is a symmetric matrix, dependent on k.
then the unique equilibrium point (u * , v * ) T of the system (4.3) is globally exponentially stable with convergence rate λ 2 , where I represents identity matrix, and Proof. Firstly, according to Remark 4, the conditions of Theorem 3.1 warrant that (u * , v * ) T is the unique equilibrium point of the system (4.3).
On the other hand, which together with (4.9), and (4.4)-(4.6) means that all the conditions of [1, Theorem 2.1] are satisfied. According to [2,Theorem 2.1], the null solution of the system (4.1) is globally exponentially stable with convergence rate λ 2 . And hence, the unique equilibrium point (u * , v * ) T of the system (4.3) is globally exponentially stable with convergence rate λ 2 .

Numerical examples
which means that the conditions (H1) and (H2) hold, and direct computation yields Particularly, u * = 1 = v * doesn't contradict the boundedness condition (H1). Moreover, set Then the condition (4.4) holds. Let Direct computation yields Set σ = 1.5, and then σ − λ = 1 > c. Moreover, Remark 5. Table 1 illuminates that the greater the pulse intensity, the faster the stability of the system. And Table 2 indicates that the more frequent the pulses, the faster the stability of the system.

Conclusions
At first, the author uses variational methods to derive the uniqueness of the equilibrium point of ecosystem so that the global stability can be consider. After giving two global stability criteria, the author proposes numerical examples to illuminate the effectiveness of the theorems. Numerical examples and stability criteria tell that in the face of the epidemic situation, taking a certain frequency of positive and effective epidemic prevention measures is conducive to the stability and control of the epidemic situation.