Global stability mathematical analysis for virus transmission model with latent age structure

: Background and objective: Mathematical model is a very important method for the control and prevention of disease transmissing. Based on the communication characteristics of diseases, it is necesssery to add fast and slow process into the model of infectious diseases, which more e ﬀ ectively shows the transmission mechanism of infectious diseases. Methods: This paper proposes an age structure epidemic model with fast and slow progression. We analyze the model’s dynamic properties by using the stability theory of di ﬀ erential equation under the assumption of constant population size. Results: The very important threshold R 0 was calculated. If R 0 < 1, the disease-free equilibrium is globally asymptotically stable, whereas if R 0 > 1, the Lyapunov function is used to show that endemic equilibrium is globally stable. Through more in-depth analysis for basic reproduction number, we obtain the greater the rate of slow progression of an infectious disease, the fewer the threshold results. In addition, we also provided some numerical simulations to prove our result. Conclusions: Vaccines do not provide lifelong immunity, but can reduce the mortality of those infected. By vaccinating, the rate of patients entering slow progression increases and the threshold is correspondingly reduced. Therefore, vaccination can e ﬀ ectively control the transmission of Coronavirus. The theoretical incidence predicted by mathematical model can provide evidence for prevention and controlling the spread of the epidemic.


Introduction
COVID-19 is an emerging acute infectious disease, which incubation period is 1-14 days, usually 3-7 days. COVID-19 generally have no obvious precursors, the infection will be after the Coronavirus disease early, some patients may also have no obvious symptoms, most common in patients with fever, Feng who is the deputy director of the Chinese Center for Disease Control and Prevention, said that the re-positive case caused continued transmission is rare and does not play a big role. Recently, some provinces in China have seen new cases of COVID-19 including asymptomatic patients. Therefore an infected individual ows into the exposed class after been treated but not becomes susceptible in our models. Inspired by the above discussions, we consider an SEI epidemic model which introduce the latent age, the nonlinear incidence of reactive unsaturated treatment and the saturation treatment function. where S (t), e(t, a) and I(t) respectively be the population sizes of susceptible, latent and infective classes. We assume that the population size is changeless. β represents susceptible people in contact with an infected person transmit rate, p represents susceptible people contact with an infected person enter slow propagation process called latent stage which denoted by e(t,a) where individuals are infected with disease but are not yet contagious, where a is called the age of latency progression, which is the duration of the incubation period. We denote E(t) = +∞ 0 e(t, a)da as the latent individuals' total density. 1 − p represents susceptible people and the onset of contact after rapid development for the onset of ratio, µ and d represent people natural mortality and mortality due to illness. f (x) = γx/(1 + mx) represents the saturation treatment function where the γ is cure rate of the disease.
We define X = R + × L 1 The initial condition of system (2.1) belongs to the positive cone of X, then can be rewritten as x 0 = (S 0 , e 0 (·), I 0 ) ∈ X. We can get a continuous semi-flow associated with system (2.1), that is, Θ : R + × X −→ X produced be system (2.1) adopts the following form, Let Ω = (S (t), e(t, ·), I(t)) ∈ X, S (t) + +∞ 0 e(t, a)da + I(t) ≤ Λ µ . The interior of Ω isΩ. we can easily verify the non-negative and positive invariance set of the system (2.1) with the help of article [5].
We note that if R 0 > 1, Ω is the positive invariant set for Θ, and it attracts all solutions of the system (2.1) with non-negative initial conditions.

Reproduction number
Firstly, there is a disease-free equilibrium point E 0 = (S 0 , 0, 0) in the system (2.1), where S 0 = Λ µ , and we define the basic reproduction number of the system (2.1) as following and it is easily known that if R 0 > 1, system (2.1) has an only positive endemic equilibrium point E * (S * , e * (a), I * ), where Define space , 0 e(t, ·) T Therefore systerm (2.1) can be rewritten as an abstract Canchy problem Draw on the results in Magal [16] and Magal and Thiemel [17], there exists an uniquely deterministic semiflow {U(t)} t≥0 on X 0+ which is bound dissipative and asymptotically smooth, and {U(t)} t≥0 has a global attractor T ⊂ X which attracts the bounded sets of X.

Locally asymptotic stability
Integrating the second equation of system (2.6) from 0 to a, considering the boundary condition, into the fourth equation of system (2.6), solving it, we get the characteristic equation easily know H (λ) < 0 which implies that H (λ) is a decreasing function, and Let λ = x + yi is an arbitrary complex root of H(λ) = 0, then Thus if and only if R 0 ≤ 1, all the roots of system (2.6) have negative real part.
Theorem 2.2. If R 0 > 1, the unique endemic equilibrium E * is locally asymptotically stable.
Proof. The perturbation variables are as follows Linearizing system (2.1) at E * and assuming the form of solution is as follows where y 0 1 , y 0 2 (a), y 0 3 will be determined. we get (2.8) By calculating, we can get the following eqution From the first equation of system (2.8) and the Eq (2.9), we obtain the following characteristic equation (2.10) Note M is the right side of the Eq (2.10). Assuming λ > 0, M satisfies the following inequality Since λ > 0 , therefore (βI * +µ+1)(γ+d+µ) , easily know A < 1, i.e. R 0 < 1 which is contradict with R 0 > 1. then we can get λ < 0.

Global asymptotic stability
Theorem 2.5. If R 0 > 1, the unique endemic equilibrium E * is globally asymptotically stable.   For g(x) ≥ 0, x ∈ R, therefore dV * (t) dt ≤ 0 is always true, furthermore the strict equality holds if and only if S = S * , e(t, a) = e * (a), I = I * . Therefore when R 0 > 1 the endemic equilibrium E * is globally asymptotically stable.

Simulations and summary of results
In the work, an age structure epidemic SEI model with fast and slow progression is considered. The basic reproduction number R 0 is obtained as R 0 = pβΛθ+γµθ+(1−p)βΛ (γ+d+µ)µ . We have proved the globally asymptotically stable for disease-free and endemic equilibrium respectively. In the following, we also give some numerical simulations to illustrate the global stability. Let Λ = 1; β = 0.055; γ = 0.7; µ = 0.063; d = 0.04; p = 0.8; m = 0.02; S (0) = 30, e(0, a) = 6e −0.4a , I(0) = 10. and In Figure 1, we choose τ = 12 , then R 0 < 1, it can be seen that E 0 is globally asymptotically stable. While in Figure 2, we choose τ = 1, then R 0 > 1, it can be seen that E * is globally asymptotically stable. The figures show the series of S (t) and I(t) which converge to their equilibrium values, in addition the age distribution of e(t, a).

Discussion
COVID-19 has spread rapidly around the world since 2020 with a high fatality rate. Today the epidemic in some countries is still unable to be effectively controlled, and social and economic life has been greatly disrupted. COVID-19 trend prediction has become a major research focus. Current trend prediction methods include epidemic disease prediction model, COVID-19 trend prediction model based on deep learning, etc. These models have effectively assisted medical experts and scientific research institutions to efficiently predict COVID-19. The countermeasures and suggestions for strengthening epidemic prevention and control are put forward, which have a good guiding role for accurate epidemic prevention and control.
The large-scale epidemic of COVID-19 in China has basically ended, but there are still occasional imported cases or local outbreaks caused by cold chain pollution which prevention and control enters a new phase of normalization. Since the outbreak of COVID-19, a large number of researchers have conducted extensive studies on infectious disease dynamics and prevention and control measures through various models and data analysis methods. Many scholars have built traditional dynamics models based on warehouses to explore the development trend of COVID-19 and provide scientific basis for epidemic prevention and control.

Conclusions
According to transmission characteristics of infectious diseases, the paper proposed the methods of fast and slow transmission, which more effectively reveals the transmission mechanism for infectious diseases. The global asymptotic stability of the system has analyzed with the help of the principle of dynamics, and abtained the threshold of infectious disease control. The greater the rate of slow progression of an infectious disease, the fewer the threshold results. The world is now being vaccinated which cannot provide lifelong immunity, but can reduce the mortality of those infected. By vaccinating, the rate of patients entering slow progression increases and the threshold is correspondingly reduced. Therefore, vaccination can effectively control the transmission of Coronavirus.