A regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion

In this paper, we present a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion. We utilize the Markov semigroup theory to obtain the existence of a unique stable stationary distribution. We prove that the densities of the distributions of the solutions can converge in L1 to an invariant density under certain condition. Moreover, the sufficient conditions for the extinction of the disease, which means the disease will die out with probability one, are given in two cases. Meanwhile, we obtain a threshold parameter which can be utilized in identifying the stochastic extinction and persistence of the disease. Some numerical simulations are given to illustrate the analytical results.

Based on the pioneering research 1 , mathematical model provides effective control measures for infectious diseases and is an significant tool for analyzing the epidemiological characteristics of infectious diseases 2 . In the course of the spread of disease, the transmission function plays an important role in determining disease dynamics (see e.g. 3,4 ). There are several nonlinear transmission functions proposed by authors (see e.g. [5][6][7][8]. For instance, in 5 , Capasso and Serio introduced a saturated incidence rate g(I)S into epidemic models, and the infectious force g(I) is a function of an infected individual that is applied in many classical disease models. Liu et al. 6 proposed a general incidence rate = β ρ + g I S ( ) I S I 1 p q , p, q > 0. Lahrouz et al. 7 introduced a more generalized incidence rate = g I S ( ) SI f I ( ) . In particular, Yuan and Li 8 considered a ratio-dependent nonlinear incidence rate in the following form where α is a parameter used to measure psychological or inhibitory effects. We note that if α = h = l = 1, then (1.1) becomes the frequency dependent transmission rate (or the standard incidence rate In the case of l = 1, Cai et al. 9 obtained a ratio-dependent transmission rate ( ) www.nature.com/scientificreports www.nature.com/scientificreports/ one is infected, in other words, when I S is large. Thus the ratio-related incidence of transmission (1.2) takes into account crowding effects and behavioural changes during epidemics.
In this paper, we first study the following SIR epidemic model with a ratio-dependent incidence rate , then E 0 is unstable and system (1.3) has a unique positive endemic equilibrium E * = (S * , I * ) which is globally asymptotically We have a deeper understanding to the effects of the transmission coefficient β on the basis of these useful research about deterministic epidemic models. However, in the real world, the spread of infectious diseases are always subject to random fluctuations, it is more reasonable and practical to study the influence of random factors. Usually, environmental noise can be simply divided into white noise and colored noise. The disease transmission coefficient β in the SIR model is a key parameter for disease transmission, so it is interesting to assess the effect of the perturbation parameter β on the model. In applications we usually estimate it by an average value plus errors. Assume these errors follow a normal distribution, that is, where σ 2 > 0 is the intensity of the white noise, B t is a standard Brownian motion defined on a complete probability space with a filtration  ≥ { } t t 0 satisfying the usual conditions (see 10 ). Recently, epidemic models described by stochastic differential equations have been studied by many researchers (see e.g. 9,[11][12][13]. For instance, a stochastic SIRS epidemic model with a ratio-dependent incidence rate is proposed in 9 . The results show that the reproduction number  S 0 can determine whether there is a unique disease-free stationary distribution or a unique local stationary distribution. In addition, they provide analysis of stochastic boundedness and permanent/extinction. A stochastic SIS epidemic model is considered by Gray and his coworkers 11 . They give the unique global positive solutions of the model and derive the conditions of persistence and extinction of the disease. Meng 12 investigated dynamical properties of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. And Zhou et al. 13 considered the property of ergodic stationary distribution for a stochastic SIR epidemic model. In addition to white noise, epidemic models are also subject to colored noise, namely telegraph noise, which switches the system from one environmental state to another 14,15 . Now we add telegraph noise in order to make our model be more realistic 14,16 . Telegraph noise can be interpreted as switching between two or more environmental states, which vary with factors such as humidity and temperature 17,18 . Switching between environment states is usually memory-free, and the waiting time for the next switch follows an exponential distribution 19 . Therefore, the state transition can be modeled using a continuous time markov chain (r(t)) t≥0 that takes the values in the finite state space = {1, 2, …, N}. Also, Assume that the Markov chain r(t) is irreducible and independent of the Brownian motion B(t). Thus there is a unique stationary distribution π = (π 1 , π 2 , …, π N ) of r(t) which and π i > 0 for any  ∈ i . Taking into account the above two disturbances into the model (1.3), the SIR epidemic model with a ratio-dependent incidence rate and regime-switching is as shown below www.nature.com/scientificreports www.nature.com/scientificreports/ until time τ 1 when the Markov chain jumps to i 1 ; the system parameters will then satisfy Λ i 1 , μ i 1 , β i 1 , α i 1 , γ i 1 , ε i 1 and σ i 1 from time τ 1 till time τ 2 when the Markov chain jumps to the next state i 2 . If the markov chain jumps, the system will continue to switch.
In recent years, many researchers have studied the regime-switching stochastic epidemic models especially the long-time behavior of them (see e.g. [20][21][22]. In these literatures, to prove the ergodicity of random systems, we must first prove the uniform ellipticity condition. However, the diffusion matrix of the model (1.4) is degenerate and there is no uniform ellipticity condition in this paper. As far as we know, there is little research in this area. Therefore, in this paper, we will focus on the asymptotic behavior of the solutions of stochastic systems (1.4). The method adopted in this paper is derived from the markov semigroup theory introduced in 23,24 to study the long-term behavior of the stochastic predator-predator model. Based on markov semigroup theory, Lin et al. 25 analyzed the long-term behavior of the distribution density of the solutions of random SIR epidemic model.
Throughout this paper, unless otherwise specified, let satisfying the usual conditions (i.e., it is increasing and right continuous while 0  contains all  -null sets). We use A T to denote the transpose of a vector or matrix A, set This paper is organized as follows. In Section 2, we study stochastically asymptotic stability, the existence of a unique stable stationary distribution and present sufficient conditions for extinction of the disease in two cases. Some numerical simulations are introduced to demonstrate the theoretical results in Section 3. Finally, we summarize the results of this paper. And in the Appendixes, we present some preliminaries and give the proofs of our main results.

Stationary Distribution and Extinction
In the study of SIR deterministic model, disease eradication and stability are two of the most concerns. But for stochastic model, the equilibrium does not exist. Therefore, we can not show the persistence of the infection by proving the stability of the equilibrium. In this section, we will investigate the dynamics of the stochastic epidemic model (1.4). First, we will prove that system (1.4) has a stationary distribution, which implies the disease is recurrent. We will give the conclusion that if  > 1 S 0 , the densities of the distributions of the solutions to system (1.4) can converge in L 1 to an invariant density. Theorem 2.1. Let (S t , I t , r(t)) be a solution of system (1.4) with any initial value (S 0 , I 0 , r(0)) ∈ E × , then for every t > 0 the distribution of (S t , I t , r(t)) has a density u(t, x, y, i). The proof of Theorem 2.1 will be given in the Appendix B. , the disease I is stochastic persistent. That is to say, the disease will prevail and persist in the population.
Next, we will investigate the stochastic extinction of the disease in model (1.4). To this end, we establish the following theorem.

Numerical Experiments
In this part, Milstein's high Order Method 26 was used to verify the theoretical results we obtained.
where γ ij is the right-continuous markov chain at the value of  = {1, 2}. By solving the linear Equation πΓ =  0, we obtain the unique stationary (probability) distribution That is to say, the condition (ii) in Theorem 2.2 is satisfied. According to the condition (ii) in Theorem 2.2, it can be seen that the disease I goes to extinction with probability one. We can see this phenomenon in Fig. 3. www.nature.com/scientificreports www.nature.com/scientificreports/

Conclusion
The long-time behavior of a regime-switching SIR epidemic model with a ratio-dependent incidence rate and degenerate diffusion are observed in this paper. The existence of a unique stable stationary distribution is obtained by using markov semigroup theory. It is proved that the distribution density of solutions converges to a invariant density in L 1 under the condition of  > 1 S 0 . In addition, we also establish sufficient conditions for the disease to  www.nature.com/scientificreports www.nature.com/scientificreports/ go extinct with probability one in two cases. Meanwhile, we obtain a threshold parameter which can be utilized in identifying the stochastic extinction and persistence of the disease. One of the most important findings is that large environmental noises can suppress the outbreak of the disease. More precisely,  the disease I goes to extinction exponentially with probability one.
As is known to all, in order to obtain the ergodicity, we must prove that strong Feller property and irreducibility of Markov process. But in view of the proof of Lemma 3.2, model (1.4) is not irreducible. Therefore we use the Markov semigroup theory in 23,27 . Furthermore, in literatures [19][20][21][22]28 , the condition γ ij > 0, i ≠ j is necessary, but in the present paper, this condition is not necessary and we only assume that r(t) is irreducible.