key: cord-1004844-yjodgy2v
authors: Lan, Guijie; Lin, Ziyan; Wei, Chunjin; Zhang, Shuwen
title: A stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching
date: 2019-09-13
journal: J Franklin Inst
DOI: 10.1016/j.jfranklin.2019.09.009
sha: 25ae2d928f17cf4dc0cec99d114e7ea260423a03
doc_id: 1004844
cord_uid: yjodgy2v

In this paper, we propose and discuss a stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching. First of all, we show that there is a unique positive solution, which is a prerequisite for analyzing the long-term behavior of the stochastic model. Then, a threshold dynamic determined by the basic reproduction number [Formula: see text] is established: the disease can be eradicated almost surely if [Formula: see text] and under mild extra conditions, whereas if [Formula: see text] the densities of the distributions of the solution can converge in L(1) to an invariant density by using the Markov semigroups theory. Finally, based on realistic parameters obtained from previous literatures, numerical simulations have been performed to verify our analytical results.

Mathematical model is an important tool to better understand epidemiological patterns and disease control. Since Kermack and McKendrick [1] published their insights on the underlying mechanisms of transmission and control of infectious diseases, various epidemic models (for example, deterministic models [2] [3] [4] , spatially explicit models [5] [6] [7] , models on complex network [8] [9] [10] , models based on real data [11] [12] [13] and stochastic models [15, 16, 22] ) have been extensively established and investigated, which have led to great progress in the prevention and control of diseases. In the study of infectious disease dynamics, incidence rate is an important indicator that describes the average number of new cases per unit time. It is traditionally assumed that the incidence rate of disease transmission is bilinear, i.e., g(I ) S = βI S, which appears quite unrealistic if the number of infective individuals is large enough. In practice, many infectious diseases also exhibit multiple peaks during the outbreaks. For these reasons, there is a great interest in investigating epidemic models with nonlinear incidence rates. For example, Liu et al. [3] proposed a nonlinear saturated incidence rate g(I ) S = βI l S 1+ αI h to model the effect of behavioral changes, where β, l , α, h > 0. The saturated incidence rate g(I ) S = βIS 1+ αI (a special case of βI l S 1+ αI h ) is introduced into the epidemic model by Capassoand Serio [18] . This incidence rate is more realistic than the bilinear rate βIS , as it considers the behavioral change and crowding effect of the infective individuals and also prevents unboundedness of the contact rate (see Fig. 1 

It is well known that the aggressive control measures and policies (such as border screening, mask wearing, quarantine, isolation, etc.), play an important role in administering efficient interventions which control disease spread and hopefully eliminate epidemic diseases. For instance, during the outbreak of severe acute respiratory syndrome (SARS) in 2003 [4] and the outbreak of H1N1 influenza pandemic in 2009 [17] , these two diseases have been well controlled by taking these intervention actions. Xiao and Ruan [19] in 2007 introduced a nonmonotone incidence function g(I ) S = βIS 1+ αI 2 (l = 1 , h = 2) to understand the influence which was intervened in the spread of diseases by government. Then they established the following SIRS epidemic model:

where S t , I t , R t are the number of susceptible ( S ), infectious ( I ) and recovered ( R ) at time t , respectively. is the constant input of new members into the population per unit time, μ is the natural death rate of the population, γ is the natural recovery rate of the infective individuals, δ is the disease-caused mortality rate, θ is the rate at which recovered individuals lose immunity and return to the susceptible class. The incidence βI 1+ αI 2 increases when I is small and decreases when I is large (see Fig. 1.1 ) , which explains the phenomenon where the rate of contacting between infected I and susceptible S decreases after government intervention. βI measures the infection force of the disease, β is the proportionality constant, α is the parameter measuring the inhibitory effect. In addition, all parameters of model (1.1) are assumed to be positive constants.

In the real world, the nature of epidemic growth and spread is inherently random due to the unpredictability of person-to-person contacts and population is subject to a continuous spectrum of disturbances. Gard [21] showed that the population dynamics is inevitably affected by environmental white noise which is an important component in an ecosystem. May [14] revealed the fact that due to environmental noise the parameters involved in the system exhibit random fluctuation to a greater or lesser extent. Cai et al. [15] found the presence of even a tiny amount of white noises can suppress disease outbreak. Li et al. [16] pointed out that the presence of white noise is capable of supporting the irregular recurrence of influenza epidemic, and the average level of I ( t ) always decreases with the increasing noise intensity. In addition, Allen [39] pointed out that, one of the most important differences between deterministic and stochastic models is that deterministic models predict an outcome with absolute certainty, whereas stochastic models provide only the probability of an outcome. It also has been shown that some stochastic epidemic models can provide an additional degree of realism in comparison with their deterministic system. Therefore, it is important to investigate the effect of white noise in the environment on population dynamics. Recently, Cai et al. [22] investigated the stochastic dynamics of a simple epidemic model which incorporates the mean-reverting Ornstein-Uhlenbeck process. They found that the stochastic epidemic dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion. However, they only roughly provided numerical simulations for stationary distribution, but did not prove that the existence of stationary distribution when R s 0 > 1 . Liu et al. [23] studied a stochastic multigroup SIQR epidemic model with standard incidence rates. They gave the existence of a stationary distribution, but unfortunately they did not give the ergodic analyzing of their model, as the diffusion matrix is not satisfaction the uniform ellipticity condition. In particular, Cai et al. [15] assumed that the environmental white noise mainly influence the disease transmission rate β of the populations. That is β → β + σ dB t dt , where σ denotes the intensity of white noise and dB t dt is a Gaussian white noise, i.e., B t is a standard Brownian motion. By replacing βdt with βd t + σ d B(t ) in the deterministic SIRS model (1.1) , they obtained the following stochastic SIRS model

(1.2)

When studying epidemic models, it is naturally important to know whether the models tend to a disease-free state or the disease will remain permanently. For the deterministic model (1.1) , a threshold dynamic determined by the basic reproduction number ˆ R 0 is established: if ˆ R 0 < 1 , the disease-free equilibrium E 0 = ( μ , 0, 0) is globally asymptotically stable in the feasible region, whereas if ˆ R 0 > 1 , then there has a unique endemic equilibrium E * = (S * , I * , R * ) which is globally asymptotically stable (see [19] ). For the stochastic model (1.2) , the basic reproduction number is defined by [15] ). Additionally, we also recommend readers to refer to Feng et al. [24, 25] , Lin et al. [26] , Wang et al. [27] , Cai et al. [28] , Liu et al. [29] , Li et al. [30] , Yang et al. [31] and the references therein.

On the other hand, the epidemic dynamics are usually affected by a random switching in the external environments. For instance, seasonal variations in temperature, rainfall and resource availability are ubiquitous and can exert strong pressures on disease dynamics (see refs. [32] [33] [34] [35] 37] ). Hemmes et al. [32] showed that influenza virus survives (IVS) better at low temperature and relative humidity (RH) with little dependence on absolute humidity (AH). Arundel et al. [33] studied that RH can affect the incidence of respiratory infections and allergies and they found that the survival or infectivity of viruses is minimized by exposure to relative humidities between 40% and 70%. Sobsey and Meschke [34] reviewed that generally viruses with higher lipid tend to be more persistent at lower RH, while viruses with lesser or no lipid content are more stable at higher relative humidities. Shaman and Kohn [35] used the influenza virus data to explore the effects of absolute humidity (AH) on influenza virus transmission (IVT) and influenza virus survival (IVS) and found that the relationship between AH and IVS is strongly nonlinear, and there is the appearance of a similar nonlinear relationship between AH and IVT. Rogers and Randolph [37] pointed out that weather influences the survival and reproduction rates of the vectors and rates of development, survival and reproduction of pathogens within vectors. Therefore, it is important to investigate the effect of random switching in the external environments on disease dynamics. We also recommend readers to refer to Minhaz Ud-Dean [36] , Semenza and Menne [38] and the references therein for learning more information about the effect of temperature, humidity and weather factors on transmission efficiency of disease.

However, the white noise cannot describe the phenomena that the population may suffer sudden catastrophic shocks or alien species invasion in nature [40] . The telegraph noise (namely, color noise) can be illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition or as rain falls [41] . Frequently, the switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed [42, [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] . Thus we can model the regime switching by a continuous-time Markov chain with values in a finite state space, which drives the changes of the main parameters of epidemic models with state switchings of the Markov chain. In particular, Li et al. [53] studied the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. They gave the threshold value of the disease persistence and extinction and then gave the ergodic analyzing of their model. Regrettably, they neglected the effect of the white noise for the transmission of the disease. Liu et al. [42] explored the dynamical behavior of a stochastic SIQR epidemic model with standard incidence which is perturbed by both white and telegraph noises. They only gave the threshold between persistence and extinction of the disease, but did not give the ergodic analyzing of their model. Inspired by the above facts, in present paper, we will firstly incorporate telegraph noise into stochastic model (1.2) to get a new model, then we provide the threshold value of the disease persistence and extinction in this model, which finally proves the existence of a unique ergodic stationary distribution of the new model. Here, we model the random by a continuous-time Markov chain { r ( t ), t ≥ 0} in a finite state space S = { 1 , 2, . . . , N } with generator = (γ i j ) N×N given by

Assume Markov chain r ( t ) is irreducible and independent of Brownian motion. Therefore, there exists a unique stationary distribution π = { π 1 , π 2 , . . . , π N } of r ( t ) which satisfies π = 0, N i=1 π i = 1 and π i > 0, ∀ i ∈ S . Then we can further obtain the following model with regime switching:

This article formulates a stochastic SIRS epidemic model with regime switching and the infectious forces under intervention strategies. Our main idea is to investigate the effect of the white and telegraph noises on the spread dynamics of the disease in the host population based on realistic parameters obtained from previous literatures. The rest of this article is organized as follows: In Section 2 , we present the main results of system (1.4) . In Sections 3 and 4 , we present the detailed proof of the theoretical results. In Section 5 , we perform numerical simulations to verify our analytical results based on realistic parameter values which are mainly taken from the work in [61] . Finally, we provide a brief discussion and summary of main results. As the proof of our result is based on the theory of integral Markov semigroups, for the convenience of the reader, in the appendix we present some auxiliary results concerning Markov semigroups.

Throughout this paper, unless otherwise specified, we let ( , F , { F t } t≥0 , P ) be a complete probability space with a filtration { F t } t≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets).

To investigate the dynamical behavior of stochastic model (1.4) , the first concerning thing is whether the solution is global and positive.

There exists a unique positive solution ( S t , I t , R t ) of model (1.4) on t ≥ 0 with any given initial value S 0 , I 0 , R 0 , r(0) ∈ × S , and the solution will remain in with probability 1.

In studying epidemic models, the most interesting and important issues are usually to establish the threshold condition for the extinction and persistence of the disease. Frequently, the threshold behavior for many epidemic models are defined by the basic reproduction number R 0 which is defined as the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during its entire period of infectiousness [20] . It is a common case (including models (1.1) and (1.2) ) that a disease dies out if R 0 is less than unity and the disease is established in the population if R 0 is greater than unity. Naturally, we wish to examine the dynamic behavior of model (1.4) and then we may ask what is the corresponding basic reproduction number R s 0 ? First of all, we define the basic reproduction number for the SDE model (1.4) as follows:

then the disease I t tends to zero exponentially.

Since stationary distribution of stochastic system means that all the individuals can be coexistent and persistent in the long term, many authors proved the existence of a unique ergodic stationary distribution (see [28, 29, 46, 50, 51] and the references cited therein). It is worthy to note that in those literatures the proofs depend heavily on the uniform ellipticity condition. However, in this paper the diffusion matrix of model (1.4) is given by

Obviously, A i does not satisfy the uniform ellipticity condition for it is degenerate. Hence, in this paper, we employ Markov semigroup theory to obtain a stable stationary distribution. .2) , one can find that the basic reproduction number R s 0 is also an extension of R . In particular, in the case N = 1 , then R s 0 becomes to R .

Remark 2.5. The proof of Theorem 2.1 is very similar to Theorem 4.3 of Cai et al. [15] and is omitted. Hence, in the rest of this paper, we are devoted to the proof of Theorems 2.2 and 2.3 .

In this section, we prove Theorem 2.2 about the extinction of disease for the stochastic model (1.4) .

o 's formula, one can see that

Substituting this inequality into Eq. (3.1) , one can see that

Integrating both sides from 0 to t and dividing by t , one can see that

Note that G t is a local martingales, whose quadratic variation 

Now we consider the case that σ 2

By the similar arguments, we can conclude that The proof is complete.

In this section, we will adopt Lemmas A.1 and A.2 (see the Appendix A ) to prove Theorem 2.3 . To begin with, we establish an partially integral Markov semigroup connected with model (1.4) .

Throughout this section, if B is a vector or matrix, we use B to denote its transpose.

Denote by A i the differential operators

where X is the state space of the SDE model (4.1) , is the σ -algebra of Borel subsets of X , m is the Lebesgue measure on (X , ) and

Next, we show that, for any i ∈ S , the operator A i generates an integral Markov semigroup {T i (t ) } t≥0 on the space L 1 (X , , m) and Let ( S t , I t , R t ) be the unique solution of system (1.4) with S 0 , I 0 , R 0 , r(0) ∈ X × S , then ( S t , I t , R t , r ( t )) constitutes a Markov process on X × S . In view of Section 5 in [57] , for t > 0 the distribution of the process ( S t , I t , R t , r ( t )) is absolutely continuous, denote by u = (u 1 , u 2 , . . . , u N x , y , z , i ) ) the density distribution of the process ( S t , I t , R t , r ( t )). Then the vector u satisfies the following master equation:

where A u = (A 1 u 1 , A 2 

and Q is a Markov operator on L 1 (Y , , ˆ m ) . From the Philips perturbation theorem [58] , Eq. (4.2) with the initial condition u(0, S, I , R, k) = g(S, I , R, k) generates a continuous semigroup of Markov operators { P ( t )} t ≥ 0 on the space L 1 (Y ) given by

The semigroup { P ( t )} t ≥ 0 satisfies the integral equation This completes the proof.

Proof. If the semigroup { P ( t )} t ≥ 0 has an invariant density g * , then from Lemma 4.3 it follows that g * > 0 almost everywhere. If a Markov semigroup has two different invariant densities then it has two invariant densities with disjoint supports, which is impossible in our case.

Thus the semigroup { P ( t )} t ≥ 0 has at most one invariant density. Fix t > 0 and ( S 0 , I 0 , R 0 ) ∈ . Since

where k i ( t , S , I , R ; S 0 , I 0 , R 0 ) is a continuous kernel of the Markov semigroup {T i (t ) } t≥0 , there exist an ( S 0 , I 0 , R 0 ) ∈ and a λ > 0 such that

From continuity of the kernel k i it follows that there exists an such that k i ( t , S , I , R ; 

Proof. We will construct a nonnegative C 2 -function V and a closed set U ∈ (which lies entirely in ) such that for any i ∈ S ,

In fact, A is the adjoint operator of the infinitesimal generator of the semigroup { P ( t )} t ≥ 0 . From the irreducibility of the generator matrix , one can get that for ˜ R = (R 1 , R 2 , . . . , R N ) , there exists = ( 1 , 2 , . . . , N ) satisfying the following Poisson system (see [59] , Lemma 2.3)

where 1 = (1 , 1 , . . . , 1) . Therefore, we have

It is easy to see that H ( S , I , R , i ) has a unique minimum point in and its minimum value is denoted by M 1 . We define a nonnegative C 2 -function V as follows:

Define a closed set

where 0 < ε < 1 is a sufficiently small constant such that

For convenience, we divide ࢨD ε into five domains 

Next, we will prove LV (S, I , R) ≤ −1 for any ( S , I , R ) ∈ ࢨD ε . Case 1. On domain D 1 , we get

On domain D 2 , one can see that

A V ≤ 1 M β i I + M 1 σ 2 i 2 + 2 2 β i i μ i + 2 2 σ 2 i 2 i μ 2 i I 2 − i S − γ i I R − δ i S ˜ N − l − δ i I L −˜ N − 2 ≤ 1 M β i i μ i + M 1 σ 2 i 2 + 2 2 β i i μ i + 2 2 σ 2 i 2 i μ 2 i 2 i μ 2 i − γ i ε ε 2 − 2 ≤ −1 .

A V ≤ 1 M β i I + M 1 σ 2 i 2 + 2 2 β i i μ i + 2 2 σ 2 i 2 i μ 2 i I 2 − i S − γ i I R − δ i S ˜ N − l − δ i I L −˜ N − 2 ≤ 1 M β i i μ i + M 1 σ 2 i 2 + 2 2 β i i μ i + 2 2 σ 2 i 2 i μ 2 i 2 i μ 2 i − δ i ε ε 2 − 2 ≤ −1 ,

Consequently A V (S, I , R) ≤ −1 , for ∀ (S, I , R) ∈ \ D ε . By using similar arguments to those in [55] , the existence of a Khasminski ȋ function implies that the semigroup is not sweeping from the set D ε . According to Lemma 4.4 , the semigroup { P ( t )} t ≥ 0 is asymptotically stable, which completes the proof.

In this section we provide numerical simulation results to substantiate the analytical findings for the stochastic model reported in the previous sections. Using the Milsteins Higher Order Method mentioned in Higham [60] , we obtain the following discretization equations , Table 5 .1 for details). In addition, we always assume that the initial value of system (1.4) is (S 0 , I 0 , R 0 , r(0)) = (50, 1 , 0, 1) ∈ × S . We then divide our simulations into two cases: Fig. 5 .2 , for first two cases(i.e., R s 0 > 1 ), both appear skew to the right and the skewness becomes larger as the intensities σ (1), σ (2) of the white noise increases; in the last case ( R s 0 < 1 ), the PDF of the distribution of I ( t ) is concentrated on the small neighborhood of zero. Here we can conclude that the small environmental perturbations can generate the irregular cycling phenomena of recurrent diseases, while the large ones will eradicate diseases. This means the small perturbations of the white noise can sustain the irregular recurrence of the disease (such as influenza, SARS) in humans between two pandemics, and larger ones may be beneficial, leading to the extinction of the disease. In order to further illustrate the effect of the random switching on the transmission dynamics of the disease, we carry out another case. In Fig. 5 .3 represent the sample paths of I of state 1, state 2 and hybrid system. It is clear that the path of hybrid system is between the paths of state 1 and state 2, that is, the hybrid system is stochastically permanent if the two subsystems are stochastically 

This article formulates a stochastic SIRS epidemic model with regime switching and the infectious forces under intervention strategies and applied our theoretical results to the inter-pandemic evolution dynamic of the disease in the host population based on realistic parameter values obtained from previous literatures. Combining analytical results with numerical simulations, a threshold dynamic determined by the basic reproduction number Compared with previous researches, the main contributions of the present study are as follows:

• We can see that SDE (1.4) is an extension of Eq. (1.2) and the basic reproduction number R s 0 is an extension of R . Particularly, the SDE (1.4) becomes to Eq. (1.2) , and R s 0 becomes to R when N = 1 . Mathematical deduction in present paper faces greater challenges than Cai et al. [15] . Moreover, we found the interesting fact: if one of the subsystems is extinctive, another is stochastically permanent, then the hybrid system may be stochastically permanent if the Markov chain spends enough time on the stochastically permanent state. From biological point of view, the Markov chain r ( t ) is conductive to the survival of the disease. That is to say, the Markov chain can suppress the extinction of the disease. Hence, the telegraph noise is an important factor, it has a direct effect on the evolution of the epidemic (see Figs. 5.3 and 5.4 ).

• Compared with results in Liu et al. [42] , Han et al. [43] for a stochastic epidemic model with the fluctuation by both white and telegraph noises in the transmission rate of disease, the advantages in this paper lie in: (1) the existence of endemic stationary distribution is proved in the case of persistence; (2) the effect of the white and telegraph noises on the epidemiological consequences is studied based on realistic parameter values. • The effect of the white noise was not studied in Li et al. [53] . The present paper shows that white noise play a very important role in determining persistence or extinction of diseases in the transmission of infectious disease. And it also reveals an interesting result: the small perturbations of the white noise can sustain the irregular recurrence of the disease in the host population during two pandemics, and larger ones may be beneficial, leading to the extinction of the disease (see Figs. 5.1 and 5.2 ).

Some interesting topics deserve further investigation. In this paper, the threshold for stochastic model (1.2) under consideration is determined merely when the intensities of white noise is relatively small. In fact, we also attempts to study the problem which if R s 0 < 1 and without any other condition, the disease I t tends to zero exponentially. Unfortunately, there are some technical obstacles that cannot be overcome at present stage. On the other hand, one may propose some more realistic but complex models, such as considering the impact of the number of hospital beds [62] on system (1.4) . We leave these investigations for future research.

This work was supported by Fujian provincial Natural science of China ( 2018J01418 ).

As the proof of our result is based on the theory of integral Markov semigroups, we need some auxiliary definitions and results concerning Markov semigroups [54] [55] [56] [57] . For the convenience of the reader, we present these definitions and results in the appendix. for almost all y ∈ X , then P g(x) = X k(x, y) g(y) m(dy) is an integral Markov operator. The function k is called a kernel of the Markov operator P. for each density g .

Definition A.5 (Asymptotically stable) . For a Markov semigroup { P ( t )} t ≥ 0 , a density g * is called invariant, if P (t ) g * = g * for t > 0. The Markov semigroup is called asymptotically stable if there is an invariant density g * such that lim t→∞ P (t ) g − g * = 0 for g ∈ D .

Remark A.1. If the Markov semigroup { P ( t )} t ≥ 0 is formed by a differential equation (e.g. model (1.2) ), then the asymptotic stability of Markov semigroup implies that all of solutions of the equation starting from a density converge to the invariant density. 

Contributions to the mathematical theory of epidemics -I

Dynamic models of infectious diseases as regulators of population sizes

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models

A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China

Disease spreading on populations structured by groups

Pattern transitions in spatial epidemics: mechanisms and emergent properties

Patch invasion in a spatial epidemic model

Analysis of transmission dynamics for zika virus on networks

Coupling dynamics of epidemic spreading and information diffusion on complex networks

Coupled disease-behavior dynamics on complex networks: a review

A vector-borne contamination model to assess food-borne outbreak intervention strategies

Hemorrhagic fever with renal syndrome in china: Mechanisms on two distinct annual peaks and control measures

Transmission dynamics of cholera: mathematical modeling and control strategies

Stability and Complexity in Model Ecosystems

A stochastic SIRS epidemic model with infectious force under intervention strategies

The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate

Campus quarantine (Fengxiao) for curbing emergent infectious diseases: lessons from mitigating a/h1n1 in Xian

A generalization of the Kermack-Mckendrick deterministic epidemic model

Global analysis of an epidemic model with nonmonotone incidence rate

Basic reproduction numbers for reaction-diffusion epidemic models

Persistence in stochastic food web models

Environmental variability in a stochastic epidemic model

Dynamics of a stochastic multigroup SIQR epidemic model with standard incidence rates

Global analysis of a vector-host epidemic model in stochastic environments

Global analysis of a stochastic TB model with vaccination and treatment

Threshold behavior in a stochastic SIS epidemic model with standard incidence

Periodic behavior in a FIV model with seasonality as well as environment fluctuations

A stochastic SIRS epidemic model with nonlinear incidence rate

Stationary distribution and extinction of a stochastic dengue epidemic model

Stochastic dynamics of feline immunodeficiency virus within cat populations

Global threshold dynamics of a stochastic epidemic model incorporating media coverage

Virus survival as a seasonal factor in influenza and poliomyelitis

Indirect health effects of relative humidity in indoor environments

Virus survival in the environment with special attention to survival in sewage droplets and other environmental media of fecal or respiratory origin, in: Report for the World Health Organization

Absolute humidity modulates influenza survival, transmission, and seasonality

Structural explanation for the effect of humidity on persistence of airborne virus: Seasonality of influenza

Limate change and vector-borne diseases

Climate change and infectious diseases in europe

An Introduction to Stochastic Processes with Applications to Biology

Persistence and ergodicity of a stochastic single species model with Allee effect under regime switching

The dynamics of a population in a Markovian environment

Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching

Stochastic SIRS model under regime switching

Long-time behavior of a regime-switching susceptible-infective epidemic model with degenerate diffusion

Asymptotic behavior of a regime-switching SIR epidemic model with degenerate diffusion

Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching

Dynamics of a stochastic regime-switching predator-prey model with harvesting and distributed delays

Persistence and extinction of a stochastic delay logistic equation under regime switching

A stochastic ratio-dependent predator-prey model under regime switching

The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching

Persistence and ergodicity of a stochastic single species model with allee effect under regime switching

Average break-even concentration in a simple chemostat model with telegraph noise

Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Long-time behaviour of a stochastic prey-predator model

Stability of Markov semigroups and applications to parabolic systems

Continuous Markov semigroups and stability of transport equations

On asymptotic stability and sweeping for Markov operators

Stability of regime-switching diffusions

An algorithmic introduction to numerical simulation of stochastic differential equations

Population biology of infectious diseases: part I

Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds