Dynamical behavior of a stochastic SEIQRV infectious model with an Ornstein-Uhlenbeck process and general incidence

In this paper, we construct a backward difference scheme for a class of SIR epidemic model with general incidence f . The step sizeτ used in our discretization is one. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, the general incidence function f must satisfy certain assumptions, under which, we establish the global stability of endemic equilibrium when R0 >1. The global stability of diseases-free equilibrium is also established when R0 ≤1. In addition we present numerical results of the continuous and discrete model of the different class according to the value of basic reproduction number R0.

In daily lives, when emergencies occur, rumors will spread widely on the internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens’ behavior and attitude, which make the transmission rates of these information among social network groups be not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation. Firstly, increasing noise intensity can effectively reduce rumor propagation when . That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens’ opinions, thus reducing the degree of rumor propagation. Secondly, speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen (i.e., increase the value of β and decrease the value of α and γ), which can effectively curb the rumor propagation.

Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations

In this study, a cholera infection model with a bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at an arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to a relevant linearized system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show the impact of regression speed and fluctuation intensity on stochastic systems.

Dynamical behaviors of a stochastic HTLV-I infection model with general infection form and Ornstein–Uhlenbeck process

Stationary distribution, extinction and probability density function of a stochastic SEIV epidemic model with general incidence and Ornstein–Uhlenbeck process

<abstract><p>In this paper, we propose a novel stochastic SEIQ model of a disease with the general incidence rate and temporary immunity. We first investigate the existence and uniqueness of a global positive solution for the model by constructing a suitable Lyapunov function. Then, we discuss the extinction of the SEIQ epidemic model. Furthermore, a stationary distribution for the model is obtained and the ergodic holds by using the method of Khasminskii. Finally, the theoretical results are verified by some numerical simulations. The simulation results show that the noise intensity has a strong influence on the epidemic spreading.</p></abstract>

Analysis of a reaction–diffusion HCV model with general cell-to-cell incidence function incorporating B cell activation and cure rate

In this paper, we investigate a mathematical model of malaria transmission dynamics with maturation delay of a vector population in a periodic environment. The incidence rate between vector and human hosts is modeled by a general nonlinear incidence function which satisfies a set of conditions. Thus, the model is formulated as a system of retarded functional differential equations. Furthermore, through dynamical systems theory, we rigorously analyze the global behavior of the model. Therefore, we prove that the basic reproduction number of the model denoted by R0 is the threshold between the uniform persistence and the extinction of malaria virus transmission. More precisely, we show that if R0 is less than unity, then the disease-free periodic solution is globally asymptotically stable. Otherwise, the system exhibits at least one positive periodic solution if R0 is greater than unity. Finally, we perform some numerical simulations to illustrate our mathematical results and to analyze the impact of the delay on the disease transmission.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Compartmental systems like the well‐famed SIR, SEIR, SQIR, SVIR, and their variants are efficient tools for the mathematical modeling of infectious illnesses, and they permit us to get a clear picture of how they proliferate. In actuality, the aleatory fluctuations factors present in the natural environment like storm surges, weather changes, and seismic tremors make the dissemination of epidemics susceptible to some randomness. This calls for a stronger mathematical formulation that takes into consideration this stochasticity effect. From this perspective, and in order to highlight in the same time the effect of the vaccination strategy, we survey in this paper an SVIR model with general incidence rates that is disturbed by both Brownian motions and Lévy jumps. Initially, we establish its well‐posedness in the sense that it has a unique positive and global‐in‐time solution. Then, we rely on some assumptions and nonstandard analytic techniques, to derive sufficient and almost necessary conditions for extinction, persistence in the mean, and also weak persistence. More explicitly, we identify firstly under some hypotheses a threshold between extinction and persistence in the mean. In other phrases, if , the infected population dies out while it persists in the mean when . Then, and by modifying the hypothetical framework in order to cover more incidence rates, we prove that can act also as a threshold between extinction and weak persistence. At last, we provide some numerical simulations to corroborate our findings and cover some particular cases of response functions.

Cholera is a global epidemic infectious disease that seriously endangers human life. It is disturbed by random factors in the process of transmission. Therefore, in this paper, a class of stochastic SIRB cholera model with Ornstein–Uhlenbeck process is established. On the basis of verifying that the model exists a unique global solution to any initial value, a sufficient criterion for the existence of a stationary distribution of the positive solution of the random model is established by constructing an appropriate random Lyapunov function. Furthermore, under the same condition that there is a stationary distribution, the specific expression of the probability density function of the random model around the positive equilibrium point is calculated. Finally, the theoretical results are verified by numerical model.

Dynamical analysis of a stochastic epidemic HBV model with log-normal Ornstein–Uhlenbeck process and vertical transmission term

In this paper, we investigate the global dynamics of a Two-Strain Disease Model with a general nonlinear incidence rate and treatment term. The two basic reproduction numbers, [Formula: see text] associated to the drug-susceptible strain and [Formula: see text] associated to the drug resistant strain, are derived using the next generation matrix method. We give a complete study of the existence and global stability of the steady states for the model so that we discuss four cases: the trivial steady state [Formula: see text], the endemic equilibrium [Formula: see text] regarding the drug-susceptible strain, the endemic equilibrium [Formula: see text] concerning the drug resistant strain, and the coexistence equilibrium [Formula: see text] relative to both drug-susceptible and drug-resistant strains. By using Lyapunov theory, we prove the global stability of the equilibria under some hypothesis on the basic reproduction numbers and the general incidence rate. Moreover, we prove that epidemics occur regarding any strain when the associated basic reproduction number remains above one. We also study the sensitivity of parameters that influence the basic reproduction number [Formula: see text]. Finally, to illustrate our analytical results, we give some numerical simulations using the ode45 Matlab routine.

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.