SIS Epidemic Model Research Articles

Stationary Distribution and Density Function for a High-Dimensional Stochastic SIS Epidemic Model with Mean-Reverting Stochastic Process

This paper explores a high-dimensional stochastic SIS epidemic model characterized by a mean-reverting, stochastic process. Firstly, we establish the existence and uniqueness of a global solution to the stochastic system. Additionally, by constructing a series of appropriate Lyapunov functions, we confirm the presence of a stationary distribution of the solution under R0s>1. Taking 3D as an example, we analyze the local stability of the endemic equilibrium in the stochastic SIS epidemic model. We introduce a quasi-endemic equilibrium associated with the endemic equilibrium of the deterministic system. The exact probability density function around the quasi-stable equilibrium is determined by solving the corresponding Fokker–Planck equation. Finally, we conduct several numerical simulations and parameter analyses to demonstrate the theoretical findings and elucidate the impact of stochastic perturbations on disease transmission.

Intelligent Adaptive Networks for Chaos Analysis in Stochastic SIS Differential Epidemic Model with the Impact of Unpredictable Environmental Factors

In this study, an intelligent computing framework is presented to explore the influence of unpredictable environmental factors on the spread of infections. A stochastic non-linear SIS epidemic model (SISEM) is examined that incorporates a non-linear incidence rate, and impact of random fluctuations on the disease transmission. The knacks of artificial neural networks Levenberg–Marquardt (ANNLMA) technique are exploited to illustrate the behavioral complexities of stochastic non-linear SISEM, which explores the dynamics of disease transmission in a stochastic environment and provides insights into the interactions between susceptible and infectious populations. The Milstein approach of order 1 generates a dataset for the proposed ANNLMA technique by varying integrated parameters, such as the rates of recovery, death, disease transmission, noise standard deviation and the entire population, for various SISEM scenarios. The referenced dataset is randomly divided into test, validation and train sets for the weight learning of the proposed ANNLMA. The viability of the ANNLMA is measured using intensive simulation-based investigations through analysis of mean squared errors, histograms, regression and autocorrelation studies. The investigations highlighted that the ANNLMA results closely match the reference results, illustrating convergence within the 10[Formula: see text] to 10[Formula: see text] range, indicating the validity of the proposed methodology.

Dynamics of a generalized nonlocal dispersion SIS epidemic model

Dynamics of a generalized nonlocal dispersion SIS epidemic model

Dynamics of an age-structured SIS epidemic model with local dispersal and general incidence functions

In this paper, an age-structured SIS epidemic model with local dispersal and general incidence functions is formulated. We first analytically derive the well-posedness of solutions, the basic reproduction number R0 and the existence of steady states. Then, we show that the disease-free steady state is global asymptotically stable if R0≤1 and disease in the model is uniformly persistent if R0>1. Moreover, we show global asymptotic stability of the endemic steady state under some cases. Finally, infection age, media coverage and spatial diffusion on the epidemic of infectious diseases are demonstrated by numerical simulations.

Saturation recovery leads to cusp type Bogdanov–Takens bifurcations of codimensions 3

We reconsider an SIS epidemic model studied by Cui et al. [1]. The model is shown that saturation recovery leads to backward bifurcation, Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. However, for the case when the Bogdanov–Takens bifurcation of codimension 2 is degenerate, the types and codimensions of Bogdanov–Takens bifurcation have not been investigated. In this paper we prove that this same model can undergo cusp type Bogdanov–Takens bifurcations of codimensions 3. Hence, more complex new phenomena, including degenerate Hopf bifurcation, degenerate homoclinic bifurcation and saddle–node bifurcation of limit cycles, exhibit. Furthermore, we get the bifurcation diagram of codimension 3 Bogdanov–Takens bifurcation with cusp type of the SIS epidemic model.

Behavior-induced phase transitions with far from equilibrium patterning in a SIS epidemic model: Global vs non-local feedback

Here, we explore the phase transitions triggered by the implementation of social distancing in a basic spatiotemporal model of a qualitative SIS-type infectious disease. We consider human decisions made based on spatiotemporal information regarding the disease spread. This information can be either local, nonlocal with a finite range, or global in scope.We show that nonlocal and global feedbacks, while resulting in the same spatially homogeneous equilibria, lead to a dynamic behavior that is fundamentally distinct from what is observed when decisions are made based on local information.Various phenomena arise due to the nonlocal nature of the feedback: (i) Instabilization of Otherwise Stable Homogeneous Equilibria; (ii) Nucleation/Invasion Phenomena; (iii) Onset of Standard and Generalized Traveling Waves, which can incur in wave-pinning; iv) in case of Global Information Feedback, onset of locally stable Far From Equilibrium Patterns that coexist with a locally stable disease-elimination equilibrium. Thus, the nonlocal nature of the human behavior-related feedback introduces a rich array of dynamic behaviors and patterns in the system.

Laplace Adomian Decomposition Method for Fractional Order SIS Epidemic Model

This research manuscript addresses finding a solution to the SIS Epidemic model using the Caputo derivative. To achieve the required results, Laplace transform together with the Adomian Decomposition Method (LADM) were utilized. This method is a powerful tool for dealing with various linear and nonlinear problems of fractional-order differential equations (FODEs). Additionally, some results related to the qualitative theory of the model of interest were studied. The computational results showed the verification of the performed analysis. The investigation of the approximate solution computed in the form of an infinite series was carried out. The results were graphically displayed to analyze the adopted procedure for solving nonlinear FODEs using the Caputo fractional derivative.

Untangling the memory and inhibitory effects on SIS-epidemic model with Beddington–DeAngelis infection rate

The dynamical behaviors of an epidemic model based on the susceptible–infected–susceptible (SIS) model are investigated. The Beddington–DeAngelis functional response is used for the infection rate to present the dependence of the transmission of the infection on the ratio of both susceptible and infected populations. A Caputo fractional derivative is applied to show the existence of memory in nature affects population dynamics. The disease-free and endemic equilibrium points are obtained as the equilibrium points that describe the condition when the population is free from the disease or the disease exists in the population throughout time. The existence, uniqueness, non-negativity, and boundedness are proven which state the biological validity of the mathematical model. The local and global stability of each equilibrium point is studied including the basic reproduction number and its influence on the dynamical behaviors. Some numerical simulations are portrayed to explore more about the dynamics of the model which are relevant to the analytical findings. The partial rank correlation coefficient is presented to investigate the dominant parameter with respect to the basic reproduction number and the density of susceptible and infected populations. The parameter continuations are demonstrated to show the impact of infection rate and the inhibitory effect which lead to the occurrence of forward bifurcations. The memory effect is also demonstrated numerically to show the changes in convergence rate due to the changes in memory strength.

Dynamics of a spatiotemporal SIS epidemic model with distinct mobility range

In this study, we consider an epidemic model for susceptible-infected-susceptible populations with nonlocal dffusion, described by convolution operators. Our main objective is to model different diffusion strategies for the susceptible and infected populations by using distinct kernel functions in the convolution operator to model the different movements of individuals for each class. This approach eliminates the simplification of reducing the stationary problem to a single equation, which makes the analysis more challenging. However, we have shown the existence of at least one positive endemic steady state when the basic reproduction number is greater than one, using the index degree theory of Leray-Schauder. Additionally, we investigated the asymptotic profile of the endemic equilibrium state for both, large and small diffusion rates, to demonstrate the persistence or extinction of the disease. Overall, our study indicates that restricting individual movements only partially cannot eliminate the disease unless the total population size is small.

Bifurcation and Stability of an Discrete-time SIS Epidemic Model with Treatment

Mathematical models are useful in examining the effect of an infection on populations. Conditions involving the spread and control of the disease are calculated by analyzing mathematical models, so that it is possible to have information about the behavior of the infection. This article includes the dynamic behavior of a discrete-time SIS epidemic model with treatment. Existence conditions of the fixed points of the model are obtained, and stability analysis is performed for these fixed points. The stability and bifurcation conditions of the obtained endemic fixed point are investigated. Depending on the infection coefficient, the flip bifurcation condition is obtained. At the same time, it is determined in which situation Neimark Sacker bifurcation may occur depending on the step size, and bifurcation is controlled. Rich dynamic behaviors are given to support our theoretical results.

ROLE OF INTERVENTION STRATEGIES AND PSYCHOLOGICAL EFFECT ON THE CONTROL OF INFECTIOUS DISEASES IN THE RANDOM ENVIRONMENT

Infectious disease outbreaks have caused significant losses to populations worldwide in terms of morbidity, social and economic costs. This has prompted governments to seek suitable intervention policies to suppress the outbreaks. This paper aims to explore the intrinsic impact of intervention strategies and the psychological behavior of people on the spread of infectious diseases. Employing the finite state Markov chain, we formulate a new stochastic SIS epidemic model under regime switching by using the Beddington–DeAngelis transmission function, in the presence of some governmental control measures. The study of the model is centered on three different scenarios in terms of varying two control parameters. We analyze the deterministic system for the global dynamics in terms of basic reproductive number. Some new threshold parameters are defined to discuss the extinction and the persistence of the disease in the stochastic setting. Numerical assessments were undertaken to validate theoretical findings for the deterministic and stochastic systems. Our findings strongly suggest that heightened awareness among susceptible individuals contributes significantly to the eradication of infection and fosters a stable epidemiological situation. As levels of public awareness increase, particularly within the susceptible population, a discernible trend toward the eradication of the infection emerges, ultimately ensuring sustained stability within the epidemiological landscape.

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switching environment

Large population asymptotics for a multitype stochastic SIS epidemic model in randomly switching environment

Dynamical Behaviors of Stochastic SIS Epidemic Model with Ornstein–Uhlenbeck Process

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.

Dynamic analysis of a SIS epidemic model with nonlinear incidence and ratio dependent pulse control

Dynamic analysis of a SIS epidemic model with nonlinear incidence and ratio dependent pulse control

CAN INFECTIOUS DISEASES THAT DO NOT CONFER IMMUNITY ERADICATE HOST SPECIES?

In SIS epidemic models with host population dynamics, two classes of incidence functions are identified for their potential to be linked to host extinction. Mildly upper density-dependent incidences are never linked to host extinction. Incidences that are positively homogeneous of degree one are linked to host extinction for certain parameter constellations and to mere host decline for others; a threshold parameter is identified that separates host decline from host extinction.

Multiplicity of Endemic Equilibria for a Diffusive SIS Epidemic Model with Mass-Action

Multiplicity of Endemic Equilibria for a Diffusive SIS Epidemic Model with Mass-Action

Qualitative analysis on a reaction–diffusion SIS epidemic model with nonlinear incidence and Dirichlet boundary

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R0 are studied. That is, when R0<1 the disease-free steady state is globally asymptotically stable, and when R0>1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R0 are established, including the calculating formula of R0, the asymptotic profiles of R0 with respect to diffusion rate dI, and the monotonicity of R0 with diffusion rate dI and domain Ω. In addition, the bang–bang-type configuration optimization of R0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition.

SIS Epidemic Spreading Under Multilayer Population Dispersal in Patchy Environments

We study SIS epidemic spreading models under population dispersal on multi-layer networks. We consider a patchy environment in which each patch comprises individuals belonging to different classes. Individuals disperse to other patches on a multi-layer network in which each layer corresponds to a class. The dispersal on each layer is modeled by a Continuous Time Markov Chain (CTMC). At each time, individuals disperse according to their CTMC and subsequently interact with the local individuals in the patch according to an SIS model. We establish the existence of various equilibria under different parameter regimes and establish their (almost) global asymptotic stability using Lyapunov techniques. We also derive simple conditions that highlight the influence of the multi-layer network on the stability of these equilibria. For this model, we study optimal intervention strategies using a convex optimization framework. Finally, we numerically illustrate the influence of the multi-layer network structure and the effectiveness of the optimal intervention strategies.

Analysis on a SIS epidemic model with diffusion and large sign-changing convection

Analysis on a SIS epidemic model with diffusion and large sign-changing convection