Susceptible-infected-susceptible Epidemic Model Research Articles

On an SIS epidemic model with power‐like nonlinear incidence and with/without cross‐diffusion

AbstractWe study global existence, boundedness, and convergence of nonnegative classical solutions to a Neumann initial‐boundary value problem for the following possibly cross‐diffusive SIS (susceptible–infected–susceptible) epidemic model with power‐like infection mechanism generalizing the standard mass action incidence: in a bounded smooth domain . The infection force of the form with is a natural extension of the classical mass action type , and the cross‐diffusive term with describes the effect that susceptible individuals tend to move away from higher density of infected populations. Global existence and boundedness of classical solutions are established in certain parameter ranges, and threshold/nonthreshold long‐time behaviors of global bounded solutions are also detected. Our findings significantly improve and extend previous related studies.

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.

Leader‐following consensus and qualitative analysis of a new multi‐agent‐based epidemic model

AbstractThis paper addresses an investigation into a leader‐following consensus problem in a novel multi‐agent‐based SIS (Susceptible‐Infected‐Susceptible) epidemic model. As infectious diseases are easily spread through travel, the model considers each agent to represent a city in an infected country, interacting with neighbouring cities towards achieving disease eradication through consensus. To make the model more realistic, the Effect Terms (ETs) arising from the effects of agent states on each other are also considered. The differential equations' positivity and existence of solutions are examined, and the proposed model's equilibrium points are obtained, with stability analysed using new local and global basic reproduction numbers defined in this study. Additionally, a leader‐following consensus control for the multi‐agent‐based system is proposed to drive each agent to follow a virtual leader agent. Finally, numerical simulations are included to validate the theoretical findings corresponding to the proposed scheme.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with Neumann boundary condition

This article studies a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition in spatially heterogeneous environment, where the spatial movement of individuals is described by dσm∫ΩJσ(x−y)[u(y)−u(x)]dy,σ is the scaling of the size of the support of dispersal kernel J, and m represents the cost parameter. We firstly derive the basic reproduction number R0 for this model, and show that 1−R0 has the same sign as the generalised principal eigenvalue of a linear nonlocal operator. We also investigate asymptotic profiles of R0 with respect to σ,m and diffusion rates. Next, the existence and uniqueness of endemic equilibrium state of this model is obtained as R0>1. Finally, we consider the impact of small dispersal spread on the persistence criteria for disease. Particularly, our results illustrate that, even though ∫Ωβ(x)dx>∫Ωγ(x)dx, there exists the possibility of disease control only when m∈[0,2)andσ→0+.

Long time numerical properties analysis of a diffusive SIS epidemic model under a linear external source

This paper deals with the numerical properties of a reaction-diffusion susceptible infected susceptible (SIS) epidemic model under a linear external source. A numerical scheme is constructed with a finite difference scheme for the space discretization and an Implicit-Explicit (IMEX) method in time integration. A threshold value, numerical basic reproduction number, is proposed in the long-time stability analysis of numerical solutions. Differently from previous works on the same model, the numerical basic reproduction number can preserve the behaviors of the basic reproduction number of the model, towards which it converges when the spatial stepsize vanishes. Moreover, it plays a role for the discrete dynamics similar to the one played by its continuous counterpart. Some numerical experiments are given in the end to confirm the conclusions and detect the conjecture on the stability of endemic equilibrium (EE) in general case.

Loss and premium calculation of network nodes under the spread of SIS virus

In the big data and “Internet+” era, the research related cybersecurity risk has attracted much attention. However, Premium pricing for cybersecurity insurance remains in its early days. In this paper, we established a premium pricing method for cybersecurity risks. Firstly, the losses during the cyber infection is modeled by an interacting Markov SIS (Susceptible-Infected-Susceptible) epidemic model. we also proposed a premium simulation method called the Gillespie algorithm, which can be used for simulation of a continuous-time stochastic process. At last, as an example, we calculated the premiums by using premium principles and simulation in a simple network respectively. The numerical case studies demonstrate the premium pricing model performs well, and the premiums based on simulations are rather conservative, and recommended using in practice by comparing the results of premiums.

Bifurcation analysis of an SIS epidemic model with a generalized non-monotonic and saturated incidence rate

In this paper, a new generalized non-monotonic and saturated incidence rate was introduced into a susceptible-infected-susceptible (SIS) epidemic model to account for inhibitory effect and crowding effect. The dynamic properties of the model were studied by qualitative theory and bifurcation theory. It is shown that when the influence of psychological factors is large, the model has only disease-free equilibrium point, and this disease-free equilibrium point is globally asymptotically stable; when the influence of psychological factors is small, for some parameter conditions, the model has a unique endemic equilibrium point, which is a cusp point of co-dimension two, and for other parameter conditions the model has two endemic equilibrium points, one of which could be weak focus or center. In addition, the results of the model undergoing saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation as the parameters vary were also proved. These results shed light on the impact of psychological behavior of susceptible people on the disease transmission.

A pedestrian evacuation model based on ant colony algorithm considering dynamic panic spread

<p style='text-indent:20px;'>To explain the evacuation behavior characteristics of people in panic, an improved cellular automata pedestrian evacuation model is proposed by combining with the SIS (Susceptible-Infected-Susceptible) epidemic model and ant colony algorithm. First, to explain the spread of panic in dynamic pedestrian evacuation, an improved SIS infection model that considers individual movement is proposed. At the same time, a quantitative formula for panic mood is given. The infection critical value of the SIS model is analyzed by mean field theory. Secondly, the ant colony algorithm is introduced based on considering exit distance, obstacles, panic, and other factors. These factors are mapped with heuristic function and pheromone concentration in the ant colony algorithm. Finally, the evacuation model and parameters such as evacuation time, pedestrian density, and panic spread are analyzed and discussed through simulation experiments. The simulation experiment results show that the evacuation model proposed in this research can better reflect the actual situation. Simultaneously, the critical rate of effective propagation is closely related to the crowd density in the evacuation area, and individual movement has a significant impact on the panic transmission behavior.</p>

An SIS epidemic model with mass action infection mechanism in a patchy environment

AbstractIn this paper, we study an SIS (susceptible–infected–susceptible) epidemic model with mass action infection mechanism in a patchy environment. We first analyze the long‐time dynamics of the model in terms of the basic reproduction number , and prove under certain conditions that the disease‐free equilibrium (DFE) is globally attractive if whereas the endemic equilibrium (EE) is globally attractive if . Then we establish the existence and uniqueness of EE provided , which is achieved, among other ingredients, by reducing the equilibrium problem into the one involving only the component of infected population. We further investigate the asymptotic profile of EE as the motility rate of the susceptible and/or infected population becomes small. We also perform the theoretical analysis on the corresponding model with linear birth/death effect. By comparing the findings of the models considered here with those of other closely related ones, our results suggest that the variation of total population, individual motility, infection mechanism, and spatial heterogeneity all can play significant roles in the dynamics of disease transmission, which may shed light on the practical implications for predicting the patterns of disease outbreak and designing optimal control strategies.

Dynamics and asymptotic profiles of a nonlocal dispersal SIS epidemic model with bilinear incidence and Neumann boundary conditions

This paper is concerned with a nonlocal (convolution) dispersal susceptible-infected-susceptible (SIS) epidemic model with bilinear incidence and Neumann boundary conditions. First we establish the existence and uniqueness of stationary solutions by reducing the system to a single equation. Then we study the asymptotic profiles of the endemic steady states for large and small diffusion rates to illustrate the persistence or extinction of the infectious disease. The lack of regularity of the endemic steady state makes it more difficult to obtain the limit function of the sequence of endemic steady states. We also observe the concentration phenomenon which occurs when the diffusion rate of the infected individuals tends to zero. Our analytical results demonstrate that limiting the movement of susceptible individuals is not effective in eliminating the infectious disease unless the total population size is relatively small.

A note about the invariance of the basic reproduction number for stochastically perturbed SIS models

AbstractIn Gray et al. [A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 (3) (2011) 876–902] a susceptible‐infected‐susceptible (SIS) stochastic differential equation (SDE), obtained via a suitable random perturbation of the disease transmission coefficient in the classic SIS model, has been studied. Such random perturbation enters via an informal manipulation of stochastic differentials and leads to an Itô's‐type SDE. The authors identify a stochastic reproduction number, which differs from the standard one for the presence of those additional parameters that describe the employed random perturbation, and show that, similarly to the deterministic case, the stochastic reproduction number rules the asymptotic behavior of the solution. Aiming to make that random perturbation rigorous, we suggest an alternative approach based on a Wong–Zakai approximation argument thus arriving at a different stochastic model corresponding to the Stratonovich version of the Itô equation analyzed in Gray et al. Rather surprisingly, the asymptotic behavior of this alternative model turns out to be governed by the same reproduction number as the deterministic SIS equation. In other words, the random perturbation does not modify the threshold for extinction and persistence of the disease.

Revealing the relationship of topics popularity and bursty human activity patterns in social temporal networks

In social networks, how human activity patterns affect the popularity of topics has always been the focus of research. In this paper, a quantitative temporal analysis of the dynamics of topics popularity in Sina Weibo system was provided. Firstly, the popularity time series of 1167 topics were clustered into four clusters by K-Spectral Centroid (K-SC) clustering algorithm. Secondly, for each cluster, we calculated the exponents of topic popularity decay distribution α and the exponents of inter-activity time distribution β, respectively. Two interesting results were found: one is that the peak fraction F of topics popularity positively correlated with the topics popularity decay exponent α; the other is that bursty activity patterns in social network significantly affect topics popularity dynamics: there is a positive correlation between exponent α and exponent β. Finally, we proposed an extended SI (susceptible–infected) epidemic model with incorporate bursty human activity and verified the results by simulation.

Bifurcation and stability of a delayed SIS epidemic model with saturated incidence and treatment rates in heterogeneous networks

In this paper, to characterize the limited availability of medical resources, we incorporate a saturated treatment rate into a network-based susceptible-infected-susceptible (SIS) epidemic model with time delay and nonlinear incidence rate. Analytical study shows the boundedness of solutions, the basic reproduction number R0 and equilibrium points of the proposed system. For any infection delay, we perform both local and global stability analyses for the disease-free equilibrium point by analyzing the characteristic equation and using Lyapunov functional. Furthermore, this system exhibits bifurcation behavior at R0=1 due to the introduction of saturated treatment. More precisely, a backward bifurcation takes place from the disease-free equilibrium point when the saturation constant β is sufficiently large. Under the given conditions, the unique disease-spreading equilibrium point is also proved to be locally asymptotically stable. In addition, we analyze an optimal control problem with consideration of two time-dependent control measures. Several numerical simulations are presented to validate the obtained theoretical results.

Dynamical analysis of the SIS epidemic model in cluster events

Considering individuals’ aggregation behavior, we study the SIS (Susceptible-Infected-Susceptible) epidemic model in a modified activity driven network (ADN). The relations among individuals are divided into three categories: strong links (SLs), the first kind of weak links (FWLs) and the second kind of weak links (SWLs). Based on the mean field approximation, we theoretically calculate the epidemic threshold condition while the strong links structure is a homogeneous or a heterogeneous network, respectively. Finally, we provide some numerical examples that support our theoretical results.

SIS Epidemic Model Birth-and-Death Markov Chain Approach

We are interested in describing the dynamics of the infected size of the SIS Epidemic model using the Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size M; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter α; the disease spreads within the population is modelled by a Poisson process with a rate λ_I. λ_I=βI(1-I/M)  is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number (R=β/α) is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows that the infected size dynamics converge to a stationary stochastic process. And the analytical results determine the distribution of the stationary stochastic process as a normal distribution with mean (1-1/R)M and Variance M/R when M becomes larger.

Transmission dynamics, global stability and control strategies of a modified SIS epidemic model on complex networks with an infective medium

In this paper, a new SIS (susceptible–infected–susceptible) epidemic model on complex networks with an infective medium is proposed. The infection rate is modified by applying the theory of probability. By using mean-field approximation, iterative analysis method and mathematical analysis, the epidemic threshold, stabilities of the disease-free equilibrium and the endemic equilibrium are studied. Moreover, some disease control strategies are proposed. The results show that the epidemic threshold plays an important role in the spread of disease. Finally, the theoretical results are confirmed by some simple numerical examples.

Stability analysis and control strategies for a new SIS epidemic model in heterogeneous networks

Abstract In this research, a new SIS (susceptible-infected-susceptible) epidemic model is proposed. Unlike most other models, our model considers the infection rate of multiple edges interfering with one another in complex networks. The stability of the disease-free equilibrium, the basic reproduction number R0, the uniqueness and the global stability of the endemic equilibrium are studied by using the concepts of next-generation matrix, reduction to absurdity, induction and piecewise continuous Lyapunov function. Moreover, some disease control strategies are given. Finally, some numerical simulations are given to confirm the theoretical analysis.

Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment

In the recent paper [ 29 ], a susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model with a mass action infection mechanism and linear birth-death growth with no flux boundary condition was studied. It has been recognized that spontaneous infection is an important factor in disease epidemics, in addition to disease transmission [ 43 ]. In this paper, we investigate the SIS model in [ 29 ] with spontaneous infection. We establish the global boundedness and uniform persistence in the general heterogeneous environment, and derive the global stability of the unique constant endemic equilibrium in the homogeneous environment case. Moreover, we analyze the asymptotic behavior of the endemic equilibrium when the movement (migration) rate of the susceptible or infected population tends to zero. Compared to the case that there is no spontaneous infection, our study suggests that spontaneous infection can enhance persistence of infectious disease, and hence the disease becomes more threatening.

Analysis of a Stochastic Susceptible-Infective Epidemic Model in a Polluted Atmospheric Environment

Atmospheric pollution is deteriorating, which has affected the evolution of respiratory disease for the exposed human worldwide. Thus, exploring the influence of air pollution on the evolution of disease transmission dynamics is a significant issue. In this article, a stochastic susceptible-infective (SI) epidemic model in a polluted atmospheric environment is investigated. The existence and uniqueness of the global positive solution are established. In virtue of the aggregation methods and Lyapunov function, the sufficient conditions of disease extinction, persistence, and existence of the stationary distribution are established, respectively. TakingPM2.5concentration as the air pollutant index, numerical simulations are carried out to support these results. Our results indicated that the disease transmission dynamics (extinction, persistence, and stationary distribution) are significantly associated with the environmental atmospheric pollution and fluctuation.

Dynamical analysis of a discrete-time SIS epidemic model on complex networks

In this letter, a discrete SIS (susceptible–infected–susceptible) epidemic model on complex networks is presented. Firstly, the non-negativity and the boundedness of solutions are studied. Secondly, the basic reproduction number R0 is calculated. Thirdly, applying the Lyapunov direct method of difference equations, the global asymptotic stability of disease free equilibrium is investigated. Finally, there give some simulations.