Extinction Of Infectious Disease Research Articles

Stochastic dynamic effects of media coverage and incubation on a distributed delayed epidemic system with Lévy jumps

A stochastic SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic system with Lévy jumps and distributed time delay is established, which is utilized to investigate hybrid dynamic effects of media coverage and incubation on infectious disease transmission. Stochastically ultimate boundedness of the positive solution is discussed. Existence of a unique global positive solution is studied. By constructing appropriate stochastic Lyapunov functions, existence of a unique ergodic stationary distribution is investigated. Sufficient conditions are derived to discuss exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, sufficient conditions for persistence in mean and extinction of infectious disease are investigated. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

Ergodic Stationary Distribution and Threshold Dynamics of a Stochastic Nonautonomous SIAM Epidemic Model with Media Coverage and Markov Chain

A stochastic nonautonomous SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Lévy jumps on infectious disease transmission. The uniform upper bound and lower bound of the positive solution are studied. Based on defining suitable random Lyapunov functions, we researched the existence of a nontrival positive T-periodic solution. Sufficient conditions are derived to discuss the exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, the persistence in the average sense and extinction of infectious disease are investigated using stochastic analysis techniques. Finally, numerical simulations are utilized to provide evidence for the dynamical properties of the stochastic nonautonomous SIAM.

Classifying the level set of principal eigenvalue for time-periodic parabolic operators and applications

We study the effect of diffusion and frequency on the principal eigenvalue of linear time-periodic parabolic operators with zero Neumann boundary conditions. Monotonocity of the principal eigenvalue and its asymptotic behavior, as diffusion rate and frequency approach zero or infinity, are established. This leads to a classification of the topological structures of level sets for the principal eigenvalue, as a function of diffusion rate and frequency. As applications, we investigate a susceptible-infected-susceptible reaction-diffusion model in spatially heterogeneous and time-periodic environment. We characterize the parameter regions for the persistence and extinction of infectious disease by the basic reproduction number. The asymptotic profiles of endemic periodic solutions are determined when the diffusion rate of susceptible population is small. Our results suggest that fast movement of infected populations and high frequency of oscillation tend to eliminate the disease. Even if the disease persists, it can be controlled by limiting the movement of susceptible populations.

Network synchronization, stability and rhythmic processes in a diffusive mean-field coupled SEIR model

Connectivity and rates of movement have profound effect on the persistence and extinction of infectious diseases. The emerging disease spread rapidly, due to the movement of infectious persons to some other regions, which has been witnessed in case of novel coronavirus disease 2019 (COVID-19). So, the networks and the epidemiology of directly transmitted infectious diseases are fundamentally linked. Motivated by the recent empirical evidence on the dispersal of infected individuals among the patches, we present the epidemic model SEIR (Susceptible-Exposed-Infected-Recovered) in which the population is divided into patches which form a network and the patches are connected through mean-field diffusive coupling. The corresponding unstable epidemiology classes will be synchronized and achieve stable state when the patches are coupled. Apart from synchronization and stability, the coupled model enables a range of rhythmic processes such as birhythmicity and rhythmogenesis which have not been investigated in epidemiology. The stability of Disease Free Equilibrium (or Endemic Equilibrium) is attained through cessation of oscillation mechanism namely Oscillation Death (OD) and Amplitude Death (AD). Corresponding to identical and non-identical epidemiology classes of patches, the different steady states are obtained and its transition is taking place through Hopf and transcritical bifurcation.

On extinction of infectious diseases for multi-group SIRS models with saturated incidence rate

We generalize the deterministic and the stochastic single-group SIRS epidemic models with saturated incidence rate introduced by Lahrouz, Omari, and Kiouach to the multi-group versions. In the deterministic multi-group model, the fact is highlighted that if the threshold $\mathscr{R}_{0}\leq1$ , then the infective condition disappears and it means the extinction of the disease. If $\mathscr{R}_{0}>1$ , then there exists an endemic equilibrium in a feasible region. Allowing the noise perturbation, for the stochastic version, we utilize stochastic Lyapunov functions to show the stability of the disease-free equilibrium of system. A detailed analysis is performed on almost surely exponential stability and pth moment exponential stability of the disease-free equilibrium. We also go into several numerical simulations to illustrate how exactly the theoretical results are verified. Good agreement was observed between our theoretical results and numerical simulations. A comprehensive conclusion is provided.

Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infected-susceptible) epidemic model with n patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. The basic reproduction number ${\cal R}_0$ for the model is determined. It is then shown that when the disease-free equilibrium is stable (${\cal R}_0 < 1$) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (${\cal R}_0 > 1$) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (${\cal R}_0 < 1$) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (${\cal R}_0 > 1$). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.