Qualitative analysis of a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity

In this paper,a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed.The well-posedness of the solution is firstly established.Then the basic reproduction number R0 is defined and a threshold dynamics is obtained.That is,when R_(0)<1,the disease-free steady state is locally stable,which implies that the disease is extinct,when R_(0)>1,the disease is permanent,and there exists at least one positive steady state solution.Finally,the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated.Furthermore,as an application of theoretical analysis,a numerical example involving the spread of influenza is discussed.Based on the numerical simulations,we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation,and the increase of diffusion rate,saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation.Therefore,we propose to reduce the flow of people to lower the effect of spatial hetero-geneity,increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate,and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.

Analysis of SVEIR worm attack model with saturated incidence and partial immunization

Internet worms can propagate across networks at terrifying speeds,reduce network security to a remarkable extent,and cause heavy economic losses.Thus,the rapid elimination of Internet worms using partial immunization becomes a significant matter for sustaining Internet infrastructure.This paper addresses this issue by presenting a novel worm susceptible-vaccinated-exposed-infectious-recovered model,named the SVEIR model.The SVEIR model extends the classical susceptible-exposed-infectious-recovered model(refer to SEIR model)through incorporating a saturated incidence rate and a partial immunization rate.The basic reproduction number in the SVEIR model is obtained.By virtue of the basic reproduction number,we prove the global stabilities of an infection-free equilibrium point and a unique endemic equilibrium point.Numerical methods are used to verify the proposed SVEIR model.Simulation results show that partial immunization is highly effective for eliminating worms,and the SVEIR model is viable for controlling and forecasting Internet worms.