Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate
Mathematics and Computers in Simulation
Verified Solution Method for Population Epidemiology Models with Uncertainty
International Journal of Applied Mathematics and Computer Science - Verified Methods: Applications in Medicine and Engineering
Analysis and control of an SEIR epidemic system with nonlinear transmission rate
Mathematical and Computer Modelling: An International Journal
Asymptotic behavior of an SEI epidemic model with diffusion
Mathematical and Computer Modelling: An International Journal
Qualitative analyses of SIS epidemic model with vaccination and varying total population size
Mathematical and Computer Modelling: An International Journal
Hopf bifurcation in two SIRS density dependent epidemic models
Mathematical and Computer Modelling: An International Journal
Global analysis of SIS epidemic models with variable total population size
Mathematical and Computer Modelling: An International Journal
A mathematical model for endemic malaria with variable human and mosquito populations
Mathematical and Computer Modelling: An International Journal
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In this paper, some SEIRS epidemiological models with vaccination and temporary immunity are considered. First of all, previously published work is reviewed. In the next section, a general model with a constant contact rate and a density-dependent death rate is examined. The model is reformulated in terms of the proportions of susceptible, incubating, infectious, and immune individuals. Next the equilibrium and stability properties of this model are examined, assuming that the average duration of immunity exceeds the infectious period. There is a threshold parameter R"o and the disease can persist if and only if R"o exceeds one. The disease-free equilibrium always exists and is locally stable if R"o 1. Conditions are derived for the global stability of the disease-free equilibrium. For R"o 1, the endemic equilibrium is unique and locally asymptotically stable. For the full model dealing with numbers of individuals, there are two critical contact rates. These give conditions for the disease, respectively, to drive a population which would otherwise persist at a finite level or explode to extinction and to cause a population that would otherwise explode to be regulated at a finite level. If the contact rate @b(N) is a monotone increasing function of the population size, then we find that there are now three threshold parameters which determine whether or not the disease can persist proportionally. Moreover, the endemic equilibrium need no longer be locally asymptotically stable. Instead stable limit cycles can arise by supercritical Hopf bifurcation from the endemic equilibrium as this equilibrium loses its stability. This is confirmed numerically.