On mathematical modeling and stability analysis of chickenpox models in the presence of weakened-immune individuals in a population

The varicella-zoster virus (VZV) also known as chickenpox is one of the most contagious diseases. Individuals who have never had VZV, have never been vaccinated, or have a compromised immune (which is refers to as immunocompromised) systems, stand the highest risk of VZV infection. This paper considers susceptible-exposed-infectious-weaken immune-recovered-vaccinated (SEIWRV) epidemic model for chickenpox infectious disease, in the presence of treatment. The basic reproduction number, denoted by {\cal R}_o for the model is obtained, and found to be re-enforce by two classes of individuals: -spread from the first-time infected and unvaccinated individuals, and spread by the weaken-immune individuals. This basic reproduction number depends on incidence rate from the susceptible and weaken-immune individuals as well as treatment rate. It is shown in this paper that the model exhibits two equilibria, which include, the disease-free and the endemic equilibriums. By constructing a suitable Lyapunov function, it is observed that the global asymptotic stability of the disease-free equilibrium depends on number of infectious, {\cal R}_o and the treatment rate. The global endemic equilibrium is established using geometric approach, which is applied to a five-dimensional system of differential equations. We found that chickenpox will remain endemic as long as weaken-immune individuals remain in the population. Numerical simulations are also presented to illustrate our main results. It is found that it is possible to eradicate chickenpox from the population, only if the medical practitioners and researchers understand the role of weaken-immune individuals in the spread of chickenpox.

کلیدواژه ها:

Mathematical model ، SEIWRV ، chickenpox ، weakened-immune individuals ، Vaccination ، Stability analysis ، endemic equilibrium ، basic reproduction number