Journal of Applied Nonlinear Dynamics 15(2) (2026) 409--438 | DOI:10.5890/JAND.2026.06.012

C. I. Nkeki$^1$, I. A. Mbarie$^2$

$^1$ Department of Mathematics, Faculty of Physical Sciences, University of Benin, Benin City, Edo State, Nigeria

$^{2}$ Institute of Child Health, College of Medical Sciences, University of Benin, Benin City, Edo State, Nigeria

Abstract

Chickenpox is an acute and rapidly spread infectious disease. In this paper, we examine mathematical modelling and stability analysis of chickenpox disease model in the presence of vaccination in a homogeneous population. The model is structured into eight compartments, which represents the actual structure of chickenpox transmission dynamics. As a result, a nonlinear dynamical system is obtained. In this paper, the vaccinated susceptible individual immediately moves to the recovered class with immunity, while a fraction of the inflow of individuals into the population who are unvaccinated entered the susceptible class and the remaining who are vaccinated move to the immune class. The basic reproduction number of our chickenpox model is determined. A suitable Lyapunov function is constructed for our model, and it is observed that the global asymptotic stability of the disease-free equilibrium depends on the recruitment rate, rate of inflow of individuals that are vaccinated and basic reproduction number of the disease. We found that if all the inflows into the population are vaccinated, the Lyapunov function will be a constant term, which simply means that the population will totally be free from chickenpox infection over time. The global stability of the disease-free equilibrium and the local stability of the endemic equilibrium of the chickenpox model are discussed, in this paper. Using a geometric approach on our seven-dimensional sub-systems, we established the global stability of our chickenpox endemic equilibrium. Some numerical simulations are also carried out in this paper, to determine the role of vaccination and recovery rate in the control and prevention of the spread of chickenpox in a public primary school in a Central City of China. It is found that individual students who are unvaccinated against chickenpox are exposed to higher risk of chickenpox infection at the incubation, prodromal and active stages of the infection periods. We also found that recovery, treatment and vaccination rates played vital roles in the control and prevention of the spread of chickenpox among the school children. We found that vaccination against chickenpox will reduce the spread of the virus up to 90\%, which is in line with WHO estimation of 87\% to 95\%.

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