MATHEMATICAL MODELING AND ANALYSIS OF AN SIR EPIDEMIC MODEL WITH THERAPY: EXISTENCE, POSITIVITY, AND STABILITY STUDY

Abstract

The paper has introduced a modified SIR epidemic model that includes therapeutic intervention and demographic effects to examine the dynamics of the disease in the closed population. The model breaks down the population into the following compartments; susceptible, infected and recovered, with recruitment, natural death, and rate of therapy. The mathematical theory is mathematically analyzed to be well-posed. The existence and uniqueness of solutions are proved through standard Lipschitz conditions in which the system is said to evolve continuously out of any admissible initial condition. Solutions are proved to be positive, and this is necessary to make population compartments biologically meaningful. The model has a positively invariant region, and this indicates that total population is also bounded as time goes by. A basic reproduction number, , is calculated, which is a threshold value that is used to identify disease outbreak or extinction. Local stability is also studied using the Jacobian matrix whereby the disease-free equilibrium is determined. Moreover, global stability of the disease-free equilibrium is exercised by applying the Lyapunov functions methods and it proves that the disease dies when . The theoretical findings give basic insights on the epidemic behavior subjected to therapeutic and demographic influence. On the whole, this paper provides a solid mathematical model of infection dynamics and guarantees biologically plausible predictors and conditions when a disease.