Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics

Abstract

Communicated by: R. Bravo de la Parra Funding information South African DST/NRF SARChI ; Mathematical Models and Methods in Bioengineering and Biosciences MOS Classification: 37C75; 92D30A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease-free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh-Volterra type, to be globally asymptotically stable for a special case.

Mathematical analysis of a disease-resistant model with imperfect vaccine, quarantine and treatment

“…Particularly, case iii shows the possibility of backward bifurcation. This is the phenomenon where an asymptotically stable DFE coexists with an asymptotically stable endemic equilibrium whenever R o < 1 [2,4]. To prove this fact, a critical value of R o denoted by R c o can be obtained by solving the discriminant a 2 1 − 4a o a 2 = 0 to obtain…”

“…In this case, possibility of backward bifurcation is consequently ruled out. This is because bifurcation only occur when at least two positive endemic equilibria exist whenever R p ≤ 1 [2,4].…”

“…This phenomenon can be removed by replacing the standard incidence function with mass action function. In particular, [4] applied the technique to measles model while [5] applied the technique to dengue fever model. Here, we shall use the same technique to analyze the removal of backward bifurcation in a disease-resistant mathematical model.…”

“…And the third one is waning vaccine which works only for a short period of time. This is for us to grasp the effect of imperfect vaccine in literature which ensures its inclusion in this model and mostly because it has been studied only by few researchers [1,4,5] in mathematical modeling.…”

“…In order to further extend, complement and contribute to the work of [6,7], a new comprehensive model has been designed. We also picked insights in [2,4] to improve the quality of this work. The model we present here is suitable for infectious diseases like influenza virus, severe acute respiratory syndrome (SARS) and others.…”