Abstract
This paper considers a deterministic model for the dynamics of measles transmission in a population divided into six classes with respect to the disease states: susceptible, vaccinated, exposed, infected, treated, and recovered. First, we investigate the dynamical properties of the SVEITR model such as its equilibrium points, their stability, and parameter sensitivity by applying constant controls. Criteria for determining the stability of disease-free and endemic equilibrium points are provided in terms of basic reproduction number. The model is then extended by incorporating vaccination, therapy, and treatment rates as time-dependent control variables representing the level of coverages. Application of Pontryagin’s maximum principle provides the necessary conditions that must be satisfied for the existence of optimal controls aiming at minimization of the number of exposed and infected individuals simultaneously with the control effort. Numerical simulations that were carried out using the backward sweep method and Runge–Kutta scheme suggest that optimal controls under moderate and high scenarios can effectively reduce the cases of measles. In particular, the moderate scenario that utilizes the existing coverage level of 86% for MCV1 and 69% for MCV2 can degrade the cost functional by 47% of the low scenario. Meanwhile, high scenario that takes the 2020 target of 96% as coverage only makes a slight difference in reducing the number of exposed and infected individuals.
Highlights
Measles as a highly contagious and serious viral disease has been known for centuries. e formal story of measles has begun when the first written accounts of this disease were published by a Persian physician in the ninth century
Motivated by the facts that vaccination and treatment are among key strategies in the management and mitigation of measles, we propose an extended compartmental model by adding vaccinated and treated classes into the standard SEIR model. us, to describe the dynamics of the spread of measles, we consider a population that consists of six subpopulations, namely, susceptible (S), vaccinated (V), exposed (E), infected (I), treated (T), and recovered (R)
An autonomous nonlinear differential equation system for measles dynamics, which incorporates constant vaccination, therapy, and treatment rates, is considered first of all. e SVEITR model consists of six compartments to categorize individuals according to their disease state. e stability analysis of the model is carried out towards diseasefree and endemic equilibrium points using Routh–Hurwitz criteria and bifurcation theory
Summary
Measles as a highly contagious and serious viral disease has been known for centuries. e formal story of measles has begun when the first written accounts of this disease were published by a Persian physician in the ninth century. Is model was later developed by Momoh et al [17] to study the effect of vaccination on the dynamics of the spread of measles. Bakare et al [14] propose the use of SEIR model to portray the dynamics of the spread of measles without discussing the effects of the vaccination control variables. Pang et al [22] developed a SEIR model with vaccination to investigate its effect in controlling the transmission of measles among people. E dynamic of measles infection transmission in the form of age-stratified compartmental model was developed by Verguet et al [42] aiming at exploration of the frequency of SIAs in order to achieve measles control in selected countries. We explore the effect of the change in immunization coverage on the control objective of minimization of the number of exposed and infected individuals simultaneously with the control effort
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