A Mathematical Model of Measles with Vaccination and Two Phases of Infectiousness

Abstract

Despite the availability of the measles vaccine since 1963, the infectious disease is still endemic in many parts of the world including developing and developed nations. It was eliminated from the U.S. in 2000 but due to importation secondary transmissions has been occurring since 2008. With a greater than 90% attack rate measles is not only a health but also an economic problem as it spread very quick leading to a lot of persons been hospitalized or seeking medical help. The disease is endemic in Nigeria with an annual incidence rate of 18.2 per 100, 000 children and a case fatality rate of 1.2% as at 2008. The disease is one of the leading causes of child mortality worldwide. Its prevalence and severity in Nigeria might hamper the fourth millennium development goal by 2015 (To reduce by two thirds, between 1990 and 2015, the under five mortality rate). In this paper we proposed a mathematical model measles incorporating vaccination as a control strategy and capturing the two phases of infectiousness (i.e asymptomatic infectives and symptomatic infectives). We found the basic reproduction number using the next generation approach and proved that our system of equations is locally asymptotically stable if which means that the disease can be eradicated under such condition in finite time. Furthermore we note that in the presence of vaccination approaches 0 as p, the proportion of those vaccinated approaches 1. We reiterated the already known fact that close to 100% (at least 94%) vaccination of susceptible is required for eradication of the disease and suggests that the government should make the vaccination compulsory in other to achieve the herd immunity. We also used LHS/PRCC to support this claim.