The behaviour of measles transmission in three different populations

Universitas Negeri Surabaya, Department of Mathematics, Surabaya, Indonesia

^* Corresponding author: budiprawoto@unesa.ac.id

Abstract

SIR Model can be employed to model the transmission of either fatal or non-fatal disease within a closed population based on certain assumptions. In this paper, the behaviour of non-fatal diseases transmission model is observed from three types of population, that is (i) increasing population, (ii) constant population, (iii) decreasing population. This paper acquired two equilibria, i.e, the disease-free equilibrium point $\left(\frac{\mu }{{\mu }_{\text{S}}},0,0\right)$ and the endemic equilibrium point $(\frac{\alpha +{\mu }_{\text{I}}}{\beta },\frac{\mu \beta -{\mu }_{\text{S}}\alpha -{\mu }_{\text{S}}{\mu }_{\text{I}}}{\beta (\alpha +{\mu }_{\text{I}})},\frac{\alpha (\mu \beta -{\mu }_{\text{S}}\alpha -{\mu }_{\text{S}}{\mu }_{\text{I}})}{\beta (\alpha +{\mu }_{\text{I}}){\mu }_{\text{R}}})$. At the disease-free equilibrium, the behaviour of the model is stable when $\beta <\frac{{\mu }_{\text{S}}}{\mu }({\mu }_{\text{I}}+\alpha )$, while at the endemic equilibrium, its behaviour is stable for any positive parameters α, β, μ, μ_S, μ_I, and μ_R.