Simple MSEIR Model for Measles Transmission
Mojeeb Al-Rahman EL-Nor Osman *
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P.R. China and Department of Mathematics and Computer Science, International University of Africa, P.O.Box 2469, Khartoum, Sudan.
Appiagyei Ebenezer
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P.R. China and Department of Mathematics, Valley View University, Techiman Campus, P.O.Box 183 B/A, Ghana.
Isaac Kwasi Adu
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P.R. China and Department of Mathematics, Valley View University, Techiman Campus, P.O.Box 183 B/A, Ghana.
*Author to whom correspondence should be addressed.
Abstract
In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria. The basic reproduction number is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out. The disease was locally asymptotically stable if and unstable if . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results.
Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.
Keywords: Reproduction number, measles transmission, equilibrium states, stability analysis