STOCHASTIC MODEL OF TUBERCULOSIS WITH VACCINATION OF NEWBORNS IN CASE OF EASTREN ETHIOPIA

Abstract

In this thesis we formulated a Stochastic Model of Tuberculosis with Vaccination of Newborns for studying the dynamics of Tuberculosis (TB) by incorporating vaccination of newly born babies and the aim of this work is to develop and analyze qualitatively both the stochastic and the deterministic models of the population dynamics of tuberculosis for the purpose of studying the effect of vaccination coverage. The total population in this model is sub-divided in to four compartments, namely Susceptible, Infected, Vaccinated newborns and Recovered. First, the developed model is expressed and analyzed by deterministic approach. Since this approach neglects the randomness of the dynamics of the process, it has limitations in the modeling process. To avoid this kind of issues we transformed the deterministic approach into a stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The invariant region of the solution, conditions for positivity of the solution, existence of equilibrium points of the model and their stabilities and also sensitivity analysis are checked. Additionally we showed that in both deterministic and stochastic case the effective reproduction number is less than one, then the disease free equilibrium point is stable so that the disease die out. According to the analysis, we came to realize that the basic reproduction number for the stochastic approach is smaller than deterministic one and which shows us stochastic approach is closer to reality than the deterministic one. To conduct the thesis, secondary datas for infected, recovered and vaccinated population were collected from randomly selected Diredawa and Harar Hospitals. We have conducted various numerical experiments to analyze the collectded data using Euler’s method by MATLAB software and obtained interesting simulation results which indicate that combination of increased newborn vaccination has a great contribution in combating TB. It is worth mentioning that the simulation results confirm the conclusion drawn from the qualitative analysis of the model. Hence, we came to realize that the number of infected people keeps decreasing if one carefully combines vaccination with appropriate treatment and decrease the contct between susceptible and infected indeviduals. Therefore, we recommend that a combination of a decrease in contact between infected and susceptible individuals, increasing vaccination coverage, creating awareness to decrease contact rate and increasing recovery rate with proper treatment to effectively control TB infection.