Existence of a Stochastic Model of an Age-Structured Deterministic Model of Diphtheria Infection
In this research, a stochastic model is developed and analysed for the age structured deterministic model of diphtheria infection. The model includes specific epidemiological classes such as $S_{1}$ Susceptible infant at time $t$ (0-1years), $S_{2}$, Susceptible school children population at time $t$, $V$, Vaccinated population at time $t$, $E$, Exposed population at time $t$, $I_{1}$, Asymptomatic infected population at time $t$, $I_{2}$, symptomatic infected population at time $t$, $I_{D}$, Detected infectious humans at time $t$ (Asymptomatic and symptomatic) population through testing, and $R$, Recovered population at time $t$. The work investigates the transformation of age-structured deterministic model of diphtheria infection into an age-structured stochastic model of diphtheria infection, by applying the well established procedure proposed in [1] and also investigates the existence and uniqueness of solution of the transformed stochastic model. It is established that developed stochastic model of diphtheria infection dynamics satisfies the condition for the existence and uniqueness of solution. The numerical simulation of the work shows that increased vaccination rate will lead to Diphtheria infection extinction while low vaccination rate will produce an epidemic, demonstrates how randomness can cause the outbreak of Diphtheria die out and indicates the effect of random noise in detecting the disease hence suppressing the initial infection wave.