There are major advances which have been made to understand the epidemiology of infectious diseases. However, more than 2 million children in the developing countries still die from pneumonia each year. The efforts to promptly detect, effectively treat and control the spread of pneumonia is possible if its dynamics is understood. In this paper, we develop a mathematical model for pneumonia among children under five years of age. The model is analyzed using the theory of ordinary differential equations and dynamical systems. We derive the basic reproduction number, R 0 , analyze the stability of equilibrium points and bifurcation analysis. The results of the analysis shows that there exist a locally stable disease free equilibrium point, E f when R 0 < 1 and a unique endemic equilibrium, E e when R 0 > 1.The analysis also shows that there is a possibility of a forward bifurcation.
Deterministic models have been used in the past to understand the epidemiology of infectious diseases, most importantly to estimate the basic reproduction number, R o by using disease parameters. However, the approach overlooks variation on the disease parameter(s) which are function of R o and can introduce random effect on R o. In this paper, we estimate the R o as a random variable by first developing and analyzing a deterministic model for transmission patterns of pneumonia, and then compute the probability distribution of R o using Monte Carlo Markov Chain (MCMC) simulation approach. A detailed analysis of the simulated transmission data, leads to probability distribution of R o as opposed to a single value in the convectional deterministic modeling approach. Results indicate that there is sufficient information generated when uncertainty is considered in the computation of R o and can be used to describe the effect of parameter change in deterministic models.
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