An Application of Deterministic and Stochastic Processes to Model Evolving Epidemiology of Tuberculosis in Kenya

Abstract

Tuberculosis, a highly infectious disease which is transmitted within and between communities when infected and susceptible individuals interact. Tuberculosis at present is a major public health problem and continues to take toll on the most productive members of the community. An understanding of disease spread dynamics of infectious diseases continues to play a critical role in design of disease control strategies. Modeling of Tuberculosis is useful in understanding disease dynamics as it will guide the importance of basic science as well as public policy, prevention and control of the emerging infectious disease and modeling the spread of the disease. This study sought to establish how long under different frameworks will TB disease recede to extinction. In this study,deterministic and stochastic models for the trends of tuberculosis cases over time in Kenya were developed. Susceptible Infective (SI), Susceptible Infective and Recovered (SIR) and Susceptible Exposed Infective and Recovered (SEIR) models were considered. These models were modified in order to fit the data more precisely (age structure and predisposing factors of the incident cases).The SIR and SEIR model with non-linear incidence rates were further looked at and the stability of their solutions were evaluated. The results indicate that both deterministic and stochastic models can give not only an insight but also an integral description of TB transmission dynamics. Both deterministic and stochastic models fit well to the Kenyan TB epidemic model however with varying time periods. The models show that for deterministic model the number of infected individuals increases dramatically within three years and begins to fall quickly when the transmissible acts are 10 and 15 and falls to close to zero by 15 years but when the transmissible act is 5 the number infected peaks by the 11th year and declines to zero by year 31, while for stochastic models the number infected falls exponentially but when the transmissible acts is 15 the decline is slow and will get to zero by the 53rd year while for 10 transmissible acts to declines to zero by the 18th year. The other transmissible acts (1,3,5) decline to zero by the 9th year.From this study we conclude that if the national control program continues with the current interventions it could take them upto the next 31 years to bring the infection numbers to zero if the deterministic model is considered, while in the stochastic model with accelerated interventions and high recovery rate and assuming that there is no change in the risk factors it could take them upto 11 years to bring the infections to zero.