A model of population dynamics of TB in a prison system and application to South Africa

Abstract

BackgroundTuberculosis (TB) continues to spread in South African prisons in particular, as prisons are over-capacitated and have poor ventilation. The awaiting trial detainees are not screened on admission and are at high risk of getting infected with TB.ResultsWe propose a compartmental model to describe the population dynamics of TB disease in prisons. Our model considers the inflow of susceptible, exposed and TB infectives into the prison population. Removal of individuals out of the prison population can be either by death or by being released from prison, as compared to a general population in which removal is only by death. We describe conditions, including non-inflow of infectives into the prison, which will ensure that TB can be eradicated from the prison population. The model is calibrated for the South African prison system, by using data in existing literature. The model can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Illustrative simulations in this regard are presented. The model can be used for other prison populations too, if data is available to calculate the model parameters.ConclusionsVarious simulations generated with our model serve to illustrate how it can be utilized in making future projections of the levels of prevalence of TB, and to quantify the effect of interventions such as screening, treatment or reduction of transmission parameter values through improved living conditions for inmates. This makes it particularly useful as there are various targets set by the World Health Organization and by governments, for reduction of TB prevalence and ultimately its eradication. Towards eradication of TB from a prison system, the theorem on global stability of the disease-free state is a useful indicator.

A Stochastic TB Model for a Crowded Environment

“…The current paper considers a stochastic form of the model in [10]. It is well understood that stochastic differential equation (sde) attempts to reflect the effect of random disturbances in or on a system.…”

“…For a deterministic model of similar type, in [10] we computed parameter values pertaining to South Africa. For the stochastic model in this paper the focus is on mathematical analysis.…”

“…It is important to note that in general populations removal of individuals out of the system is only by death. In this model, as in [10], the removal is by death or by discharge from prison, and the discharge is the dominant factor. This rate of removal is denoted by .…”

“…The underlying deterministic model of (1) is the model given by the same system of equations in the special case = 0, i.e., without stochastic perturbation as in [10]. The underlying deterministic model coincides with the model of Buonomo and Lacitignola [2].…”

“…Against this background the paper [10] offers a model for the population dynamics of TB in a prison or prison system.…”