Mathematical models of tuberculosis and childhood diseases: Very different approaches for very different diseases

David James Gerberry, Purdue University

Abstract

Policies regarding the use of the Bacille Calmette-Guérin (BCG) vaccine for tuberculosis vary greatly throughout the international community. In several countries, consideration of discontinuing universal vaccination programs is currently under way. The arguments against mass vaccination are that the effectiveness of BCG in preventing tuberculosis is uncertain and that BCG vaccination can interfere with the detection and treatment of latent tuberculosis. In this work, we pose a dynamical system model for tuberculosis dynamics at the population level in order to examine the conditions which justify discontinuing BCG vaccination. The model incorporates treatment of latent infections as well as the interference caused by vaccination. Whenever possible, model parameters are defined in terms of control indicators estimated by the World Health Organization. The model is analyzed via numerical experiments and dynamical systems theory. The model is fit to TB data for eight countries with varying TB burdens. Numerical simulation shows that vaccination results in lower TB burdens for all countries considered. In the absence of immigration, mathematical analysis leads to explicit conditions which justify the discontinuation of BCG vaccination. In the second part of this thesis, we study the dynamics of childhood diseases. It has been shown that the inclusion of an isolated class in the classical SIR model for childhood diseases can be responsible for self-sustained oscillations. Hence, the recurrent outbreaks of such diseases can be caused by autonomous, deterministic factors. We extend the model to include a latent class (i.e., individuals who are infected with the disease, but are not yet able to pass the disease to others) and study the resulting dynamics. The existence of Hopf bifurcations is shown for the model, as well as a homoclinic bifurcation for a perturbation to the model. For historical data on scarlet fever in England, our model agrees with the epidemiological data much more closely than the model without the latent class. For other childhood diseases, our model suggests that isolation is unlikely to be a major factor in sustained oscillations.

Degree

Ph.D.

Advisors

Milner, Purdue University.

Subject Area

Applied Mathematics|Epidemiology

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