# Mathematical models for infectious diseases and optimal control

ABSTRACT

Following the World Health Organization, in the last decades emerging and re-emerging epidemics such as acquired immunodeficiency syndrome (AIDS), measles, malaria, and tuberculosis (TB) cause death to millions of people each year. Mathematical models for transmission dynamics of infectious diseases play an important role on prediction, assessment and control of potential outbreaks. The first mathematical model in epidemiology dates back in 1766, where Daniel Bernoulli developed a model to analyze the mortality due to smallpox in England. Since then many mathematical models have been proposed and studied [1, 2]. In this seminar, we present different models for TB, human immunodeficiency virus (HIV), AIDS and TB-HIV/AIDS co-infection, for population and cell levels [3 – 5]. We illustrate how to compute the equilibrium points and the basic reproduction number using the approach of . Some local and global stability results will be proved. Optimal control is a mathematical theory that emerged after the Second World War with the formulation of the celebrated Pontryagin maximum principle, responding to practical needs of engineering, particularly in the field of aeronautics and flight dynamics . In the last decade, optimal control has been largely applied to biomedicine, namely to models of cancer chemotherapy and also to epidemiological models. In the last part of the seminar, we apply optimal control theory to epidemiological models and show how to derive optimal solutions for the minimization of new infectious with minimal cost [3–5].

References

1.   C. I. Siettos and L. Russo, Mathematical modeling of infectious disease dynamics, Vir- ulence 4(4) (2013), 295?306.
2.   H. W. Hethcote, A Thousand and One Epidemic Models, Series Lecture Notes in Biomathematics, Vol. 100, 504-515 (1994).
3.   C. J. Silva and D. F. M. Torres, Optimal control for a tuberculosis model with reinfection and post-exposure interventions, Math. Biosci. 244, no. 2, 154–164 (2013).
1.   C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst. 35, no. 9, 4639–4663 (2015).
2.   D. Rocha, C. J. Silva and D. F. M. Torres, Stability and Optimal Control of a De- layed HIV Model, Mathematical Methods in the Applied Sciences, in press. DOI: 10.1002/mma.4207
3.   P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold en- demic equilibria for compartmental models of disease transmission, Math. Biosc. 180, 29–48 (2002).
4.   L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The mathematical theory of optimal processes, Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962.