Mathematical models of tuberculosis extension and control of it (review)

The survey considers mathematical models of epidemiological processes, which determine the dynamics of tuberculosis morbidity. The first models of tuberculosis epidemiology were developed and published in the early sixties of the last century. These works and the works of seventies have formulated and described peculiarities of tuberculosis epidemiology, which are: long-lasting latent phase of the infection, very low probability of total deliverance from the infection, possibility of quick growth of the disease after introduction of infection, and dependence of probability of activation of the infection on the condition of the vehicle of the disease and duration of the latent stage. The most important sphere of application for mathematical models of tuberculosis epidemiology is estimation of efficiency of the control strategies for this disease. A new tide of interest in mathematical models of spread of tuberculosis is connected with growth of morbidity rate in developing countries because of HIV epidemics and emergence of mycobacterium strains, which are resistant to one or several medications. The models of the 80-s and 90-s are devoted to interaction of HIV infection and mycobacteria, to formation and spread of drug-resistant strains. Much attention is given to investigation of properties of the models, to estimation of parameters, and comparison with real data. During this period models became important means for working out and argumentation of activity of both national and international organizations, which are responsible for fight against this infection. For convenience sake a unified system of designation of variables and parameters is used in the survey, block schematic diagrams of the simulated processes and estimations of values of parameters are given; assumptions and presuppositions having been employed in construction of the models are discussed. The work under consideration is the first complete survey of the models of this class up to the year 2006.