In this lesson we provided an elementary discussion about the use of mathematical models in the study of communicable diseases epidemiology and their ability to explain qualitative patterns of prevalence at population level. The use of mathematical models in the study of communicable diseases requires the use of simple descriptions or caricature that capture the relevant epidemiological aspects. These models facilitate the study of specific scientific disputes?, at the population level, either in a theoretical or practical level. The theoretical results obtained let us answer questions such as: What is the optimal age for vaccination against measles?, or What is the proportion of children that should be effectively vaccinated in order to decrease (to a certain level) or eliminate the risk that young pregnant women get infected with rubella?. Each type of question requires the development of its own model, and the effectiveness of this mathematical model is directly related to its quality and simplicity.
The most simple epidemiological situation is still very complicated, and consequently, it is essential for the researcher to be able to abstract the minimum epidemiological aspects necessary for the study of the question in turn. The process of designing the simplest model is an art, favoured by the organization of multidisciplinary research groups. It is important to observe that even with access to supercomputers, it is not possible to progress in epidemiological theory with extremely detailed models, i.e. with models that include absolutely everything, since the model becomes as complicated as reality. In addition, increasing the level of detail in a model requires the introduction of parameters. With increased detail, the analytical or numerical study of the model becomes more complex and its validation requires much more data.
The use of differential equations epidemic models has an immense historic tradition. The formulation and analysis of the first mathematical models associated to the transmission dynamics of pathogens begin with Sir Ronald Ross and with his English disciples, Kermack and McKendrick. The work of these doctors established the foundations of modern theory of the propagation of diseases at the population level. Sir Ronald Ross developed his models (systems of non linear differential equations) with the purpose of investigating and supporting new public health policies that incorporated his findings on the tranmission cycle of malaria. The mathematical models of Ross focused naturally on questions related to the vector disease dynamics. Ross observed that his mathematical models described well a process of contact between two populations (vectors and humans), incapable of transmitting the parasite of malaria between the members of the same species. Consequently, Ross mentioned that his models were also capable of modelling directly the dynamics of sexually transmitted diseases in heterosexual populations. The contact processes introduced by Ross in the modelling of transmission dynamics of malaria are currently being used for studies of transmission dynamics of HIV as a sexually transmitted disease or as a disease transmitted by sharing infected needles (vectors) (1). One of the objectives of Kermack and McKendrick’s studies between 1927 and 1939 (2) was to adapt (under Ross’s instigation) the epidemiological models to the dynamics of vector transmitted diseases to the case of diseases transmitted by members of the same species. Their achievements are summarized in their threshold theorem. The concept of threshold (basic reproductive number ) is a fundamental concept in epidemiology and in theoretical biology. The theoretical content of this lesson is based on the usefulness and applicability of this fundamental concept, as for example in estimating the rate of vaccination necessary to erradicate a disease .
We use Tuberculosis transmission dynamics as an example. The case of Tuberculosis is a non trivial example of the explanatory utility of mathematical models. Four studies on tuberculosis dynamics are presented : 1) the basic model ; 2) when a susceptible population is exposed to drug-resistant strains , 3) the effect of different control policies on Tuberculosis dynamics , and 4) the effect of endogenous reinfection . We have also included some exercises to help understand how these models work.