The model(take from Capurro et al 1998)
In epidemiological modelling, the assumption of random mixing among individuals in a population resulted in simpler and more easily analyzed models. However, this assumption lacks real consideration of the social interaction between individuals and other heterogeneities, such as geographical distances, sexual behaviour and age structure. Therefore, success in such modelling requires accounting for population heterogeneity, variation between individuals in contact rate or susceptibility, and heterogenous mixing, i.e. how the pattern of contacts depends on spatial location or on the connectivity of social networks.
The role of transportation is an important one in the transmission of diseases. This role is related to the fact that infectious people can move long distances and spread a disease far from where it begins, but the transportation vehicle could be where the transmission process occurs, e.g. in air-borne diseases. Sattenspiel and Dietz (1995), Sattenspield and Castillo-Chavez (1991), Sattenspield and Simon (1988) are good examples of how a mobility process could be modeled. Also Rvachev and Longini et al. (1985) developed a model for the global spread of influenza that incorporates a transportation matrix giving the average number of individuals that travel from one city to another. But this stochastic model takes no account of the effect of varying the pattern of contact among subpopulations (Sattenspiel and Simon 1988).
The transmission of tuberculosis in buses is a good example to show how to incorporate such heterogeneity in each subpopulation. People of different social classes or neighbourhoods ride in buses, but they use them at different rates. Also, people from the same neighbourhood or social group use buses at different rates. We developed the following model incorporating such heterogeneity:
We divided the population into n neighbourhoods. Each one contains individuals that frequently take a bus, called bus takers, and individuals, non bus takers, that always have contact with people within their neighbourhood and never take buses. Therefore, bus takers have contact with people from the same neighbourhood and people from other neighbourhoods, whereas the non bus takers only have contact with individuals of their same neighbourhood. In the following example, we will only consider 2 different neighbourhoods.
where NC stands for non-communicable individuals and C stands for communicable individuals. In each of these subgroups, the population is divided in susceptible, latent, infected and treated.
Capurro, A F, Zellner, M L, Castillo-Chavez C . Public Transportation and the Transmission of air-borne communicable diseases. Documneto de trabajo N° 20 - Departamento de investigaciones Universidad de Belgrano -Buenos Aires, Argentina(1998)
Rvachev LA, Longini IM. A mathematical model for the global spread of influenza. Math Biosc 1985, 75: 3-22.
Sattenspield L, Castillo-Chavez C. Environmental context, social interactions and the spread of HIV. Am J Hum Biol 1990, 2: 397-417.
Sattenspield L, Dietz K A structured epidemic model incorporing geographic mobility among regionsMath Biosc 1995: 128: 71-92.
Sattenspield L, Simon C. The spread and persistance of infectious diseases in structured populations. Math Biosc 1988, 90: 341-366.