Estimation of the infection rate in epidemic models with multiple populations

Many mathematical models for the spread of infectious diseases have been proposed in the literature, and it is often too easy to get lost in their details. Since models necessarily encompass a set of assumptions, it makes sense to classify them according to the hypotheses they are based on.
Assumptions roughly belong to three main categories, namely:
(1) assumptions about the disease itself;
(2) assumptions about the environment within which the disease spreads;
(3) assumptions due to mathematical modelling.

Assumptions about the disease mostly regard the possible states (also sometimes called compartments) according to which individuals are exclusively and exhaustively classified at any given time, and dynamics among them. In this work we shall restrict ourselves to contagious illnesses, under the basic assumption that diseases spread as a result of contacts between susceptible and infective (carrier) individuals. In this context, models range from the simplest S → I model, in which individuals are either susceptible to the disease (S) or forever infected with it (I), to more complex extensions that account for different stages of infection (such as incubation periods), temporary recovery, removals (which may themselves be due to different causes, such as acquired immunity, isolation, or death), vaccinations and vertical transmission and immunity.

Of course, realistic assumptions about the disease can only be formulated based on the epidemiological properties of the pathogen. However, the time scale being considered also plays a crucial role: in the case of influenza, for example, it is reasonable and often easier to assume that, in a single season, an individual can only catch the disease once, and thus earns “lifelong” immunity after the infection. Assumptions regarding the environment can be further classified into two subcategories: assumptions about its structure and about its dynamics.

As for structure, an important distinction needs to be made between population models, which deal with a single population, and meta-population models, which consider a set of populations and constrained interactions among them, such as travel of individuals. The population (or populations, in the case of meta-population models) may be considered as a single, homogeneous group, as a collection of several homogeneous strata, or else as completely heterogeneous. In all cases it is also necessary to define the population dynamics, which range from closed populations, with a constant number of individuals, to open populations considering complex birth and death processes, as well as migrations.