Model Equations

The design of the flow diagram also specifies the structure of the system of differential equations that constitutes the compartmental model. There is a direct correspondence between the flow diagram compartments and the model's state variables (representing the proportion or numbers in the population in each stage), and between the flows in the diagram and how the rates of change in the equations are specified. (The model equations may also incorporate subsidiary population structure such as different age groups or different behavioral groups which might not be included in the flow diagram.) The simplest example of a differential equation is the exponential growth curve in which the rate of increase over time of, say, a population N depends only on a constant per capita growth rate parameter r, and on the size of N (known as the equation's state variable).

dN AT

Here dN/dt is the rate at which the value of N changes with time and is equivalent to the slope of the curve of N at any point in time. This example, of course, assumes no deaths. A more realistic simple example incorporating also infection and recovery to permanent immunity would be the equation system corresponding to the flow diagram of Figure 15.1.

(NB: Addition of the equations will give the rate of change of the whole population (dN/dt) so that, if population size is constant, the sum of the equations will be zero, a useful check for errors in specifying the equations.)

In this set of equations, X (the susceptible population), Y (the infected population), and Z (the recovered population) are the ''state variables'' corresponding to the compartments in the flow diagram, and the various terms in the equations represent the different flows into and out of the compartments in the form of the product of the values of the state variables at that moment in time and the values of the parameters determining the rates of flow. (Here X represents the force of infection, a the rate of recovery from infection and ^ the per capita death rate and also, for convenience in order to ensure constant population size, the per capita birth rate.) Note that the model is expressed as rates of change of each of the state variables. In simple cases, the values of the state variables at any point in time (an analytic solution) can be found through a mathematical solution of the equation system (see Appendix). In many cases, however, it is necessary to rely on a numerical solution of the equations using either computer modeling tools, or in the more complex cases involving partial differential equations (see below) computer programs written ad hoc. Various types of intervention can also be incorporated into such models, perhaps by explicitly adding further compartments/variables (e.g., for individuals who are vaccinated and thus at reduced risk of infection, or for those receiving some therapeutic intervention speeding recovery or reducing infectiousness) together with a corresponding further flow from the appropriate compartment (e.g., from the susceptible compartment in the case of prophylactic intervention or from an infection compartment for a therapeutic intervention). In the case of vaccination at or shortly after birth, the entry flow into the population could be divided into those entering as susceptible and those entering as immune through vaccination (in other cases intervention might be included implicitly by assuming that only a proportion of the population might be susceptible to infection) (Figures 15.2, 15.3, and 15.4) (Anderson and May, 199l).

The model population can also be stratified into different behavioral or other relevant subgroups (it may not be necessary to represent this stratification in the flow diagram), and contact matrices specifying contacts between the subgroups can then become a component of the force of infection. A simple example of the latter is for sexually transmitted infections in which the force of infection term experienced by individuals in subgroup i (X¡ ) is composed of a term describing probability of infection, p, as a result of a sexual contact with an

Figure 15.2. A flow diagram with vaccination. As Figure 15.1 but with the addition of a latent stage following infection prior to becoming infective at the per capita rate 01, with recovery at rate 02, and a compartment representing vaccination with permanent immunity of susceptible individuals at a per capita rate v. Note that, with the possible exception of lambda, X, representing the force of infection, there are no rules about which symbol should be used to label transitions in the model (equivalent to parameters in the equations).

Figure 15.2. A flow diagram with vaccination. As Figure 15.1 but with the addition of a latent stage following infection prior to becoming infective at the per capita rate 01, with recovery at rate 02, and a compartment representing vaccination with permanent immunity of susceptible individuals at a per capita rate v. Note that, with the possible exception of lambda, X, representing the force of infection, there are no rules about which symbol should be used to label transitions in the model (equivalent to parameters in the equations).

infected individual, the contact matrix itself, p j, specifying the proportion of sexual contacts that an individual in group i might have with an individual in group j, and the number of sexual contacts per year, c, by individuals in group i:

Xi = CiPpij