## Age Structured Models

Simple compartmental models (i.e., ODE models), as described above, in effect represent populations in which the risk of mortality and infection processes does not change with age, which as a result would have an age distribution that is exponentially declining with increasing age. The mean age at death for such a population is 1/^, where ^ is the per capita death rate. However, with a mean age at death of 75, this distribution would imply a substantial proportion of the population would be over 100 years old and some would be very much olderâ€” several hundreds of years with the oldest tending to infinite age. Nevertheless, despite this lack of realism in terms of age distribution, such models are extremely useful for preliminary investigations of transmission dynamics.

Simple representations of age can be introduced into ODE models by the straightforward means of adding further series of compartments, such as for children, young adults and older adults (Figure 15.6), with flows representing aging between, for example, compartments for children and young adults and between young adults and older adults; by this means some age-related infection processes can be incorporated, so that risk of infection, rate of recovery, mortality etc. can vary according to the subset of compartments representing a particular age grouping; also contact patterns between the different age groupings can be specified using a contact or mixing matrix.

Figure 15.6. Simple model with different ages. A development of the simple model in Figure 15.1 representing three successive age groups, e.g., children (X1, Y1, Z1), young adults (X2, Y2, Z2) and older adults (X3, Y3, Z3) and the rate of aging a1 and a2 from each age group to the next. In this instance the value of the force of infection is shown to differ from age group to age group (A1, A-2, A3) although the rate of recovery from infection, a, does not change.

Figure 15.6. Simple model with different ages. A development of the simple model in Figure 15.1 representing three successive age groups, e.g., children (X1, Y1, Z1), young adults (X2, Y2, Z2) and older adults (X3, Y3, Z3) and the rate of aging a1 and a2 from each age group to the next. In this instance the value of the force of infection is shown to differ from age group to age group (A1, A-2, A3) although the rate of recovery from infection, a, does not change.

There is no restriction on the number of compartments that can be added in this way, but the model will become increasingly cumbersome, and it needs to be borne in mind that in each ''age grouping'' so created there will effectively be an exponential distribution of ages, so that, for example, although the average time spent in a compartment representing the first 5 years of life may be 5 years (corresponding to a rate of 0.2 per year = 1/5 years), as with a single compartment representing all age groups, a proportion of this exponentially distributed population will be far older with no limit on maximum age. (A further drawback of this scheme is that the rate of flow from one compartment to another in the same age group, say, upon infection or upon recovery from infection, corresponds to an exponentially distributed waiting time in the first compartment. For a flow, for example, from the susceptible children compartment via infected and recovered children's compartments to that of recovered adolescents, the overall mean time to reach adolescence is greater than that for a flow from susceptible children to susceptible adolescentâ€”the mean time to reach the oldest age group represented is not the same for all routes through the system.)

A great improvement on this strategy is achieved by the use of a more complex class of models using partial differential equations (PDE). In essence, while ODE models describe how the state variables change over time, PDE models describe how they change both with time and with age; that is, whereas ODE models can be visualized as epidemiology evolving in one dimension along the arrow of time, PDE models incorporating age structure can be thought of as epidemiology evolving two-dimensionally through both time and by age. (NB: Such PDE models are a special case of PDE because population age and time advance in step at the same rate as in Figure 15.7; this greatly facilitates approaches to solution of the equations.)

When using PDE models, in addition to specifying how entries into the population at birth (i.e., at age = 0) are distributed between the various states of susceptibility, infection and immunity, the initial age distribution (i.e., at time = 0) must also be specified (these are known as the boundary conditions) as inputs to the model. Using PDE models allows achievement of much more realistic age distributions through the use of age-related mortality and fertility rates. An age-structured version of the simple ODE model of Figure 15.1 might be

Here the differential on the left-hand side of the equation represents the rate of change of the state

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