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Figure 15.7. Schematic diagram of an age-structured model. A schematic diagram of a partial differential equation (PDE) model with full age structure. As before, X, Y and Z represent susceptibles, infecteds and recovereds, with horizontal arrows representing infection and recovery; however, with each time step individuals also increase in age by one age step (typically time steps and age steps are of equal duration), thus gradually progressing vertically downwards until reaching maximum age, when death occurs.

deaths deaths deaths

Figure 15.7. Schematic diagram of an age-structured model. A schematic diagram of a partial differential equation (PDE) model with full age structure. As before, X, Y and Z represent susceptibles, infecteds and recovereds, with horizontal arrows representing infection and recovery; however, with each time step individuals also increase in age by one age step (typically time steps and age steps are of equal duration), thus gradually progressing vertically downwards until reaching maximum age, when death occurs.

variable with age, a, and time, t, and the parameters all vary with age. Typically, however, when modeling growing populations in developing countries, mortality rates may be kept constant over age and time (commonly referred as Type II mortality, a usage adopted by epidemiological modelers from demography and ecology). This results in an exponentially decreasing age distribution, but, to avoid unrealistically high ages, it is truncated at a predetermined maximum age (e.g., 75 years) so that when reaching this age all die. For populations in the so-called developed world an alternative scheme is often adopted in which it is assumed that no one dies until the limiting maximum age is reached so that the age distribution, in the absence of disease induced mortality, is completely uniform (referred to as Type I mortality). Use of these distributions providing a stable age distribution makes it easier to understand the transmission dynamics of a system than it is with one in which the age distribution is changing over time at the same time as processes of infection and disease and behaviors may also be changing with age. However, in the real world, of course, age distributions do indeed change with time, and once a good understanding of the nature of the dynamics has been achieved through the use of theoretical rates, it is relatively simple to incorporate age-specific mortality rates derived from data. In a similar way, age-specific fertility rates can be incorporated into the model so that a much better representation of demography may be obtained. It should be borne in mind, however, that it is necessary also to specify the initial age distribution of the population. Clearly, if the majority of the population is in the fertile age range, the shape of the age distribution of the population will, over the short and medium term, evolve in a different way to that observed if the majority of the adult population have passed their fertile years. Provided that the per capita age-related fertility and mortality rates are constant over time, the final equilibrium age distribution is determined simply by the rates specified. This will, however, be attained only over a generational time scale. It should also be borne in mind that even though a stable age distribution may be achieved, this does not necessarily mean that population size itself is at equilibrium. A declining population may have attained a stable age distribution skewed towards upper ages (Figure 15.5) but, in the absence of migration, population numbers will continue to decline as fewer individuals are entering the population through births than are leaving it through deaths. Age-specific migration also may be incorporated into the model through additional terms in the state variable equations, but this does increase the challenge of interpreting model results and of maintaining a suitable age distribution. In theoretical terms, also, this adds a further layer of difficulty in terms of gaining theoretical insight into the PDE equation system, although in practice PDE models are solved numerically using a computer so that migra-tional flows into and out of the model population can be dealt with in a relatively straightforward way. For practical considerations, much epidemiological modeling has been and remains concerned with populations at demographic equilibrium. With the prospect of substantial demographic change occurring in many parts of the world, however, often leading to a significantly older population age distribution, it becomes desirable to consider possible epidemiological scenarios under such conditions of change, as there may be the potential for these demographic changes to drive epidemiological dynamics in unexpected ways (Figures 15.8 and 15.9) (Williams and Manfredi, 2004).

Often the population in PDE (and ODE) models is not divided between the two sexes—although in many instances this is not of great significance particularly if we are concerned solely with population levels of infection and disease, as control or treatment issues that relate to males or females can be simply considered by assuming a 50:50 split. Even in the case, for example, of congenital rubella in which the concern is the proportion of females reaching fertile age ranges without prior exposure or immunization, the impact of a given universal vaccination strategy can usefully be assessed without specific representation of males and females, as the age at infection remains the same for both. In some circumstances, however, it will be useful to explicitly divide the population into male and female. Examples are health provisions specifically directed at males or females, and cases of significant differences in epidemiology between males and females. One instance might be the modeling of sexually transmitted infections (STI) in heterosexual populations in which transmission risks or proportions

Figure 15.8. Examples of age-structured model outputs for measles incidence under two regimes of measles vaccination in an aging population. Outputs from fully age-structured (PDE) model with realistic demography. For a population with declining birth rate leading to an increasingly aging population the incidence of measles infection over time is shown under two scenarios: (i) from 1976 onwards moderate continuous infant vaccination coverage insufficient to achieve elimination (thin line) and (ii) the same level of vaccination but with additional vaccination campaigns every 4 years from 1989 onwards, which achieves elimination (thick line), albeit after a significant epidemic in 1995 and a miniscule one in 2015. (Williams and Manfredi, 2004)

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Figure 15.8. Examples of age-structured model outputs for measles incidence under two regimes of measles vaccination in an aging population. Outputs from fully age-structured (PDE) model with realistic demography. For a population with declining birth rate leading to an increasingly aging population the incidence of measles infection over time is shown under two scenarios: (i) from 1976 onwards moderate continuous infant vaccination coverage insufficient to achieve elimination (thin line) and (ii) the same level of vaccination but with additional vaccination campaigns every 4 years from 1989 onwards, which achieves elimination (thick line), albeit after a significant epidemic in 1995 and a miniscule one in 2015. (Williams and Manfredi, 2004)

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Figure 15.9. Examples of age-structured model outputs showing the influence of suboptimal measles vaccination on age at infection in an aging population. Further outputs from the same PDE model as shown in Figure 15.8. These show for a population with declining birth rate leading to an increasingly aging population how the median and upper and lower quartiles of the distribution of the age at infection change through time under a scenario of moderate vaccination coverage insufficient to achieve elimination. This scenario highlights the potential for a substantial proportion of cases of ostensibly childhood infection to occur among older individuals. (Williams and Manfredi, 2004)

of asymptomatic (and hence probably untreated) infection may differ between males and females. In the case of heterosexual transmission of HIV/AIDS in sub-Saharan Africa while the risk of transmission from an infectious partner for women may be significantly higher, males in general may tend to have more partners and so may be more likely to have sex with an infectious partner. The interaction between the two effects is difficult to tease out without incorporating both sexes in the model. Note that when a model population is explicitly divided into males and females for modeling STI epidemiology, it is necessary to ensure that the numbers of partnerships experienced by females with males corresponds to the number of partnerships that females have with males; where numbers in male or female populations change at different rates, a mechanism must be included in the model to allow the balance in the numbers of partnerships to be maintained over time.

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