these individual and population-level processes could occur on similar or very different time scales.
For many biologists, the utility of mathematical models for real-world infectious disease problems is not always obvious. This lack of appreciation can be partly attributed to the challenges of translating model parameters into phenomena that can be measured in the field. These challenges can be overcome, however, and many benefits arise from using epidemiological models. Specifically, models demonstrate the mechanisms by which parasites can regulate host populations, and they can highlight processes that predict disease spread among groups or populations. Models also can be used to evaluate the costs and benefits of intervention strategies aimed at limiting disease risks to threatened host species, such as vaccination, quarantine, and culling of reservoir hosts (Chapter 7). Models of host-parasite interactions share several features in common with other mathematical models in population ecology, including a set of clearly stated assumptions, reliance well-defined variables, quantitative expression of processes influencing biological events, and a set of predictions regarding dynamical outcomes and equilibrium conditions. To that end, collaboration between modelers and field biologists is crucial, with field biologists providing key information required to parameterize epidemiological models and models pointing to predictions that are testable with field data.
Two general classes of mathematical models have been used to describe the dynamics of infectious disease in microparasites and macroparasites. For microparasites such as viruses and bacteria, where researchers are generally not concerned with the numbers of parasites per host, a compartment model structure divides the host population into susceptible, infectious, and recovered (or resistant) individuals. For macroparasites such as helminths and arthropods, models must also account for parasite eggs or larvae that persist outside of the host, as well as the frequency distribution of the number of parasites per host. Both types of models generally rely on a framework of coupled differential equations, and these can be complicated by factors such as latency or carrier states and the details of the transmission process. A full description of epidemiological models for both micro- and macroparasites is provided in Anderson and May (1991).
A broad array of microparasites can infect primate hosts (Chapter 2). Mathematical models for directly transmitted microparasites typically divide the host population into susceptible (S), infected (I), and recovered/immune (R) classes and track changes in the number of hosts within each category (Box 4.1). This type of compartment model (often called an SIR model) has been developed and analyzed extensively by Anderson and May (1979, 1991), Getz and Pickering (1983), and others, drawing on classical approaches of Ross (1911) and Kermack and McKendrick (1927). For cases where hosts do not acquire immunity to re-infection (e.g. some STDs and chronic diseases, such as tuberculosis and brucellosis), the resistant class is eliminated and the equations simplify to an SI model (or SIS, for susceptible-infected-susceptible). Other complications can be added to the simple compartment model, some of which are addressed later in this chapter.
Box 4.1 Compartment models for directly transmitted microparasites
Mathematical models for microparasites divide the host population into susceptible (S), infected (I), and recovered/immune (R) classes and track changes in the number of hosts within each category. Total host population size, N, is the sum of S + I + R. Susceptible hosts arise from new births (where a is the per capita birth rate, here arising from each host class) or loss of immunity from the recovered class (y). Individuals leave the susceptible class through natural mortality (b) or through infection after encountering an infected host (at rate fiSI). Infected hosts are lost through natural death (b), disease-induced mortality (a) or through recovery (v) to an immune state. Arrows indicate movement between host states, and the differential equations express these processes in mathematical terms. This model assumes that hosts are uninfected at birth, that pathogens do not affect host fecundity, and that host populations are large enough that stochastic processes can be ignored. The simple SIR model shown here is useful for parasites with density dependent transmission, which is a "mass action" process where transmission increases directly with host population density.
Many complications can be added to the simple compartment model framework. For example, a disease may reduce the fecundity of infected hosts, or be associated with a long latent period. Age or social structure may complicate among-host contact rates and parasite transfer. In addition, the density-dependent mixing assumed by this model is often inappropriate to describe the transmission dynamics of many pathogens. Other transmission modes can have profound effects on the invasion, persistence, and temporal dynamics of disease, and their consequences have been explored in theoretical and comparative studies (described in Box 4.3). Additional factors that increase the realism and complexity of host-parasite interactions are described in Section 4.4.
-d- =a(S + I + R)-bS-ßSI+ gR -f = ßSI-(a +b + v)I
Fig. 4.4 Schematic diagram and differential equations for a typical SIR compartment model for a directly transmitted microparasite.
The basic SIR model gives rise to several key principles that characterize host-pathogen interactions and have important consequences for infectious disease dynamics in wild populations. Probably the most important issue for any infectious disease is whether it will invade and establish in the host population. A related question concerns how fast it spreads. Both of these issues can be addressed by the basic reproductive number, R0, which sets the conditions under which pathogens can increase in prevalence when the disease is initially rare. Formally defined, R0 is the number of secondary infections produced by a single index case introduced into an entirely susceptible host population (Anderson and May 1991; Dietz 1993; Heesterbeek 2002). This is estimated by multiplying the expected number of new infections from a single infected host (fiS, where initially S ~ N, the total population size) by the average duration of infectiousness, D, where D = 1/(a + b+v). Thus, for the SIR model in Box 4.1, fiS
As in Box 4.1, fi corresponds to the pathogen transmission parameter, a denotes disease-induced mortality rate, b captures host background mortality rate, and v corresponds to host recovery rate from infection. In a deterministic system, R0 defines a break-even point above which the pathogen will establish in the population and below which the pathogen will decline to extinction. In other words, R0 must exceed 1.0 for the disease to invade (although in a stochastic system, the pathogen could go extinct even if R0 > 1, and could increase if R0 < 1, albeit with low probability; Lloyd-Smith et al. 2005).
Values for R0 clearly differ among infectious diseases and also can change over space and time for the same pathogen (Dietz 1993). For the model in Box 4.1, the form of Equation (4.1) suggests that pathogens with high transmission rates (fi), low virulence (a), and low host recovery (v) should have the highest R0 values, and that pathogen spread will also be favored by high host population size and low host background mortality. Approaches for deriving expressions for R0 in more complicated systems (such as when host populations are structured by age, sex, or other heterogeneities) have been developed by Diekmann et al. (1990), Hasibeder and Dye (1988), and others.
Once an epidemic has started, the larger the size of R0, the faster the disease will spread, although this will depend on whether R0 is large due to a longer infectious period or because of greater transmission potential. Thus, pathogens with low values of R0 should generally cause longer epidemics with lower peak prevalence, and pathogens with high R0 values should initiate rapid epidemics with higher peak prevalence. Once an epidemic is underway, a parameter defined as R measures the number of subsequent cases following the initial secondary infections. In the absence of control measures, R = R0s, where s is the remaining proportion of susceptible hosts in the population. This is an important concept and shows that the per capita rate of spread of a pathogen will decrease over time as animals are removed from the susceptible class. For most directly transmitted pathogens, the expectation is that the susceptible class will eventually be reduced to a break-even point that roughly corresponds to R = 1, and this also corresponds to the threshold population size described in Equation (4.2) below.
Estimating R0 is an important step toward management and intervention of epidemic pathogens (Table 4.2), but it has rarely been calculated in wild primate populations. In a perfect scenario, known parameter values for transmission (¡), host population size, and the duration of the infectious period (D) can be used to evaluate expressions such as Equation (4.1). On the other hand, this information is usually incomplete and reliable values for ¡ are extremely elusive (Heesterbeek 2002). Several methods for estimating the reproductive number rely on detailed knowledge of host longevity or retrospective studies of epidemics that recently subsided. For well-studied endemic
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