Introduction
Mathematical models have long provided important insight into disease dynamics and control (e.g. Anderson & May 1991; Hethcote 2000). As emerging and re-emerging infectious diseases increase in outbreak frequency, there is a compelling interest in understanding their dynamics (Daszak et al. 2000; Dobson & Foufopoulos 2001; Castillo-Chavez et al. 2002; Gubler 2002; Chomel 2003; Enserink 2004). In all disease modelling, a model's mathematical structure is determined by its underlying biological assumptions, which therefore influence the model's predictions. Here, we show how a central assumption in epidemiological modelling, the form of the disease-transmission term, affects a central prediction, the basic reproduction number. We also illustrate the effects of other epidemiological features on model predictions.
The disease-transmission term represents the contact between host individuals in directly transmitted diseases, or between host and vector individuals in host–vector diseases (see Fig. 1a). The choice of which transmission term to use has been extensively discussed, particularly for directly transmitted diseases (Getz & Pickering 1983; Anderson & May 1991; Thrall et al. 1993; McCallum et al. 2001; Begon et al. 2002; Keeling 2005; Rudolf & Antonovics 2005). However, the implications of this choice for disease prediction and control have received much less attention (e.g. Wood & Thomas 1999; McCallum et al. 2001).
A dynamical disease model generates the basic reproduction number, ℛ_{0}, which represents the average number of secondary infections caused by the introduction of a typical infected individual into an otherwise entirely susceptible population (Anderson & May 1991; Heesterbeek 2002). This number serves as an invasion threshold both for predicting outbreaks and for evaluating control strategies. In recent epidemiological modelling, uncertainty and sensitivity analyses have been used to evaluate the effect of different model parameters on ℛ_{0} for directly transmitted diseases (e.g. Blower & Dowlatabadi 1994; Sanchez & Blower 1997; Chowell et al. 2004). Again, however, the effect of transmission-term assumptions on ℛ_{0} has only briefly been mentioned (McCallum et al. 2001), and has not been quantitatively assessed. We connect these central two elements of disease modelling by showing how different transmission terms influence ℛ_{0}, both qualitatively and quantitatively. We focus on host–vector disease systems, in which the added complexity of two interacting populations introduces a wider range of model features. We also show how a range of assumptions about epidemiological features, species’ parameter values, and life-history features influence ℛ_{0}.
Our analysis is motivated by a particular emerging infectious disease system, the North American outbreak of West Nile virus. This arboviral encephalitis amplifies in a transmission cycle between vector mosquitoes and reservoir birds, and is secondarily transmitted to mammals, including humans (Gubler et al. 2000; Bernard et al. 2001; Peterson et al. 2004). Since its initial North American report in New York City in 1999, West Nile virus has spread across the continent and prompted at least five dynamical mathematical modelling studies (Table 1). Although these models share a common structure, they differ in their biological assumptions and therefore in their predictions. To compare the effects of these different assumptions, we first develop a core model that contains the elements common to all the published models. We then systematically alter the core model to consider the qualitative and quantitative effects of different assumptions on ℛ_{0}. To gain further insight into this type of disease dynamic, we include in our review two similar models of other mosquito-borne pathogens, Japanese encephalitis and Ross River virus (Table 1).
Disease system | West Nile virus | Japanese encephalitis | Ross River virus | ||||
---|---|---|---|---|---|---|---|
Model | WN1^{1} | WN2^{2} | WN3^{3} | WN4^{4} | WN5^{5} | JE^{6} | RR^{7} |
| |||||||
No. equations | 8 (4) | 6 (3) | 7 (3) | 4 (2) | 5 (2) | 5 (2) | 11 (5) |
Disease classes | |||||||
Vector | S _{ V }, E _{ V }, I _{ V } | S _{ V }, E _{ V }, I _{ V } | L _{ V }, S _{ V }, E _{ V }, I _{ V } | S _{ V }, I _{ V } | S _{ V }, I _{ V } | S _{ V }, I _{ V } | S _{ V1}, E _{ V1}, I _{ V1}, R _{ V1}, S _{ V2}, E _{ V2}, I _{ V2}, R _{ V2} |
Reservoir | S _{ R }, S _{ r } , I _{ R }, I _{ r } , R _{ R } | S _{ R }, I _{ R }, R _{ R } | S _{ R }, I _{ R }, R _{ R } | S _{ R }, I _{ R } | S _{ R }, I _{ R }, R _{ R } | S _{ R }, I _{ R }, R _{ R } | S _{ R }, I _{ R }, R _{ R } |
Transmission dynamics | FR | MA | FR | FR | FR | FS | MA, C |
Epidemiological features | |||||||
Transmission probabilities | Y | · | Y | Y | Y | Y | · |
V viral incubation period | Y | Y | Y | · | · | · | Y |
V loss of infectivity | · | · | · | · | · | Y | · |
V vertical transmission | · | Y | · | · | Y | · | · |
R death from virus | Y | · | Y | Y | Y | · | · |
R recovery to immune | Y | Y | Y | · | Y | Y | Y |
R loss of immunity | · | · | · | · | · | Y | · |
Vital dynamics | |||||||
Vector | Y | Y | Y | Y | Y | Y | Y |
Reservoir | Y | Y | · | Y | Y | Y | · |
Age structure | |||||||
Vector | · | · | Y | · | · | · | · |
Reservoir | Y | · | · | · | · | · | · |
West Nile virus models
All seven arboviral models reviewed here share a standard susceptible–infectious (S–I) structure for vector and host populations. Table 1 summarizes their key features, and the Appendix presents their equations and ℛ_{0} expressions in a common notation. For simplicity, we abbreviate the models as follows: West Nile virus, WN1–5, Japanese encephalitis, JE, and Ross River virus, RR (Table 1). In an additional series of studies, Japanese and Murray Valley encephalitis and RR virus are modelled using a cyclic representation of vector feeding behaviour (Kay et al. 1987; Saul et al. 1990; Saul 2003; Glass 2005). However, as these models cannot readily be converted to a system of ordinary differential equations for comparison with the other seven, they are not included in our review.