Transmission assumptions generate conflicting predictions in host–vector disease models: a case study in West Nile virus

Authors

  • Marjorie J. Wonham,

    Corresponding author
    1. Department of Biological Sciences and Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, CAB 632, Edmonton, AB, Canada T6G 2G1
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  • Mark A. Lewis,

    1. Department of Biological Sciences and Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, CAB 632, Edmonton, AB, Canada T6G 2G1
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  • Joanna Rencławowicz,

    1. Department of Biological Sciences and Department of Mathematical and Statistical Sciences, Centre for Mathematical Biology, University of Alberta, CAB 632, Edmonton, AB, Canada T6G 2G1
    2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3P4
    3. Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland
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  • P. Van Den Driessche

    1. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3P4
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* E-mail: mwonham@ualberta.ca

Abstract

This review synthesizes the conflicting outbreak predictions generated by different biological assumptions in host–vector disease models. It is motivated by the North American outbreak of West Nile virus, an emerging infectious disease that has prompted at least five dynamical modelling studies. Mathematical models have long proven successful in investigating the dynamics and control of infectious disease systems. The underlying assumptions in these epidemiological models determine their mathematical structure, and therefore influence their predictions. A crucial assumption is the host–vector interaction encapsulated in the disease-transmission term, and a key prediction is the basic reproduction number, ℛ0. We connect these two model elements by demonstrating how the choice of transmission term qualitatively and quantitatively alters ℛ0 and therefore alters predicted disease dynamics and control implications. Whereas some transmission terms predict that reducing the host population will reduce disease outbreaks, others predict that this will exacerbate infection risk. These conflicting predictions are reconciled by understanding that different transmission terms apply biologically only at certain population densities, outside which they can generate erroneous predictions. For West Nile virus, ℛ0 estimates for six common North American bird species indicate that all would be effective outbreak hosts.

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).

Figure 1.

 Flow diagrams of (a) the core West Nile virus model, (b) the core model with added epidemiological dynamics, and (c) the core model with added vital dynamics and population stage structure. Shaded boxes highlight the core model; solid lines indicate movement of individuals in and out of classes; dashed lines indicate disease transmission by mosquito bite. Variables and parameters are defined in Table 2.

A dynamical disease model generates the basic reproduction number, ℛ0, 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 ℛ0 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 ℛ0 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 ℛ0, 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 ℛ0.

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 ℛ0. 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).

Table 1.   Comparison of seven dynamical models of mosquito-borne virus transmission
Disease systemWest Nile virusJapanese encephalitisRoss River virus
ModelWN11 WN22 WN33 WN44 WN55 JE6 RR7
  1. Classes for vectors (V) and reservoirs (adult R, juvenile r) determine the number of model equations, of which only the subset in parentheses contribute to the calculation of ℛ0. Disease transmission functions are reservoir frequency dependence (FR), mass action (MA), susceptible frequency dependence (FS), and a constant rate from hosts outside the model (C). Inclusion of epidemiological and vital dynamics features indicated as Y, yes or ·, no.

  2. Sources: 1 Lord & Day (2001a,b); 2 Thomas & Urena (2001) as converted to continuous time ODEs in Lewis et al. (2006b); 3 Wonham et al. (2004); 4 Bowman et al. (2005); 5Cruz-Pacheco et al. (2005); 6 Tapaswi et al. (1995); 7 Choi et al. (2002).

No. equations8 (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 V1E V1I V1R V1S V2E V2I V2R 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 dynamicsFRMAFRFRFRFSMA, C
Epidemiological features
 Transmission probabilitiesY·YYYY·
 V viral incubation periodYYY···Y
 V loss of infectivity·····Y·
 V vertical transmission·Y··Y··
 R death from virusY·YYY··
 R recovery to immuneYYY·YYY
 R loss of immunity·····Y·
Vital dynamics
 VectorYYYYYYY
 ReservoirYY·YYY·
Age structure
 Vector··Y····
 ReservoirY······

West Nile virus models

All seven arboviral models reviewed here share a standard susceptible–infectious (SI) structure for vector and host populations. Table 1 summarizes their key features, and the Appendix presents their equations and ℛ0 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.

Core model

The three central populations of an arboviral encephalitis system like West Nile virus are the arthropod vectors, reservoir hosts and secondary hosts (Gubler 2002). The dynamical behaviour of the disease is determined by the vector and reservoir populations alone, from which the risk to secondary hosts may be inferred (but see Higgs et al. 2005).

The core vector–reservoir structure common to all seven arboviral models comprises four compartments: susceptible vectors (S V ) infectious vectors (I V ), susceptible reservoirs (S R ), and infectious reservoirs (I R ) (Fig. 1a). The vector equations describe the dynamics of adult female mosquitoes; the reservoir equations describe the dynamics primarily of birds for West Nile virus, and mammals for JE and RR virus. Cross-infection occurs when an infectious vector bites a susceptible reservoir or a susceptible vector bites an infectious reservoir. The mosquito lifecycle, which is on the order of 1 month, is represented by birth and death rates. For the bird lifecycle, which is one to two orders of magnitude longer, the birth and death rates are correspondingly lower and can reasonably be omitted for a single-season model. However, as the model requires at least one reservoir loss parameter, we begin with the natural death rate, which is used in all but two of the original models. (WN3 and RR use other loss terms.) The core model can be expressed as a system of four ordinary differential equations,

image(1)

where the total adult female vector population density N V  =S V  + I V , and the total reservoir population density N R  =S R  + I R . See Table 2 for parameter definitions. At the disease-free equilibrium (DFE), the vector and reservoir population densities are denoted inline image and inline image respectively. In this model (eqn 1) we assume reservoir frequency-dependent disease transmission (after Anderson & May 1991); the choice of transmission terms is treated in the following section.

Table 2.   Common notation for the arboviral encephalitis models reviewed here
ParameterSymbolMeanRangeSources* or equation
  1. Subscript V refers to vectors, and R to reservoirs. All rates are per capita per day, except for the mosquito biting parameter for mass action inline image, which is bites per day per unit density birds. Values indicated with – not used in ℛ0 calculations. Species-specific values for three additional reservoir parameters, π R , τ R , and σ R , shown in Table 3.

  2. *Sources: 1, Bugbee & Forte (2004); 2, Colton et al. (2005); 3, Cornell Lab of Ornithology, http://www.birds.cornell.edu/; 4, Dohm et al. (2002); 5, Dyer et al. (1977); 6, Goddard et al. (2002); 7, Goddard et al. (2003); 8, Griffith & Turner (1996); 9, Hayes & Hsi (1975); 10, Hickey & Brittingham (1991); 11, Milby & Wright (1976); 12, Mpho et al. (2002); 13, Oda et al. (1999); 14, Sardelis & Turell (2001); 15, Suleman & Reisen (1979); 16, Tiawsirisup et al. (2004, 2005); 17, Turell et al. (2000); 18, Turell et al. (2001); 19, Vanlandingham et al. (2004); 20, Walter & Hacker (1974); 21, Work et al. (1955).

  3. †Approximated from a range of rates reported for passerines.

Vector
 Recruitment rate a V
 Maturation rate m L 0.070.05–0.099, 12
 Natural death rate, adults d V 0.030.02–0.0713, 15, 20
 Natural death rate, larvae d L 0.020.01–0.0613, 15, 20
 Birth rate assuming no larval class b V CalcCalc b V  = d V
 Birth rate assuming larval class b L CalcCalc b L  = d V (m L  + d L )/m L
 Infected proportion of births ρ V 0.0010.000–0.0021, 4, 7, 18
 Probability of virus transmission to vector α V 0.690.23–1.002, 7, 14, 16, 17, 18, 21
 Virus incubation rate κ V 0.100.09–0.1214
 Proportion surviving viral incubation period ϕ V CalcCalc ϕ V  = κ V /(d V  + κ V )
 Loss of infectivity η V 0.0519
 Biting rate β R 0.440.34–0.538
 Biting rate per unit reservoir density inline image CalcCalc inline image (see text)
Reservoirs
 Recruitment rate a R
 Maturation rate m R
 Natural death rate† d R 0.00150.001–0.0023, 5, 10, 11
 Birth rate b R  = d R b R CalcCalc
 Probability of virus transmission to reservoir α R 0.740.27–1.002, 6, 16, 17, 18, 19
 Death rate from virus δ R CalcCalc inline image
 Recovery rate to immunity γ R CalcCalc inline image
 Loss of immunity rate η R

Transmission terms

Disease transmission between vectors and reservoirs in arbovirus systems depends at its simplest on the mosquito biting rate. The term used to capture the biting rate in an SI model may take one of several forms. We compare the assumptions and effects of three forms that have been used in these seven models: reservoir frequency dependence, mass action, and susceptible frequency dependence. The dynamics of reservoir frequency dependence can be thought of as intermediate between the extremes captured by the other two terms.

We will examine the biting rate from two perspectives: the number of bites per unit time by an individual vector, and the number of bites per unit time on an individual reservoir. We show how each transmission term is valid only within a range of vector and reservoir population densities, beyond which it assumes an unrealistically high or low biting rate. We introduce notation for the vector and reservoir population densities at the DFE, at which the biting rates, and therefore the ℛ0 values, coincide for the different transmission terms. Reservoir frequency dependence coincides with mass action at inline image and inline image, the two forms of frequency dependence coincide at inline image and inline image, and mass action coincides with susceptible frequency dependence at inline image and inline image (Fig. 2). The implications for the disease reproduction number, ℛ0, are treated in the following sections.

Figure 2.

 Different disease-transmission terms assume different biting rates (a,c) and lead to different reproduction numbers, ℛ0 (b,d) in arboviral models. Each transmission term (FR, reservoir frequency dependence; MA, mass action; FS, susceptible frequency dependence) can appropriately be applied only at the population densities indicated by the solid lines. If a term is applied to populations in the dotted line regions, it will give misleadingly high or low ℛ0 values. The number of bites per day by a single vector mosquito is shown as a function of reservoir density in (a), and the number of bites per day on a single reservoir is shown as a function of vector density in (c). The reproduction number ℛ0 is shown as a function of reservoir density in (b) and vector density in (d). Threshold densities marked on the x-axes indicate intersection points for the three transmission terms. At mid-reservoir densities inline image , the biting rates (a,c) and the ℛ0 (b,d) of MA and FR coincide. At high reservoir densities inline image , the biting rate on reservoirs for FR lies below that of MA, whereas at low reservoir densities inline image , it lies above (c). At high reservoir densities, ℛ0 for MA lies above that of FR (d); at low reservoir densities the relative positions of the two curves are swapped (not shown).

Reservoir frequency dependence

The commonly used reservoir frequency-dependent transmission (eqn 1) follows Anderson & May (1991) in assuming that the vector biting rate is saturated, and not limited by reservoir density. In other words, the biting rate by vectors is constant across reservoir densities, and the biting rate experienced by reservoirs increases with vector density (Fig. 2a,c). Over the same range of vector densities, the biting rate on reservoirs increases faster for a lower reservoir density than for a higher one (Fig. 2c).

These biological assumptions are evident in the mathematical formulation of the transmission terms, in which the proportional susceptible and infectious reservoir densities appear (eqn 1). At the DFE, both populations are entirely susceptible, i.e. S R  = N R and S V  = N V . Thus, at the DFE, the vector-to-reservoir transmission rate β R I V S R /N R depends only on the vector biting rate, while the reservoir-to-vector transmission rate β R S V I R /N R depends also on the ratio of vector and reservoir densities (eqn 1).

Under reservoir frequency dependence, the biting rate by vector mosquitoes is taken to be the maximal rate allowed by the gonotrophic cycle, i.e. the minimum time required between blood meals for a female to produce and lay eggs. This biting rate, β R , has units per time, and can be thought of as the maximum possible number of bites per day made by a single mosquito. In contrast, the biting parameter for mass-action transmission has different dimensions. (See Begon et al. 2002 for further explanation of transmission term units.) Reservoir frequency dependence is used in models WN1, WN3, WN4 and WN5.

Mass action

A second common disease transmission term is mass action (e.g. McCallum et al. 2001). Mass action assumes that biting rates are limited by the densities of both vectors and reservoirs (Fig. 2a,c). At the DFE, the vector-to-reservoir transmission rate inline image is thus a function of reservoir density, whereas the reservoir-to-vector transmission rate inline image is a function of vector density (Fig. 2a,c).

Although the assumption of mass action is widely used, it is valid only up to the threshold reservoir density denoted inline image. We can understand this limit by examining the transmission term, in which the biting parameter inline image has units per time per density, and can be thought of as the number of bites per day made by a single vector, per unit density of reservoirs. It is calculated as inline image. Above inline image, the vector biting rate in units of bites per time, inline image, would exceed β R , and therefore by definition would exceed the physiological capacity of the vector (Fig. 2a). The vector biting rates under mass action and reservoir frequency dependence coincide when inline image (Fig. 2a). At lower reservoir densities where inline image, the biting rate of reservoir frequency dependence is unrealistically high. At higher densities where inline image, the biting rate of mass action is unrealistically high. Mass action is used in models WN2 and RR.

Susceptible frequency dependence

A third type of transmission, susceptible frequency dependence, assumes that the biting rate is not limited by either reservoir or vector density. The resulting vector biting rate β R is the same as for reservoir frequency dependence (Fig. 2a). However, the biting rate on reservoirs is assumed to be at some maximum, denoted inline image, which is reached at a threshold vector density (Fig. 2c). Even if the vector density exceeds this threshold, the biting rate experienced by reservoirs does not exceed inline image. We consider this latter assumption unlikely to hold in an ecological system. However, as it is used in model JE and subsequent analyses (Tapaswi et al. 1995, Ghosh & Tapaswi 1999), we examine its structure and implications.

Mathematically, the transmission terms for susceptible frequency dependence are formulated as a function of the proportional susceptible population, whether reservoir or vector. The vector-to-reservoir term, β R I V S R /N R , is the same as for reservoir frequency dependence, but the reservoir-to-vector term β R I R S V /N V is not. Under susceptible frequency dependence, the vector biting rate β R intersects that of mass action at inline image (Fig. 2a). The biting rate on reservoirs, inline image, intersects that of mass action at the threshold vector density inline image, and that of reservoir frequency dependence at inline image (Fig. 2c). Susceptible frequency dependence would clearly overestimate the biting rates below these threshold population densities (Fig. 2a,c). In the absence of empirical evidence for a vector-density threshold, this is probably not a biologically applicable transmission term.

Each model in this review uses one of these three transmission terms (Table 1). More complex models could incorporate saturating or other transmission functions to capture a range of spatial and temporal variation in host and vector population densities (e.g. McCallum et al. 2001; Keeling 2005).

Effect of transmission terms on ℛ0

The disease basic reproduction number, ℛ0, provides key insights into disease outbreak and control (Anderson & May 1991; Hethcote 2000; Dobson & Foufopoulos 2001; Heesterbeek 2002; Heffernan et al. 2005). It represents the average number of secondary infections deriving from the introduction of an infected individual into an otherwise susceptible population. Quantitatively, it has a threshold value of 1: when ℛ0 > 1 a disease outbreak can occur, and when ℛ0 < 1 it will not. Qualitatively, the expression for ℛ0 indicates which elements of the disease system can be manipulated to reduce the chance of an outbreak.

To evaluate the effects of different model features on ℛ0, we first determine analytical expressions for ℛ0 using the next generation matrix method (Diekmann & Heesterbeek 2000; van den Driessche & Watmough 2002). For details, see the Appendix. The three disease-transmission terms, reservoir frequency dependence, mass action and susceptible frequency dependence, generate different expressions for ℛ0, with different implications for disease outbreak prediction and control (Fig. 2c,d).

Reservoir frequency dependence

The core model with reservoir frequency-dependent transmission (eqn 1; Fig. 1a) has the basic reproduction number

image(2)

This expression consists of two elements under the square root sign. The first represents the number of secondary reservoir infections caused by one infected vector. The second represents the number of secondary vector infections caused by one infected reservoir. Taking the square root gives the geometric mean of these two terms, which can be interpreted as ℛ0 for the addition of an average infected individual, whether vector or reservoir, to an otherwise susceptible system.

This ℛ0 expression (eqn 2) with frequency-dependent transmission is notable in that it contains the ratio of the susceptible vector and reservoir densities at the DFE. By inspection, we can see that reducing the vector density inline image will reduce ℛ0 and therefore help prevent an outbreak (Fig. 2d). In contrast, reducing the reservoir density inline image will increase ℛ0 and therefore increase the chance of outbreak (Fig. 2b). Although this latter result may seem initially counterintuitive, it follows directly from the biological assumption that the vector biting rate is not affected by reservoir density (Fig. 2a). Consequently, a reduction in reservoir density means the remaining individuals are bitten more frequently. This focuses disease transmission on a few highly bitten individuals that are more likely to become infected and more likely to re-infect the vectors. Setting ℛ0 = 1 under reservoir frequency dependence gives the threshold vector density for a disease outbreak, inline image, which is an increasing function of reservoir density.

Mass action

Using mass-action transmission in the core model gives a different reproduction number, namely

image(3)

In this case, ℛ0 is sensitive not to the ratio, but to the absolute densities of vectors and hosts. Thus, the model predicts that reducing either vector or reservoir density will reduce ℛ0 and reduce the chance of disease outbreak (Fig. 2b,d). In terms of the reservoir population, this prediction is opposite to that of reservoir frequency dependence.

The ℛ0 for mass action and reservoir frequency dependence are equal when the reservoir density inline image (Fig. 2b,d). The consequence of misapplying mass action at higher reservoir density is that ℛ0 is artificially high (Fig. 2b). In contrast, the ℛ0 of reservoir frequency dependence is artificially high at lower reservoir density (Fig. 2b). With respect to vector density, the ℛ0 curves for both transmission terms overlap when inline image (Fig. 2d). When inline image, the curve for mass action is higher and that for reservoir frequency dependence is lower (Fig. 2d). In the opposite case of inline image, the relative positions of the two curves are reversed.

Setting ℛ0 = 1 under mass action gives the threshold vector density inline image, which is a decreasing function of reservoir density. This threshold for vector–host models can be compared with that resulting from mass-action transmission in directly transmitted disease models (McCallum et al. 2001).

Susceptible frequency dependence

When disease transmission is assumed to be susceptible frequency dependent, the vector and host densities cancel out during the calculation of ℛ0. The resulting reproduction number is a constant value that does not vary with the density of either species,

image(4)

This formulation differs importantly from the previous two, in that it predicts that the chance of disease outbreak is not influenced by controlling either reservoir or vector densities (Fig. 2b,d). The lack of density threshold for disease outbreak in susceptible frequency-dependent transmission can be compared with the case for frequency dependence in a directly transmitted disease model (McCallum et al. 2001).

With respect to the reservoir density, ℛ0 for susceptible frequency dependence intersects that of mass action at inline image (Fig. 2b), which is also when the densities inline image (compare eqns 3 and 4). The ℛ0 values for the two frequency-dependent terms intersect at inline image (Fig. 2b), which is also when the densities inline image (compare eqns 2 and 4). With respect to vector density, ℛ0 for susceptible frequency dependence intersects that of mass action at inline image, and that of reservoir frequency dependence at inline image (Fig. 2c). When inline image, the vector density thresholds coincide for all three transmission terms (Fig. 2d).

If the assumptions of susceptible frequency dependence held in a natural system, mass action and reservoir frequency dependence would give an artificially high ℛ0 if misapplied above the threshold vector density (Fig. 2d). However, since these assumptions seem unlikely to hold, we anticipate that susceptible frequency dependence will almost always give an incorrect ℛ0.

The appropriate choice of disease-transmission term is clearly determined by the vector and host population densities. For a given vector density, it is appropriate to assume mass-action transmission when low reservoir densities limit the biting rate, and reservoir frequency dependence at higher reservoir densities where the biting rate is saturated (Fig. 2a). For a given reservoir density, it seems appropriate to assume mass action or reservoir frequency dependence across all vector densities (Fig. 2c). Choosing an inappropriate transmission term can lead to a misleadingly high or low ℛ0, and correspondingly inaccurate disease predictions (Fig. 2b,d).

Numerical values of ℛ0

The qualitative analysis above illustrates how different transmission terms alter the expressions and interpretations of ℛ0. Do these results translate into significant differences in the numerical estimates of ℛ0? To address this question, we generate quantitative ℛ0 estimates that incorporate the underlying variation in the constituent model parameters. We follow a standard methodology of constructing a triangular distribution for each parameter, based on reported minimum, mean and maximum estimates. We then use 10 000 Monte Carlo realizations from each triangular distribution to generate an estimated distribution of ℛ0 (e.g. Blower & Dowlatabadi 1994; Sanchez & Blower 1997; Chowell et al. 2004). Distributions that do not overlap at the fifth or 95th percentiles are taken to be significantly different.

For parameter estimates from a single study, we used the minimum, mean, and maximum values reported in that study. When more than one study was available, we took the overall mean of the average values across studies, and the minimum and maximum from all studies combined (Tables 2 and 3). For parameters with only a single point estimate, we used that value as a constant. The reported values for most parameters were based on small sample sizes, so we consider the estimates and distributions used here to be preliminary.

Table 3.   Mean (and range) of infection and mortality parameter values for six West Nile virus bird reservoir species
ParameterProbability of surviving infectionDays infectiousDays to death
  1. Sources: Komar et al. (2003, 2005); Brault et al. (2004); Langevin et al. 2005; Reisen et al. 2005; Work et al. (1955).

  2. *Mean and maximum values from Komar et al. (2005); minimum value of 1 day selected for simulation purposes.

Symbol π R σ R τ R
American crow0.003.25 (3–5)5.1 (4–6)
American robin1.003 (4–5)n.a.
Blue jay0.253.75 (3–5)4.7 (4–5)
House sparrow0.47 (0.16–0.90)3 (2–6)4.7 (3–6)
Northern mockingbird1.001.25 (1–2)*n.a.
Northern cardinal0.781.5 (1–3)*4

For vectors, we preferentially used parameter values from the mosquitoes Culex pipiens sspp., and used related species (Culex quinquefasciatus and Culex tritaeniorhynchus) when necessary. Most reservoir data, including virus transfer estimates between vectors and reservoirs, were based on experiments with domestic chickens (Gallus gallus), but in some cases multiple species were used. To compare ℛ0 for different model structures, we used mortality and recovery rates for house sparrows Passer domesticus. We later compare a single model using mortality and recovery rates for six North American reservoir species.

Different transmission functions in the core model led to significantly different numerical estimates of ℛ0, ranging over an order of magnitude or more (Fig. 3). At low reservoir density where mass action applies, the ℛ0 of reservoir frequency dependence was significantly higher, and that of susceptible frequency dependence significantly lower (Fig. 3a). At the threshold reservoir density where both mass action and reservoir frequency dependence apply, the ℛ0 of susceptible frequency dependence remained significantly lower. At higher reservoir density where only reservoir frequency dependence applies, the ℛ0 of mass action was significantly higher, and that of susceptible frequency dependence significantly lower (Fig. 3a).

Figure 3.

 Numerical estimates of ℛ0 differ significantly for the core arboviral model under three different transmission assumptions: MA, mass action; FR, reservoir frequency dependence; FS, susceptible frequency dependence. Vertical dotted lines separate regions of low inline image , medium inline image , and high inline image reservoir densities in (a) and low inline image , medium inline image , and high inline image vector densities in (b). Shaded boxes indicate the biologically appropriate transmission terms for each population density range (all FS boxes in (a) are unshaded). Sample population densities chosen to illustrate different regions of ℛ0 curves, expressed as number km−2, are (a) inline image , and inline image (low), 100 (mid), and 500 (high), and (b) inline image , inline image , and inline image (low), 450 (mid), and 5000 (high). Parameter values as in Tables 2 and 3, using disease mortality and recovery rates for house sparrows. Boxes show median and 25th and 75th percentiles, bars show 10th and 90th percentiles, and dots show fifth and 95th percentiles.

At lower vector densities where mass action and reservoir frequency dependence apply, the ℛ0 of susceptible frequency dependence tended to be higher (Fig. 3b). At the intermediate threshold vector density where all three transmission terms are equivalent, the ℛ0 distributions overlapped substantially. In the hypothetical case of susceptible frequency dependence, the ℛ0 of mass action and reservoir frequency dependence were significantly higher when inline image (Fig. 3b). These numerical results show that, for these parameter values, a transmission term applied to an inappropriate host or vector population density can over- or underestimate ℛ0 by an order of magnitude or more.

Effects of other model features

We now return to the core model (eqn 1) and investigate the effects on ℛ0 of different epidemiological features, species parameter values and life history features.

Epidemiological features

The seven arboviral models include a range of epidemiological parameters (Table 1, Fig. 1b). For vectors, some models add an exposed class (E V ) with an associated transition rate to account for the observed viral incubation period in mosquitoes. Additional vector modifications include a return rate from infectious to susceptible, and a vertical (transovarial) transmission probability from infected vectors E V and I V to their offspring (Table 1, Fig. 1b).

For reservoirs, most models add a recovered reservoir class (R R ), which allows immune reservoirs to remain in the disease system. Most models also incorporate disease-induced mortality in infectious individuals, and one adds loss of immunity in recovered individuals (Table 1, Fig. 1b). Horizontal transmission among reservoir hosts has been reported empirically (Kuno 2001), but its scale and importance remain to be determined and we have not yet seen it incorporated in a published model. The above epidemiological features can be added to the core model by using the following system of ordinary differential equations:

image
image(5)

This expanded model (eqn 5) incorporates the exposed vector class as in models WN1, WN2, WN3 and RR, vector vertical transmission as in WN2 and WN5, reservoir disease mortality as in WN1, WN3, WN4 and WN5, reservoir recovery as in all but WN4, and reservoir loss of immunity and vector loss of infectivity as in JE.

The basic reproduction number for this model, in the absence of vector vertical disease transmission, is

image(6)

With vertical disease transmission, i.e. with ρ V  > 0,

image(7)

where

image(8)

As ℛ0 is a disease outbreak threshold, we can assess the role of each model element in eqn 7 in contributing to the chance of an outbreak. Only the probability of vertical transmission from female vectors to their offspring, ρ V  > 0, increases ℛ0. All the other epidemiological features decrease ℛ0. The exposed vector compartment reduces ℛ0 by introducing into the numerator the proportion of vectors surviving the viral incubation period to become infectious (ϕ V ). The virus transmission probabilities in the numerator (α V , α R ), and the added vector and reservoir loss rates in the denominator (η V , δ R , γ R ), also reduce ℛ0.

An additional term that might intuitively be expected to increase the chance of disease outbreak is reservoir loss of immunity, η R . However, this term does not appear in ℛ0. Its absence results from the definition of ℛ0, which is calculated at the DFE using only the equations for infected individuals, and is valid only during the earliest stages of an outbreak. Terms such as η R that do not appear in the equations for infected individuals will not appear in ℛ0, although they may still influence the predicted long-term disease trajectory.

These different epidemiological features also affect the numerical estimate of ℛ0. For the sample parameter values used here, ℛ0 values ranged over two orders of magnitude (Fig. 4a). The two parameters with the greatest individual influence on ℛ0 were the added reservoir loss terms: the disease-induced death rate δ R and the recovery rate γ R (Fig. 4a). Both rates are two to three orders of magnitude greater than the natural death rate d R (Table 3). Among the seven arboviral models, these three loss rates are incorporated in different combinations: two models use all three terms, but the others use subsets (d R  + δ R ), (d R  + γ R ), (δ R  + γ R ), and γ R alone (Table 1). For the sample parameter values used here, the choice of loss terms clearly has a significant effect on a model's ℛ0. The individual addition of the remaining five epidemiological parameters generated ℛ0 distributions overlapping that of the core model (Fig. 4a). The model with all features combined had the lowest ℛ0 (Fig. 4a).

Figure 4.

 The reproduction number ℛ0 differs for different epidemiological features added to the arboviral core model (a), and for different reservoir bird species (b,c). In (a), ℛ0 shown for (1) the core model, and the core model with added parameters; (2) ρ V  ; (3) ϕ V  ; (4) η V  ; (5) α V and α R ; (6) γ R ; (7) δ R ; (8) the core model with all additional parameters. Parameter values as in Tables 2 and 3, using disease mortality and recovery rates for house sparrows. In (b), ℛ0 as in eqn 6 shown for different reservoir bird species: (1) American crows; (2) American robins; (3) house sparrows; (4) blue jays; (5) Northern mockingbirds; (6) Northern cardinals. In (c) the ℛ0 index indicates relative reservoir capacity of the same species, accounting for their relative abundances and vector feeding preferences. Parameter values as in Tables 2 and 3; population densities ( inline image , inline image  = (10 000 km−2, 1000 km−2). Boxes show median and 25th and 75th percentiles, bars show 10th and 90th percentiles, and dots show fifth and 95th percentiles.

Variation among reservoir species

The quantitative effects of adding these different epidemiological features depend to some extent on the species chosen for model parameterization. We illustrate this effect by estimating ℛ0 values for six common North American bird species, American crows Corvus brachyrhynchos, American robins Turdus migratorius, blue jays Cyanocitta cristata, house sparrows Passer domesticus, Northern mockingbirds Mimus polyglottos and Northern cardinals Cardinalis cardinalis. We first estimate ℛ0 assuming each species constitutes simultaneously the entire blood meal source population for mosquitoes, and the entire disease reservoir population (Fig. 4b). We then estimate a more ecologically informative ℛ0 index that assumes all species are available for blood meals, but only one species can act as a disease reservoir (Fig. 4c).

For each bird, we used species-specific values of δ R and γ R (Table 3). An additional species-specific parameter reported in the empirical literature is the mean infectiousness rate, i (Komar et al. 2003), which would correspond to the transition rate from exposed to infectious classes if reservoirs were modelled with an SEIR structure. As this approach has not yet been taken in these models, we did not incorporate it here. The resulting ℛ0 distributions for the six species are all significantly above the threshold value of 1, illustrating that each species could serve as a reservoir for a West Nile virus outbreak (Fig. 4b). Under the assumption that each bird species was the only mosquito blood meal source, the ℛ0 for American crows is highest, and that of Northern cardinals the lowest.

Realistically, of course, multiple reservoir species are simultaneously present, and the biting rates they experience vary according to their abundance and to vector preference. When this is taken into consideration, the reservoir species’ relative contributions to disease outbreak alter. We explore these factors by introducing an ℛ0 index to give a relative ranking for each reservoir species that incorporates its contribution to the proportion of mosquito blood meals in a natural system. To do this, we multiply the mosquito biting rate by the reported proportion of blood meals for each species, and calculate ℛ0 according to the same formula as above. The resulting ℛ0 index values will always be lower than the true ℛ0 for each species, and cannot be used to predict a disease outbreak. They do, however, allow a relative rank comparison of reservoir species in a fuller ecological context.

In this example, we constructed triangular parameter distributions based on the mean and approximate range of Cx. pipiens spp. blood meal proportions reported for the same six bird species in one study in Tennessee: crows 2.8%, 0–8.2%; robins 11.1%, 0.8–21.4%, jays 8.3%, 0–17.3%; sparrows 2.8%, 0–8.2%; mockingbirds 30.6%, 15.6–45.6%; cardinals 19.4%, 6.5–32.3% (Apperson et al. 2004). We then multiplied the biting rate by this proportion, and calculated ℛ0 as before. We emphasize that since these blood meal data are drawn only from a single case study, the index values calculated here are specific to this Tennessee season and locale.

The ℛ0 index distributions indicate that the six reservoir species are comparable in their potential contribution to a disease outbreak (Fig. 4c). The difference between species ranks in Fig 4a,b arises because species with a higher ℛ0 in Fig. 4b had a lower proportion of blood meals (e.g. crows) and those with a lower ℛ0 in Fig. 4b had a higher proportion of blood meals (e.g. cardinals). The advantage of using this ℛ0 index is that it incorporates both disease transmission and blood meal data, providing a more realistic assessment of the relative role each reservoir species would be likely to play in a natural disease outbreak.

Life history features

We return again to the core arboviral model to consider the effects of adding vital dynamics and stage structure on ℛ0 (Table 1, Fig. 1c). The vital rates for vector mosquitoes, which have a relatively short lifecycle on the order of 1 month, are already incorporated in the core model. More complex seasonal or density-dependent recruitment functions can also be used (e.g. WN1 and JE) but limit the model's analytical tractability. Larval and pupal stages, which may represent up to a quarter of the vector lifespan, but are not involved in vector–host disease transmission, may be added as an additional class (as in WN3).

For reservoirs, a birth rate must be considered if the model is to apply beyond a single season. Again, this may be added as a constant term or as a more complex function. Reservoir stage structure can be used to account for different disease susceptibilities of, for example, juveniles and adults (e.g. WN1). The core model can be modified to include these vital dynamics and stage structures as follows (Fig. 1c):

image
image(9)

This model (eqn 9) incorporates a vector larval stage L V with associated birth rate b L and maturation rate m L , as in WN3. It also includes a simplified reservoir stage structure based on WN1, where the population is divided into younger and older stages denoted with subscripts r and R respectively. Equation 9 also includes parameters from eqn 5 that may differ between the two reservoir stages (α i  , β i  , d i  , δ i  , γ i ; i = r, R ), and includes the parameter α V for symmetry in disease transmission. It has the basic reproduction number

image(10)

To evaluate the effects of these life history features on ℛ0, we follow model WN1 in assuming that juvenile reservoirs are more susceptible than adults to both natural and disease mortality, and recover more slowly from infection. Relative to a homogeneous adult population, the additional terms d r  > d R and δ r  > δ R therefore decrease ℛ0, whereas α r  > α R and γ r  < γ R increase ℛ0. The net effect of adding a juvenile stage thus depends on the differences in parameter values and on the relative densities of the two stages (see for example Lord & Day 2001a,b). The addition of vector-stage structure does not influence ℛ0 or the disease trajectory. This is because of the steady-state assumption at the DFE that b L  = d V (m L  + d L )/m L (Wonham et al. 2004), which means that vector replacement in the population is instantaneous.

Incorporating two or more reservoir species (e.g. Dobson 2004) or vector species (e.g. Choi et al. 2002) would similarly introduce additional equations and parameter sets into ℛ0. The net effect of these additions, in terms of disease dilution or amplification (Dobson 2004), would again depend on the relative densities of each species and on the specific parameter values.

Implications and future directions

Implications of model comparison

In this review, we have synthesized a group of mathematical models to illustrate how biological assumptions alter model predictions in host–vector diseases. Our analytical results show that different disease-transmission assumptions generate fundamentally different disease basic reproduction numbers. The qualitative differences in ℛ0 expressions can lead to diametrically opposed predictions for disease control, and the quantitative ℛ0 values can span orders of magnitude. These conflicting predictions can be reconciled by appreciating that each disease-transmission term can realistically be applied only to a certain range of vector and reservoir population densities. If a term is misapplied outside this range, the resulting ℛ0 values can be misleadingly low or high, and predicted control strategies may backfire.

The assumption of reservoir frequency-dependent transmission applies when the biting rate by vectors is saturated but the biting rate on reservoirs is not. It predicts that controlling the vector density decreases the chance of disease outbreak, whereas controlling the reservoir density increases the chance of outbreak. This is in stark contrast to the other transmission terms. Mass-action transmission, which applies when both biting rates are limited by low vector and reservoir densities, predicts that controlling either vector or host density will reduce the chance of disease outbreak. Susceptible frequency dependence, which assumes that the biting rates by vectors and on reservoirs are both saturated, predicts that control of neither vector nor host density will influence the chance of disease outbreak, but this assumption seems unlikely to hold in a natural system. Quantitatively, the use of an inappropriate transmission term can incorrectly increase or decrease the estimate of ℛ0 by a factor of two or more orders of magnitude in the examples we calculated. As each of the three transmission terms is in use in West Nile virus and related disease modelling (Table 1), it is essential to establish which is appropriate for a given system before making model-based disease predictions and control recommendations.

Additional assumptions about disease epidemiology also significantly affected ℛ0. For the system we examined, simpler models with fewer parameters generated higher ℛ0 values, and therefore a greater predicted risk of outbreak, than more complex and realistic models. The ℛ0 of the most complex model was some two orders of magnitude lower than that of the simplest model.

Comparing the ℛ0 of six bird species showed that all are intrinsically effective reservoirs for a West Nile virus outbreak, with American crows the highest and Northern cardinals the lowest. We introduced the ℛ0 index to identify the relative contributions of each species in an ecological context that includes the proportion of mosquito blood meals that each species supplies. The ℛ0 index values for all six species were indistinguishable, showing that all could play an approximately equal role in an outbreak.

For clarity, we separately synthesized the effects of transmission terms, epidemiological features, and life history features, but for a given model the effects of all these assumptions should be assessed together. Most of the seven original arboviral encephalitis models presented numerical simulations of the disease trajectory, but in only one case were these validated against observed outbreak data (WN3). With the seven models now synthesized into a systematic array, reasonable next steps would include model parameterization, validation against independent data, and model selection. As most of the parameter estimates for West Nile virus are currently based on small sample sizes, parameterization would be more robust with more accurate and precise distributions for all relevant species. Validation against outbreak data would be enhanced by field estimates of these same vector and reservoir species’ densities at the beginning of the outbreak. With the resulting parameter estimates and quantified uncertainty in hand, appropriate transmission terms could better be selected, the models could better be tested and compared, and the sensitivity of ℛ0 more accurately assessed.

Future directions in arboviral encephalitis modelling

The model variations that we synthesized here are sufficiently simple to remain analytically tractable. However, important extensions in environmental, ecological, and evolutionary model complexity could also be developed and tested for host–vector models in general, and for West Nile virus in particular.

Extrinsic environmental factors influence seasonal and interannual population dynamics of both vectors and hosts (Shaman et al. 2002; Hosseini et al. 2004; Pascual & Dobson 2005) and could be incorporated into the model structure both analytically and numerically (e.g. Lord & Day 2001a,b; Koelle & Pascual 2004; Wonham et al. 2004). Spatial models (e.g. Hastings et al. 2005; Lewis et al. 2006a) could incorporate both environmental variability and animal dispersal patterns (e.g. Yaremych et al. 2004). Extending the temporal and spatial model scales would also allow dynamical models to be integrated with statistical environmental models of habitat quality and climate change for more detailed disease forecasting (e.g. Randolph & Rogers 2000; McCallum & Dobson 2002; Rogers et al. 2002).

The importance of many additional ecological features of the vectors, reservoirs, and disease, could also be tested. For example, arboviral pathogens can influence vector behaviour and survival (e.g. Grimstad et al. 1980; Moncayo et al. 2000; Lacroix et al. 2005) in ways that have yet to be incorporated into these models. The importance of disease transmission horizontally among reservoir individuals (Kuno 2001) and through secondary hosts (Higgs et al. 2005) should also be evaluated.

Although arboviral models thus far have generally been developed to consider single generic species, empirical studies show that parameter values can vary among interacting vector species (e.g. Turell et al. 2001; Goddard et al. 2002; Tiawsirisup et al. 2004, 2005), host species (e.g. Bernard et al. 2001; Komar et al. 2003) and viral strains (e.g. Brault et al. 2004; Langevin et al. 2005). The effect on ℛ0 of modelling multiple species (e.g. Choi et al. 2002; Dobson 2004), including vector biting preference for different host stages and species (e.g. Lord & Day 2001a,b; Apperson et al. 2004) also warrants further investigation. Mathematically, most of the existing models use constant rate parameters for transition from one population class to the next. This standard mathematical simplification can underestimate ℛ0 (Wearing et al. 2005), and its importance for host–vector systems should be established.

These environmental and ecological factors scale up to the evolutionary level, where mosquito hybridization (Fonseca et al. 2004), coevolution among viruses, vectors and hosts (Anderson & Roitberg 1999; Myers et al. 2000; Grenfell et al. 2004) and the interactive effects of multiple diseases (e.g. Hunter et al. 2003; Rohani et al. 2003) all play out. Although these extensions inevitably increase model complexity and reduce tractability, it remains important to assess their relative roles in linking disease dynamics and control at local and global scales.

Acknowledgements

We are grateful for support from NSERC and the Killam Foundation (MJW), NSERC Discovery and CRO grants and a Canada Research Chair (MAL), a PIMS postdoctoral fellowship and KBN grant no. 2 P03A 02 23 (JR), and NSERC and MITACS (PvdD). We thank C. Jerde, W. Nelson, and two anonymous referees for valuable input into earlier versions of this paper.

Appendix

Here we summarize the formulation and analysis of the seven arboviral encephalitis models treated in this review. For all models, we present the vector and reservoir equations using the common notation in Table 2. We also calculate the basic reproduction number ℛ0 following the next-generation matrix method (Diekmann & Heesterbeek 2000; van den Driessche & Watmough 2002). (Although convention dictates the use of a different symbol for the reproduction number when a model explicitly includes control, we use ℛ0 throughout for all models.) We include the calculation details only for models in which ℛ0 has not previously been reported in the literature from this method. For the other models, we simply give the formula and the reference. For simplicity of ℛ0 calculation, we treat all rate parameters in the original equations as constants. Although this approach leads to some simplification of the original models, particularly WN1 and JE, it allows a comparison of their basic structures. At the DFE in all models, I i  = E i  = R i  = 0, inline image, and inline image.

WN1 (simplified model II from Lord & Day 2001a,b)

These two papers model a single West Nile virus season in Florida, with vectors and juvenile and adult reservoirs. The model is extensively parameterized to allow seasonal variation in environmental factors, which influence mosquito population dynamics, feeding preference, and bird reproductive success and age structure. This is the only study to conduct a statistical analysis of parameter intercorrelation and model sensitivity. However, ℛ0 is not calculated and the full system appears too complex for theoretical stability analysis. The same model is also applied to St Louis encephalitis by setting reservoir mortality to zero.

Vectors (Lord & Day 2001a, equation 5):

image((1.1))

Reservoirs (Lord & Day 2001a, p. 304):

image((1.2))

This model treats birds that are more susceptible (juvenile, subscript r) and less susceptible (adult, subscript R) separately in compartments S i and I i  , where i = r, R. In the original, the parameters β i are functions of juvenile and adult reservoir abundances and mosquito preference, d V and κ V are linear functions of temperature, which is a cosine function of time, and a V and a r are environmentally forced functions of time. The parameter m r is originally a time delay function accounting for the survival and maturation of uninfected juvenile birds to adults. We inserted the parameter δ i into the equations dI i /dt as per the text and figures in Lord & Day (2001b), and made two minor notational corrections: in the original equation for dJ i /dt we replaced m J J s with m J J i (which in our notation is d r I r ), and in the original equation for dA in/dt we replaced r A A s with r A A in (which in our notation is γ R I R ).

Under the simplifying assumption that all parameters are constant per capita rates, the DFE (S V ,E V , inline image) or for the simpler case where all birds are adults (inline image), where inline image and inline image. Then ℛ0 is calculated as follows. The infected equations for E V I V I r  , and I R can be rewritten in matrix form, separating new infection terms (f) from vital dynamics terms (v):

image((1.3))

Calculating the respective linearized matrices at the DFE gives:

image((1.4))
image((1.5))

Thus,

image((1.6))

and the next generation matrix is

image((1.7))

There is a double zero eigenvalue of FV −1 and the remaining two eigenvalues satisfy a quadratic, so the spectral radius of FV −1 is

image((1.8))

This can be compared with ℛ0 in eqn 10. For a homogeneous reservoir population where all birds are adults, the expression reduces to

image((1.9))

which can be compared with text eqn 6.

WN2 (based on Thomas & Urena 2001 rewritten as the ODE system in Lewis et al. 2006b)

This West Nile virus model was originally formulated as a system of discrete-time difference equations. The model incorporates vectors, reservoirs and humans, and evaluates the effects of periodic spraying as a mosquito control measure. Theoretical model analysis provides parameter ranges on the amount of spraying needed to force the mosquito population to die out, but ℛ0 is not calculated. Numerical simulations are provided using unspecified parameter values. This continuous-time formulation from Lewis et al. (2006b) of the discrete-time model (Thomas & Urena 2001) omits the original mosquito spraying parameter c(t).

Vectors:

image((2.1))

Reservoirs:

image((2.2))

For 0 < ρ V  ≤ 1, this model incorporates vertical disease transmission in the vector population. The DFE with parameter constraints b V  = d V and b R  = d R is inline image. The basic reproduction number is

image((2.3))

and

image((2.4))

as calculated in Lewis et al. (2006b). These can be compared with ℛ0 in text eqns 3 and 6. The behaviour of the original model with different mosquito-spraying functions is explored by Darensburg & Kocic (2004).

WN3 (Wonham et al. 2004)

This model characterizes vector and reservoir populations in a single West Nile virus season in North America. The model is analysed theoretically to obtain disease-free equilibria, evaluate local stability and calculate ℛ0. Parameter values are obtained from the literature, the model is validated using reported outbreak data, and numerical simulations are shown. Model extensions are presented to calculate threshold mosquito densities for outbreak and to consider seasonal variation in mosquito levels. The model structure, excluding the original equation for dead birds which does not influence the dynamics, is as follows.

Vectors:

image((3.1))

Reservoirs:

image((3.2))

The parameter constraint for a constant mosquito population b L  = d V (m L  + d L )/m L gives the DFE as (L V ,S V ,E V , inline image. The disease basic reproduction number is

image((3.3))

as calculated in Wonham et al. (2004). Compare text eqn 6.

WN4 (Bowman et al. 2005)

This model treats a single West Nile season in North America for vectors, reservoirs and humans. The authors evaluate preventive strategies (mosquito spraying vs. personal prevention) using a detailed representation of five human classes. The model is analysed theoretically to obtain disease-free and endemic equilibria, evaluate local and global stability, and calculate ℛ0. Numerical results are also shown.

Vectors:

image((4.1))

Reservoirs

image((4.2))

In this model, the mosquito biting rate β R is initially presented as a general function of the total mosquito, bird and human populations, and later is assumed for simplicity to be a constant rate parameter. The system has a unique DFE, namely inline image, where inline image and inline image. The disease basic reproduction number is

image((4.3))

as calculated in Bowman et al. (2005); compare text eqn 6.

WN5 (Cruz-Pacheco et al. 2005)

This West Nile virus model is analysed theoretically to obtain disease-free and endemic equilibria and to evaluate local stability and, for the case of no reservoir-induced disease mortality, global stability. The authors define an expression for ℛ0, which is then compared for eight North American bird species. Numerical outbreak simulations are also shown for different species.

Vectors:

image((5.1))

Reservoirs:

image((5.2))

The DFE is inline image, where inline image and the vector vital rates b V  = d V . We recalculate ℛ0 from the infected equations for I V and I R written as

image((5.3))

Consequently,

image((5.4))

so that

image((5.5))

The spectral radius is then

image((5.6))

compare text eqn 7. The expression for ℛ0 obtained by our method differs from that of the corresponding term inline image defined in Cruz-Pacheco et al. (2005), p. 1170). This difference arises when ρ V is correctly treated as causing new infections and therefore entering in f, rather than as a vital rate term appearing in v (see van den Driessche & Watmough 2002). When we set ρ V  = 0, both ℛ0 above and inline image reduce to

image((5.7))

JE (simplified from Tapaswi et al. 1995)

Japanese encephalitis is a mosquito-borne arbovirus endemic to southeast Asia. It differs from West Nile virus in having important mammalian as well as bird reservoirs, and in not causing reservoir mortality. This model of JE in India includes vectors, reservoirs, and humans. It is analysed theoretically to obtain disease-free and endemic equilibria, discuss local and global stability, and calculate ℛ0. The reservoir–human subset of this model is simulated numerically in Ghosh & Tapaswi (1999).

Vectors:

image((6.1))

Reservoirs:

image((6.2))

In JE the original vector recruitment term a V  =(b V  − (b V  − d V )N V /K V )N V , where K V is the carrying capacity of adult mosquitoes. Under the simplifying assumption that a V is a constant rate parameter, the DFE is inline image, where inline image and the reservoir birth and death rates are assumed to be equal.

The ℛ0 defined in Tapaswi et al. (1995, p. 298) corresponds to our inline image, which we calculate from the infected equations written as

image((6.3))

Consequently,

image((6.4))

so that

image((6.5))

The spectral radius is then

image((6.6))

compare text eqns 4 and 6.

RR (Simplified from Choi et al. 2002)

Ross River virus is the most common mosquito-borne arbovirus in Australia. The majority of hosts are marsupials. This model considers Western Gray kangaroo reservoirs and humans, and is the only one of the seven arboviral models to incorporate two different vector species. Since the disease is endemic, disease transmission incorporates the added parameter ψ V , which accounts for vectors newly infected by reservoirs other than kangaroos and humans.

As the two sets of vector equations are identical, we show just one for ease of model comparison.

Vectors:

image((7.1))

Reservoirs (kangaroos):

image((7.2))

The original study presents numerical simulations and analytical disease outbreak thresholds, but ℛ0 cannot be calculated because there is no DFE. To allow the basic model structure to be compared, we set ψ V  = 0 and assume inline image, giving the DFE as inline image. The equations for infected individuals are then

image((7.3))

Consequently,

image((7.4))

The next generation matrix is then

image((7.5))

with spectral radius

image((7.6))

compare text eqns 3 and 6.

Ancillary