The control of vector-borne disease epidemics

Abstract

The theoretical underpinning of our struggle with vector-borne disease, and still our strongest tool, remains the basic reproduction number, ${\mathcal{R}}_0$, the measure of long term endemicity. Despite its widespread application, ${\mathcal{R}}_0$ does not address the dynamics of epidemics in a model that has an endemic equilibrium. We use the concept of reactivity to derive a threshold index for epidemicity, ${\mathcal{E}}_0$, which gives the maximum number of new infections produced by an infective individual at a disease free equilibrium. This index describes the transitory behavior of disease following a temporary perturbation in prevalence. We demonstrate that if the threshold for epidemicity is surpassed, then an epidemic peak can occur, that is, prevalence can increase further, even when the disease is not endemic and so dies out. The relative influence of parameters on ${\mathcal{E}}_0$ and ${\mathcal{R}}_0$ may differ and lead to different strategies for control. We apply this new threshold index for epidemicity to models of vector-borne disease because these models have a long history of mathematical analysis and application. We find that both the transmission efficiency from hosts to vectors and the vector–host ratio may have a stronger effect on epidemicity than endemicity. The duration of the extrinsic incubation period required by the pathogen to transform an infected vector to an infectious vector, however, may have a stronger effect on endemicity than epidemicity. We use the index ${\mathcal{E}}_0$ to examine how vector behavior affects epidemicity. We find that parasite modified behavior, feeding bias by vectors for infected hosts, and heterogeneous host attractiveness contribute significantly to transitory epidemics. We anticipate that the epidemicity index will lead to a reevaluation of control strategies for vector-borne disease and be applicable to other disease transmission models.

Introduction

Soon after proving the transmission of malaria by biting mosquitoes, Ross (1911) demonstrated that the prevalence of malaria tends to a fixed limit depending on the rates of transmission, recovery, and mortality within the host and vector populations. Ross's model showed that the prevalence of malaria tends toward zero, in the long-term, if a control strategy holds the number of vectors at a sufficiently low level. The realization that malaria transmission could be halted by reducing, but not eradicating, the vector population greatly influenced the modeling and control of infectious disease (Bailey, 1982). Macdonald (1952) suggested that targeting the rate of vector mortality more effectively reduced malaria prevalence than targeting other rates. Macdonald (1952) placed these rates into an index that we denote by ${\mathcal{R}}_0$. If ${\mathcal{R}}_0<1$ then malaria prevalence declines to zero in the long-term. Otherwise, malaria can become endemic. The index ${\mathcal{R}}_0$ was used to develop and evaluate control strategies meant to reduce malaria prevalence (Garrett-Jones, 1964, World Health Organization, 1975).

The index ${\mathcal{R}}_0$ is widely applied to the mathematical modeling of diseases in general (Diekmann and Heesterbeek, 2000, van den Driessche and Watmough, 2002, Thieme, 2003). Early in the mathematical study of malaria, however, Lotka (1923) used Ross's model to show that significant, but transitory, changes in prevalence often occur before reaching the long-term equilibrium determined by ${\mathcal{R}}_0$. We use this model to investigate potential causes of transitory epidemics within vector-borne disease.

The model first proposed by Ross (1911) and subsequently modified by Macdonald (1952) has influenced both the modeling and the application of control strategies to vector-borne disease. Models of malaria that investigate complications arising from host superinfection, immunity, and other factors are based on this fundamental model (Aron and May, 1982, Dietz, 1988, Koella and Antia, 2003). The model has also influenced the mathematical analysis of many other vector-borne diseases (Dye, 1992), including dengue fever (Feng and Velasco-Hernandez, 1997), rickettsia in cattle (Yonow et al., 1998), trypanosomiasis (McDermott and Coleman, 2001), and West Nile Virus (Foppa and Spielman, 2007). A standard model consistent with the assumptions of Ross (1911) and Macdonald (1952) that relates the proportion of hosts that are infected x to the proportion of vectors that are infected $y$ is given by

$$\frac{\mathrm{d}x}{\mathrm{d}t}={\mathit{ab}}_x\mathit{my}(1-x)-\mathit{rx}\text{,}\frac{\mathrm{d}y}{\mathrm{d}t}={\mathit{ab}}_{y}x({\mathrm{e}}^{-\mu n}-y)-\mu y\text{.}$$

The standard model consists of two coupled ordinary differential equations that describe the change in prevalence (i.e., the proportion of a population that is infected) for the vector and host populations. This model uses several assumptions to simplify the complexity of malaria transmission and identify the important aspects of malaria transmission between vector and host (Bailey, 1982, Aron and May, 1982, Smith and McKenzie, 2004). Each equation corresponds to a susceptible–infected–susceptible model for the vector and host populations. Infected vectors do not recover but die at rate $\mu$, and newly born vectors are susceptible. The specific effects of immunity and superinfection on hosts are ignored such that infected hosts recover at rate r and again become immediately susceptible. Vectors bite hosts at rate a and transmit infection with efficiency $b_x$. The proportion of vectors that acquire infection by biting infected hosts is $b_y$. The number of vectors per host is m, where the population sizes of hosts and vectors are assumed constant, and n is the length of the extrinsic incubation period.

In vector-borne disease, the period of time between a vector becoming infected to becoming infectious may be long compared to the lifespan of the vector. Vector mortality during this extrinsic incubation period of the pathogen within the vector has important consequences for endemicity (Macdonald, 1952). The growth equation for prevalence within vectors includes vector mortality during the extrinsic incubation period. In the standard model (1), the probability of an infected vector surviving the extrinsic incubation period is given by the quantity ${\mathrm{e}}^{-\mu n}$ (see Smith and McKenzie, 2004, Smith et al., 2007). Other formulations have explicitly modeled the time lag between when a vector becomes infected and when it becomes infectious (Aron and May, 1982, Ruan et al., 2008) but we do not consider this here. We refer to the model given by Eq. (1) as the standard model because it reflects the assumptions of both Ross (1911) and Macdonald (1952) as contained in the original index ${\mathcal{R}}_0$ for malaria.

Unlike the long-term case for endemicity, no index has summarized the short-term behavior of transitory epidemics in vector-borne disease. We initially focus on systems where long-term endemicity is not sustainable (${\mathcal{R}}_0<1$) and define an epidemic as a transitory increase in prevalence following a perturbation (Sections 2 and 3). We derive a threshold index for transitory epidemics ${\mathcal{E}}_0$, such that if ${\mathcal{E}}_0>1$ then transitory epidemics are possible; otherwise, transitory epidemics are not possible (Section 2). The index ${\mathcal{E}}_0$ contains the same parameters as ${\mathcal{R}}_0$, but the manner in which these rates affect each index differs (Section 3). For example, we demonstrate how the vector–host ratio m has a greater influence on epidemicity than endemicity. We investigate how the relative importance of potential target rates differs for control strategies that reduce epidemicity versus endemicity.

We determine the conditions for which epidemics preferentially elevate prevalence within the host population versus the vector population (Section 4). We also show that epidemics, although transitory, have lasting effects on disease prevalence. The index ${\mathcal{E}}_0$ permits comparison of epidemicity across models, and we use it to show how different assumptions of vector feeding behavior and host susceptibility affect epidemicity (Section 5). Epidemicity in systems that permit long-term endemicity is explored (Section 6). The models in this paper assume constant population sizes and can be formulated in terms of either prevalence or prevalent number (i.e., the number of infected individuals) over time. We apply these indices to prevalence and in Section 6 discuss how the alternative formulation for prevalent number affects the results.