Spatial spreading of West Nile Virus described by traveling waves

Abstract

In this work, we propose a spatial model to analyze the West Nile Virus propagation across the USA, from east to west. West Nile Virus is an arthropod-borne flavivirus that appeared for the first time in New York City in the summer of 1999 and then spread prolifically among birds. Mammals, such as humans and horses, do not develop sufficiently high bloodstream titers to play a significant role in the transmission, which is the reason to consider the mosquito–bird cycle. The model aims to study this propagation based on a system of partial differential reaction–diffusion equations taking the mosquito and the avian populations into account. Diffusion and advection movements are allowed for both populations, being greater in the avian than in the mosquito population. The traveling wave solutions of the model are studied to determine the speed of disease dissemination. This wave speed is obtained as a function of the model's parameters, in order to assess the control strategies. The propagation of West Nile Virus from New York City to California state is established as a consequence of the diffusion and advection movements of birds. Mosquito movements do not play an important role in the disease dissemination, while bird advection becomes an important factor for lower mosquito biting rates.

Introduction

West Nile Virus (WNV) is an arthropod-borne flavivirus. The primary vectors of WNV are Culex spp. mosquitoes, although the virus has been isolated from at least 29 more species of 10 genera (Campbell et al., 2002). When an infected mosquito bites a bird, the virus is transmitted. A mosquito is infected when it bites an infected bird. Also, the virus can be vertically transmitted from a mosquito to its offspring.

The intensity of transmission to human depends on the abundance and feeding patterns of infected mosquitoes, on the local ecology and behaviors that influence human exposure to mosquitoes (Hayes et al., 2005). Mammals, such as humans and horses, do not develop sufficiently high bloodstream titers to play a significant role in the transmission (DeBiasi and Tyler, 2006, Hayes, 1989), and this is a reason to consider the mosquito–bird cycle.

One major feature of WNV spatial dissemination is the high velocity of geographic invasion and colonization. This is due to the long distance flight of birds, and to the ubiquitous presence of mosquitoes. For instance, WNV was introduced in New York City in 1999, and then propagated across the USA. After five years, WNV was detected among birds in California, western USA.

Mathematical models which did not encompass spatial dynamics were developed by Kenkre et al. (2005), Wonham et al. (2004), Cruz-Pacheco et al. (2005) and Bowman et al. (2005). Those models considered different aspects of the WNV disease and determined threshold conditions with respect to control strategies. Kenkre et al. (2005) studied the periodicity of the infection considering vertical transmission, increase in mortality due to infection and time scale disparity. Wonham et al. (2004) considered the whole mosquito's life cycle. Cruz-Pacheco et al. (2005) took into account experimental data from the literature to estimate threshold values regarding several species of birds. The effects of vertical transmission on the disease dynamics were also studied, and different recovery rates were considered for different species of birds. In Bowman et al. (2005), they added the human population in order to assess preventive strategies.

With respect to spatial models, Lewis et al. (2006) considered in their model the corresponding spatially homogeneous modeling proposed by Wonham et al. (2004). They studied the propagation of WNV using traveling wave solutions for a simplified model, which did not consider vertical transmission, as well as the mortality rate induced by WNV disease and the recovered avian subpopulation. Aiming to determine the biological invasion of WNV from the east to the west coast of the USA, we develop a spatio-temporal model to study this propagation as a consequence of the zoonotic characteristic of WNV.

Since Fisher (1937) proposed a model to study the propagation of an advantageous gene in terms of reaction–diffusion equation, a large number of studies have been made about the biological invasion process using that kind of reaction–diffusion equation. The first work to describe the spatial propagation of a disease was developed by Källen et al. (1985). In that work, a directly transmitted infectious disease model was studied to describe the propagation of rabies among foxes. A simple model developed in terms of the reaction–diffusion equations was studied to determine the first front wave of propagation. More realistic models were studied after that work in order to determine the cyclic epidemic of the front wave of rabies considering the growth of susceptible fox population and immunity (Murray et al., 1986, Murray and Seward, 1992). For indirectly transmitted diseases (by vectors) recent works were developed by Lewis et al. (2006) to study the spatial propagation of WNV, and by Maidana and Ferreira (2008) to study the propagation of the hip hop disease in capybaras.

Our model concerning the spatial dynamics of WNV allows the diffusion to both avian and mosquito populations, taking into account the fact that the diffusion coefficient in the avian population is greater than the diffusion in the mosquito population. From the model we seek the traveling waves connecting two steady states, which are the disease-free and the endemic equilibria, from which we determine the wave speed of propagation of WNV disease. We also study the sensitivity of the wave speed with respect to the variations in the essential parameters of the model. In recent experiments, Komar et al. (2003) studied 25 species of birds exposed to WNV by Culex tritaniorhynchus bites in order to evaluate the transmission dynamics. We apply their estimations to our model considering the species that are more competent for WNV transmission, determining the role of different species of birds in disease propagation.

At first, we consider a biting rate of $0.5$, that is, once every two days (Cruz-Pacheco et al., 2005). Okubo (1998) estimated that the diffusion coefficient of birds ranged between 0 and $14{\mathrm{km}}^2/\mathrm{day}$. Choosing a coefficient for avian diffusion equal to $6{\mathrm{km}}^2/\mathrm{day}$, without advection and considering parameters for two species of birds—blue jay and common grackle—we obtain approximately 3 km/day for the velocity of disease propagation, which is consistent with that observed in field data. If we consider a biting rate of $0.3$, as did Lewis et al. (2006), the wave speed decreases to $1.98$ km/day. An advection velocity of $1.6–2.37\mathrm{km}/\mathrm{day}$ is needed to match with the speed range of $3–3.5\mathrm{km}/\mathrm{day}$ observed in field data. If we consider $b=0.1$ (Wonham et al., 2004), a higher value for advection is needed to be comparable with the observed data.

The paper is structured as follows. In Section 2, a WNV spatial propagation model is presented, together with the analysis of the corresponding spatial homogeneous model. In Section 3, the minimum speed of the traveling wave is determined. The sensitivity analysis of the wave speed is assessed in Section 4, and this is used to describe the geographic spread in Section 5. In Section 6, we present numerical simulation to determine the wave speed graphically and conclusion is given in Section 7.