Vector Borne Diseases with Structured Population Implications of Nonlinear Dynamics During the Maturation Phase

Vector borne diseases such as malaria, dengue fever and West Nile virus (WNV) are infectious diseases that are carried by insects from one host to another. In many of these diseases it is the mosquito that carries the virus but ticks, fleas and midges can also be responsible. The diseases can be spread to humans, birds and other animals.

This article will focus mainly on the mathematical modeling of West Nile virus but there are many other insect-borne diseases which currently constitute significant public health issues worldwide, and the mathematical modeling of some of these raises issues not covered in this article at all. Malaria is undoubtedly the best known vector-borne disease of all, with hundreds of millions of cases each year, and control of it has been complicated by widespread drug resistance of the parasites and insecticide resistance. Another of the more serious mosquito-borne diseases is dengue fever, epidemics of which are becoming larger and more frequent. Dengue has a global distribution comparable to that of malaria and up to 100 million cases per year, with a fatality rate in most countries of about 5% (though with proper treatment this can be reduced to below 1%). Dengue is caused by one of four closely related virus serotypes. Infection with one of these confers immunity for life but only to that particular serotype. Thus in principle a person can experience four dengue infections. Airline travel makes it possible for the viruses to disperse rapidly (in fact air travel to virtually any place in the world can take place nowadays in less time than the incubation period of most infectious diseases). In former times dengue could only be spread between continents if viruses and mosquito vectors could survive the long sea journeys, leading to long intervals between epidemics. There is currently no vaccine for dengue. Yellow fever is another example of a mosquito borne disease, endemic in South America and parts of Africa. A vaccine has been available for it for several decades.

Ticks can act as the vector for certain diseases, notably Lyme disease. Bluetongue is a midge-borne disease that can affect all ruminants with sheep being the most severely affected. Cattle act as an important reservoir species. Bluetongue appears to have originated in South Africa but has been reported in many European countries (including as far north as the Netherlands and Belgium), in the USA and in British Columbia and other places. Northern Europe is currently at the limits of the climatic conditions favorable for the spread of the disease (13-35° C) but climate change has the potential to change this. Since midges can be carried considerable distances by wind, it has been speculated that, in areas where bluetongue occurs only sporadically, it originates from the carriage by wind of midges from distant endemic areas.

It is widely accepted that climate change has considerable potential implications for the world wide distribution of vector-borne diseases in general, and that global warming is likely to create suitable new vector habitats. Other issues potentially contributing to the resurgence of some vector borne diseases are the excessive use of insecticide sprays to kill adult mosquitoes, the problem of insecticide and drug resistance and the general decline of the worldwide infrastructure for the surveillance, prevention and control of vector-borne diseases because the public health threat was perceived to have dropped due to previous control programs (for example, malaria had been nearly eliminated in Sri-Lanka in the 1960s).

Much has been done in terms of the mathematical modeling and analysis of the transmission dynamics and spatial spread of vector-borne diseases. However, one important biological aspect of the hosts, the stage structure, seems to have received little attention although structured population models have been intensively studied. In the context of population dynamics and spatial ecology the interaction of stage-structure with spatial dispersal has been receiving considerable attention as discussed in the last section.

The developmental stages of hosts have an important impact on the transmission dynamics of vector-borne diseases. In the case of West Nile virus the transmission cycle involves both mosquitoes and birds (the crow, jay and raven species being particularly important). Nestling crows are crows that have hatched but are helpless and stay in the nest, receiving more or less continuous care from the mother for up to two weeks and less continuous care thereafter. Fledgling crows are old enough to have left the nest (they leave it after about five weeks) but they cannot fly very well. After three or four months these fledglings will be old enough to obtain all of their food by themselves. As these facts demonstrate, the maturation stages of adult birds, fledglings and nestlings are all very different from a biological and an epidemi-ological perspective and a realistic model needs to take these different stages into account. For example, in comparison with grown birds, the nestlings and fledglings have much higher disease induced death rate, much poorer ability to avoid being bitten by mosquitoes, and much less spatial mobility [1,15,24].

Recently, Gourley, Liu and Wu [7] developed a model for the evolution of a general vector-borne disease with special emphasis on the transmission dynamics and spatial spread of West Nile virus. They started with the classical McKendrick von-Foerster equations for an age-structured reservoir population (birds, in the case of WNV) divided into two epidemiological compartments of susceptible and infected (and infectious), coupled with a scalar delay differential equation for the adult vector (mosquito) population under the assumption that the total vector population is maintained at a constant level. As a result, they obtained a system of delay differential equations describing the interaction of five sub-populations, namely susceptible and infected adult and juvenile reservoirs and infected adult vectors, for a vector-borne disease with particular reference to West Nile virus.

4.2.1 Model Derivation and Biological Interpretation

To discuss the model in [7] we can imagine a more general mosquito-borne disease with similar characteristics to WNV. We will also refer to the reservoir as the host, and assume that the host population is age-structured. We start with a simple division of the host population as susceptible hosts s(t,a) and infected hosts i(t, a) at time t and age a. These host populations are assumed to evolve according to the McKendrick von-Foerster equations for an age-structured population:

ds ds dt + da = -ds(a)s(t, a) - ((a)s(t, a)mi(t), (4.6)

and di di

— + — = -di(a)i(t, a) + ((a)s(t, a)m^(t), (4.7)

dt da where m-j(t) is the number of infected adult mosquitoes, the functions ds(a) and di (a) are the age-dependent death rates of susceptible and infected hosts.

A host becomes infected when it is bitten by an infected mosquito. Here and in what follows, we use mass action though other incidences can be used.

The host population is split into juveniles and adults, defined respectively as those of age less than some number r, and those of age greater than r . We assume:

The subscripts in these quantities refer to disease and juvenile/adult status; thus for example the per capita death rates for susceptible juveniles and infected adults would be dsj and dia respectively. The above choices enable us to formulate a closed system of delay differential equations involving only the total numbers of hosts classified as adult susceptibles, adult infected, juvenile susceptibles and juvenile infected. These total numbers are given respectively, using self explanatory notation, by

As(t) = s(t,a) da, Ai(t) = i(t,a) da, Js(t) = s(t,a) da, Jt Jt J 0

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