1. Introduction

Dengue is a vector-borne disease caused by four distinct serotypes (DEN1–DEN4), and is endemic in most countries particularly in tropical and subtropical areas [1]. It is estimated that around 390 million cases happen each year [2]. Individuals obtain lifelong immunity to the serotype that they are infected with, but have a higher chance to get the most severe form of dengue in the subsequent infection [1]. It is estimated that around 500,000 individuals get severe dengue and require hospitalisation. Of these, about 2.5% die [3]. Without a proper treatment, the fatality rate can reach 20% [3]. Dengue is also a substantial public health and economic burden [4].

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A number of strategies have been implemented, but they are generally effective in the short term. Although some progresses have been made for dengue antiviral treatment, dengue control strategies still depend on vector control [5]. One of the strategies against dengue is by the use of Wolbachia bacterium. There are two Wolbachia strains used in the experiments: WMelPop and WMel. WMelPop strain can reduce the mosquito lifespan of more than 50% and almost 20% reduction in fecundity [6]. WMel strain reduces the lifespan of around 10% and only small reduction in fecundity [6]. Wolbachia can reduce the level of virus in the salivary glands. Wolbachia gives reproductive advantage for Wolbachia-carrying female mosquitoes known as cytoplasmic incompatibility (CI). The Wolbachia-carrying female mosquitoes can reproduce when mating with both non-Wolbachia and Wolbachia-carrying male mosquitoes. Non-Wolbachia female mosquitoes can reproduce when mating with non-Wolbachia males [7]. Field experiments showed that Wolbachia-carrying mosquitoes have established and dominated the population [8]. When Wolbachia-carrying mosquitoes persist in the field, the Wolbachia intervention can be implemented. The question that arises is that to what extend this intervention can reduce dengue transmission? To answer the above question, a number of mathematical models have been formulated and analysed. Mathematical model is a useful tool to understand complex phenomena. This can be used to understand population dynamics [9], disease transmission dynamics [10, 11], and others [12, 13]. A number of mathematical models have been developed to examine the persistence and spread of Wolbachia-carrying mosquitoes and its effects on dengue transmission dynamics. In this chapter, we review the existing mathematical models of Wolbachia-carrying mosquito population dynamics and dengue with Wolbachia intervention, give examples of the mathematical models, and show several numerical simulations to illustrate the model's solutions.

The model can then be extended to include other compartments and parameters depending on the characteristics of diseases. For example, if the disease has long incubation period, we can add exposed compartment. If the disease is transmitted via vector, we can add another system of equations describing vector dynamics. When one aims to investigate the effects of vaccination, vaccinated compartment can be included. The important principles in modelling are to know characteristics of studied phenomena and the purpose of the research. The principles have been applied when we formulate mathematical models for Wolbachia-carrying mosquito population

Mathematical Model as a Tool for the Control of Vector-Borne Diseases: *Wolbachia* Example

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In this section, we review existing mathematical model of Wolbachia-carrying mosquito popu-

Many (spatial and non-spatial) mathematical models have been formulated to analyse the persistence and spread or dispersal of Wolbachia-carrying mosquitoes in the populations [9, 16–29]. The general aim is to understand the underlying factors required for the persistence

A number of nonspatial mathematical model for Wolbachia-carrying mosquito population dynamics have been developed. Ndii et al. [19] developed a mathematical model for Wolbachiacarrying mosquito population dynamics and assessed the persistence of Wolbachia-carrying mosquito populations. They found that Wolbachia-carrying mosquitoes persist in the population given that the death rate is not too high. Zhang et al. [30] formulated a mathematical model to assess the best strategies for releasing Wolbachia-carrying mosquitoes. They found that initial quantities of non-Wolbachia and Wolbachia-carrying mosquitoes and augmentation methods (timing, quantity, and order of frequency) determine the success of the Wolbachia intervention. They also formulated birth-pulse model with different density dependent death rate functions. They found that for condition with a strong density dependent death rate, the initial ratio of non-Wolbachia and Wolbachia-carrying mosquitoes should exceed a critical threshold for Wolbachia-

The spatial mathematical models have been developed to assess the Wolbachia-carrying mosquitoes' dispersal. Chan and Kim [9] used reaction diffusion approach and incorporated slow and fast dispersal mode to assess the dynamics of the Wolbachia spread. They found that temperature affects the wavespeed of the Wolbachia-carrying Aedes aegypti, that is, Wolbachia invasion for Aedes aegypti increases when the temperature decreases within the optimal temperature rate for mosquito survival. Hancock et al. [17] developed a metapopulation model to assess the spatial dynamics of Wolbachia. They found that spatial variation in the densitydependent competition experienced by juvenile host insects can influence the spread of Wolbachia into population. In their other paper [16], they found a new expression for the threshold which takes into account the main aspects of insects' life history. They showed that

constant or pulsed immigrations affect the spread of Wolbachia-carrying mosquitoes.

3. Overview of mathematical models of Wolbachia and dengue

lation dynamics and dengue with Wolbachia intervention.

and spread of Wolbachia-carrying mosquitoes.

carrying mosquitoes to dominate the population.

dynamics and dengue with Wolbachia.
