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

Dengue is a vector-borne disease that risks two-thirds of the world ’ s population particularly in tropical and subtropical regions. Strategies have been implemented, but they are only effective in the short term. A new innovative and promising strategy against dengue is by the use of Wolbachia bacterium. This requires that Wolbachia -carrying mosquitoes should persist in the population. To assess the persistence of Wolbachia -carrying mosquitoes and its effects on dengue, a number of mathematical models have been formulated and analysed. In this chapter, we review the existing mathematical models of Wolbachia carrying mosquito population dynamics and dengue with Wolbachia intervention and provide examples of the mathematical models. Simulations of the models are presented to illustrate the model ’ s solutions.


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].
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.

Mathematical modelling
This section presents background on mathematical modelling of infectious diseases. Mathematical modelling is a useful tool to understand complex phenomena including disease transmission dynamics and their control strategies. There are several types of modelling that are generally used: deterministic, stochastic, statistical, agent-based modelling, and the others. A deterministic model is mostly used because it is easily solved and can include many parameters or variables. The model is in the form of system of differential equations.
Many mathematical models have been developed to investigate disease transmission dynamics including vector-borne diseases [14]. The model is based on a standard SIR model where the human population is divided into susceptible (S), infected (I), and recovered (R) [15]. The susceptible individuals become exposed after being contacted with infected individuals at a rate β. They then recover at a rate γ. The model is written in the following system of differential equations: 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 dynamics and dengue with Wolbachia.

Overview of mathematical models of Wolbachia and dengue
In this section, we review existing mathematical model of Wolbachia-carrying mosquito population dynamics and dengue with Wolbachia intervention.
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 Wolbachiacarrying mosquitoes to dominate the population.
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.
Hancock et al. [39] developed a mathematical model and investigated the strategies for releasing Wolbachia-carrying mosquitoes and its effects on dengue transmission dynamics. They found that male-biased releases can substantially reduce the dengue transmission. Furthermore, male-biased release can be an effective strategy that results in the persistence of Wolbachia-carrying mosquitoes. Ndii et al. [36,41,42] formulated single and two serotype dengue mathematical models to investigate the Wolbachia effectiveness in reducing dengue transmission. They found that Wolbachia can reduce primary and secondary dengue infections with higher reduction in secondary infections. Hughes and Britton [35] found that Wolbachia can reduce dengue transmission in areas where the basic reproduction number is not too high. This implies that Wolbachia can reduce dengue transmission in areas with low to moderate transmission settings, which is similar to that found Ndii et al. [36] and Ferguson et al. [37]. Supriatna et al. [40] showed that Wolbachia can reduce the value of basic reproduction number.
In their other paper, they showed that the predatory and Wolbachia can reduce primary and secondary infections [43]. Furthermore, Supriatna et al. [44] investigated the use of vaccine and Wolbachia on dengue transmission dynamics [44] and showed that the optimal dengue control is determined by the epidemiological parameters and economic factors. Furthermore, they found that introducing too many Wolbachia-carrying mosquitoes would be counter-productive.

Examples and numerical simulations of mathematical models
In this section, we present examples of mathematical models of Wolbachia-carrying mosquito population dynamics and dengue with Wolbachia intervention and their numerical simulations. Here, we present an example of the mathematical model of the Wolbachia-carrying mosquito population dynamics. We present the model by Ndii et al. [19,45] and show several numerical simulations. The mosquito population is divided into aquatic (A N and A W ), male (M N and M W ) and female (F N and F W ) mosquitoes. Note that the aquatic compartment consists of eggs, larvae and pupae, which are grouped into one compartment. Furthermore, the subscripts M and W denote non-Wolbachia and Wolbachia-carrying mosquito population.
The effect of CI is captured by the following expression. The non-Wolbachia female mosquitoes reproduce when mating with non-Wolbachia males, which is governed by the following equations: and the Wolbachia-carrying females reproduce when mating with non-Wolbachia and Wolbachiacarrying males, which is governed by the following equations: Note that the population growth is limited by carrying capacity K. The maternal transmission is not perfect [46]. This means that not all Wolbachia-carrying aquatic mature to be Wolbachiacarrying adult. There is a proportion of 1 À α ð Þthat mature to be non-Wolbachia adults that is e NW 1 À α ð Þ. Note that the ratio of male and female mosquitoes is denoted by e (e N , e W , e NW ). The model is governed by the following systems of differential equations:

Numerical simulations
In this section, numerical simulations are conducted to illustrate the solutions of the model. The parameter values used are given in Table 1. The initial conditions are Figure 1 shows the numerical solutions of the model using the parameter values given in Table 1, but the Wolbachia adult mosquito death rate is 2 Â μ N . This reflects the WMelPop Wolbachia strain which reduces the mosquito lifespan by a half. Figure 1 shows that the non-Wolbachia mosquitoes dominate the population. This means that this strain cannot be used as a strategy to reduce dengue transmission. Figure 2 shows the numerical solutions of the model   Table 1, but the parameter μ W is 2 Â μ N to reflect the WMelPop Wolbachia strain.
using the parameter values given in Table 1. The Wolbachia mosquito death rate is 1:1 Â μ N which reflects the WMel Wolbachia strain. This strain reduces the mosquito lifespan by around 10%. It shows that the Wolbachia-carrying mosquitoes dominate the population. This means that WMel strain can be used in the Wolbachia intervention. Figure 3 shows the simulation results using WMel parameter values with initial conditions of It shows that the non-Wolbachia mosquitoes dominate the populations. It implies that the initial conditions also affects the persistence of Wolbachiacarrying mosquitoes.

Dengue mathematical model in the presence of Wolbachia
In this section, we give example of two-serotype dengue mathematical model. We present the model by Ndii et al. [42]. The model consists of human, non-Wolbachia and Wolbachia-carrying mosquito population. The human population is divided into susceptible (S H ), exposed to serotype i (E i H ), infected to serotype i (I i H ), temporary immunity to the serotype i (X i H ), recovered class (R H ), susceptible, exposed and infected to j strain (S   Table 1. The parameter values reflect the WMel Wolbachia strain. exposed to serotype i (E i N and E i W ) and infected to serotype i (I i N and I i W ). The subscript N and W is for non-Wolbachia and Wolbachia-carrying mosquitoes.
The model is governed by the following system of differential equations:   Table 1. The parameter values reflect the WMel Wolbachia strain but different initial conditions. The initial conditions are Model for non-Wolbachia mosquito population Model for Wolbachia-carrying mosquito population where the force of infections are where ϕ i is the antibody-dependent enhancement factor for serotype i. Note that the susceptible human becomes exposed to dengue after being bitten by non-Wolbachia and Wolbachia-infected  Table 2. Initial conditions are The parameters values used are given in Table 2. Initial conditions are mosquitoes, which then becomes infected and have temporary immunity. After a certain period in temporary immunity class, they become susceptible to the other dengue serotype. They will have secondary infection after being bitten by infected mosquitoes carrying different dengue serotype to that they are previously infected.

Numerical simulations
This section presents numerical simulations of the model. Figures 4 and 5 show the numerical simulations of primary and secondary infections in the absence and presence of Wolbachia, respectively. 5 show that Wolbachia can reduce dengue transmission. The number of infections in the presence of Wolbachia-carrying mosquitoes (see Figure 5) is smaller than that in the absence of Wolbachia-carrying mosquitoes (see Figure 4). This means that the Wolbachia can γ H Progression rate from exposed to infectious 1/5.5 day À1 [1] γ N Progression from exposed to infectious class of non-W mosquitoes 1/10 day À1 [55] γ W Progression from exposed to infectious class of W mosquitoes potentially be used to break the cycle of dengue transmission. Note that the parameter values are largely uncertain. Therefore, large data set is needed to validate the model against data.

Discussion and conclusions
The use of Wolbachia bacterium has been proposed as a new innovative strategy against dengue. A lot of research have been conducted to look at the persistence of Wolbachia-carrying mosquitoes and the potential reduction in the number of dengue cases by the use of Wolbachia bacterium. One of the approaches is by the use of mathematical model. It can be seen that mathematical model can provide insights into the persistence and the effectiveness of the Wolbachia in reducing dengue transmission dynamics.
One of the important steps in modelling is model's validation. The model can be validated against the real data. Although several parameters can be obtained from literature, it is important to estimate the influential parameters such as transmission rate against the real data. Ferguson et al. [37] validated their model against the real data. Furthermore, most parameters are strongly uncertain, which indicate that sensitivity analysis is strongly required. This aims to find the most important parameters which can guide us in collecting appropriate data to be estimated.
Models presented in this work do not take into account the environmental factors such as temperature and rainfall. These may affect the dynamics of mosquito population and hence dengue transmission dynamics. Furthermore, in our work, the ratio of male and female mosquitoes is equal, which possibly affects the mosquito's population dynamics. It is important to consider sex-biased ratio to determine its effects on the persistence of Wolbachia-carrying mosquitoes and dengue reduction.
In this paper, we review existing mathematical models of Wolbachia-carrying mosquitoes' population dynamics and dengue with Wolbachia. Examples of the mathematical models are given. It shows that Wolbachia-carrying mosquitoes can persist in the population depending on the Wolbachia strains. Furthermore, the initial conditions also affect the persistence of Wolbachiacarrying mosquito populations. It is shown that Wolbachia can potentially reduce the primary and secondary infections with higher reduction in secondary infections. Results suggest that using Wolbachia can potentially reduce the transmission of dengue and hence minimise the public health and economic burden.
The results showed that the Wolbachia can persist in the population. When mosquitoes are infected with the WMel strain of Wolbachia. For dengue mathematical models with Wolbachia, it shows that the Wolbachia can potentially reduce the primary and secondary infections. This means that using Wolbachia can be an alternative strategy against dengue.