3. Overview of mathematical models of Wolbachia and dengue

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 numeri-

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 parame-

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

dt <sup>¼</sup> <sup>β</sup>

SI

<sup>N</sup> � <sup>γ</sup>I, dR

dt <sup>¼</sup> <sup>γ</sup>I: (1)

ters or variables. The model is in the form of system of differential equations.

SI <sup>N</sup> , dI

cal simulations to illustrate the model's solutions.

114 Dengue Fever - a Resilient Threat in the Face of Innovation

dS dt ¼ �<sup>β</sup>

2. Mathematical modelling

equations:

In this section, we review existing mathematical model of Wolbachia-carrying mosquito population dynamics and dengue with Wolbachia intervention.

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 and spread of Wolbachia-carrying mosquitoes.

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.

Mathematical models for Wolbachia-carrying mosquitoes' populations consider several important aspects. They are cytoplasmic incompatibility (CI), the maternal transmission, Wolbachiacarrying mosquito death rate, release strategies of Wolbachia-carrying mosquitoes [9, 16–29, 31, 32]. These are expressed in the parameters, variables, or simulations.

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:

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

http://dx.doi.org/10.5772/intechopen.79754

MNFN MN þ FN þ MW þ FW

and the Wolbachia-carrying females reproduce when mating with non-Wolbachia and Wolbachia-

FW ð Þ MN þ MW MN þ FN þ MW þ FW

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 1ð Þ � α that mature to be non-Wolbachia adults that is eNW ð Þ 1 � α . Note that the ratio of male and female mosquitoes is denoted by e (eN, e<sup>W</sup> , eNW ).

� μNAAN � γNAN,

� μWAAW � γWAW ,

(2)

117

(3)

(4)

rN

rW

The model is governed by the following systems of differential equations:

<sup>P</sup> <sup>1</sup> � ð Þ AN <sup>þ</sup> AW

dt <sup>¼</sup> <sup>e</sup>NγNAN � <sup>μ</sup>NMN <sup>þ</sup> <sup>e</sup>NWð Þ <sup>1</sup> � <sup>α</sup><sup>W</sup> <sup>γ</sup>WAW ,

FWð Þ MW þ MN

dt <sup>¼</sup> ð Þ <sup>1</sup> � <sup>e</sup><sup>W</sup> <sup>α</sup>Wγ<sup>W</sup> AW � <sup>μ</sup>WFW:

dt <sup>¼</sup> <sup>e</sup><sup>W</sup> <sup>α</sup><sup>W</sup> <sup>γ</sup>WAW � <sup>μ</sup><sup>W</sup> MW ,

where P ¼ MN þ FN þ MW þ FW is the total population.

K 

dt <sup>¼</sup> ð Þ <sup>1</sup> � <sup>e</sup><sup>N</sup> <sup>γ</sup>NAN � <sup>μ</sup>NFN <sup>þ</sup> ð Þ <sup>1</sup> � <sup>e</sup>NW ð Þ <sup>1</sup> � <sup>α</sup><sup>W</sup> <sup>γ</sup><sup>W</sup> AW,

<sup>P</sup> <sup>1</sup> � ð Þ AN <sup>þ</sup> AW

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 AN<sup>0</sup> ¼ 0, FN<sup>0</sup> ¼ MN<sup>0</sup> ¼ 7253,

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 � μN. 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

K 

carrying males, which is governed by the following equations:

MNFN

dAN dt <sup>¼</sup> <sup>r</sup><sup>N</sup>

dMN

dFN

dAW dt <sup>¼</sup> <sup>r</sup><sup>W</sup>

dMW

dFW

AW<sup>0</sup> ¼ 0, and MW<sup>0</sup> ¼ FW<sup>0</sup> ¼ 14200.

4.1.2. Numerical simulations

A number of mathematical models have been developed to understand dengue transmission mathematical models [33, 34]. However, little mathematical models have been developed to investigate the efficacy of Wolbachia-intervention [35–40] in reducing dengue transmission. 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.
