4.1. Mathematical model of Wolbachia-carrying mosquito population dynamics and numerical simulations

#### 4.1.1. Mathematical model of Wolbachia-carrying mosquito population dynamics

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 (AN and AW ), male (MN and MW ) and female (FN and FW) 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:

$$\rho\_N \frac{M\_N F\_N}{M\_N + F\_N + M\_W + F\_W} \tag{2}$$

and the Wolbachia-carrying females reproduce when mating with non-Wolbachia and Wolbachiacarrying males, which is governed by the following equations:

$$\rho\_W \frac{F\_W(M\_N + M\_W)}{M\_N + F\_N + M\_W + F\_W} \tag{3}$$

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

$$\begin{split} \frac{dA\_{N}}{dt} &= \rho\_{N} \frac{M\_{N} F\_{N}}{P} \left( 1 - \frac{(A\_{N} + A\_{W})}{K} \right) - \mu\_{NA} A\_{N} - \gamma\_{N} A\_{N}, \\ \frac{dM\_{N}}{dt} &= \epsilon\_{N} \gamma\_{N} A\_{N} - \mu\_{N} M\_{N} + \epsilon\_{\rm NW} (1 - \alpha\_{W}) \gamma\_{W} A\_{W} \\ \frac{dF\_{N}}{dt} &= (1 - \epsilon\_{N}) \gamma\_{N} A\_{N} - \mu\_{N} F\_{N} + (1 - \epsilon\_{\rm NW}) (1 - \alpha\_{W}) \gamma\_{W} A\_{W}, \\ \frac{dA\_{W}}{dt} &= \rho\_{W} \frac{F\_{W} (M\_{W} + M\_{N})}{P} \left( 1 - \frac{(A\_{N} + A\_{W})}{K} \right) - \mu\_{WA} A\_{W} - \gamma\_{W} A\_{W}, \\ \frac{dM\_{W}}{dt} &= \epsilon\_{W} \alpha\_{W} \gamma\_{W} A\_{W} - \mu\_{W} M\_{W} \\ \frac{dF\_{W}}{dt} &= (1 - \epsilon\_{W}) \alpha\_{W} \gamma\_{W} A\_{W} - \mu\_{W} F\_{W}. \end{split} \tag{4}$$

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

#### 4.1.2. Numerical simulations

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,

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.

4. 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 (AN and AW ), male (MN and MW ) and female (FN and FW) mosquitoes. Note that the aquatic compartment consists of eggs, larvae and pupae, which are grouped into one compartment. Furthermore, the subscripts M

4.1. Mathematical model of Wolbachia-carrying mosquito population dynamics and

4.1.1. Mathematical model of Wolbachia-carrying mosquito population dynamics

and W denote non-Wolbachia and Wolbachia-carrying mosquito population.

numerical simulations

32]. These are expressed in the parameters, variables, or simulations.

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

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

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


Table 1. Parameters, description, values and sources for the model of Wolbachia-carrying mosquitoes.

using the parameter values given in Table 1. The Wolbachia mosquito death rate is 1:1 � μ<sup>N</sup> 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 AN<sup>0</sup> ¼ 0, FN<sup>0</sup> ¼ MN<sup>0</sup> ¼ 7253, AW<sup>0</sup> ¼ 0, and MW<sup>0</sup> ¼ FW<sup>0</sup> ¼ 145. It shows that the non-Wolbachia mosquitoes dominate the populations. It implies that the initial conditions also affects the persistence of Wolbachia-

Figure 2. Numerical simulations of the Model (4). The parameter values used are given in Table 1. The parameter values

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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 (SH), exposed to

superscript ji means individuals that were previously infected by serotype i and currently infected by serotype j. The mosquito population is divided into susceptible (SN and SW),

<sup>H</sup>), temporary immunity to the serotype i (X<sup>i</sup>

H, Eji H, Iji H), recov-

<sup>H</sup>, respectively). The

i

4.2. Dengue mathematical model and numerical simulations

ered class (RH), susceptible, exposed and infected to j strain (Sji

4.2.1. Dengue mathematical model in the presence of Wolbachia

<sup>H</sup>), infected to serotype i (I

carrying mosquitoes.

reflect the WMel Wolbachia strain.

serotype i (Ei

Figure 1. Numerical simulations of the Model (4). The parameter values used are given in Table 1, but the parameter μ<sup>W</sup> is 2 μ<sup>N</sup> to reflect the WMelPop Wolbachia strain.

Mathematical Model as a Tool for the Control of Vector-Borne Diseases: *Wolbachia* Example http://dx.doi.org/10.5772/intechopen.79754 119

Figure 2. Numerical simulations of the Model (4). The parameter values used are given in Table 1. The parameter values reflect the WMel Wolbachia strain.

using the parameter values given in Table 1. The Wolbachia mosquito death rate is 1:1 � μ<sup>N</sup> 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 AN<sup>0</sup> ¼ 0, FN<sup>0</sup> ¼ MN<sup>0</sup> ¼ 7253, AW<sup>0</sup> ¼ 0, and MW<sup>0</sup> ¼ FW<sup>0</sup> ¼ 145. It shows that the non-Wolbachia mosquitoes dominate the populations. It implies that the initial conditions also affects the persistence of Wolbachiacarrying mosquitoes.

#### 4.2. Dengue mathematical model and numerical simulations

#### 4.2.1. Dengue mathematical model in the presence of Wolbachia

Symbol Description Value Unit Source r<sup>N</sup> Non-Wolbachia reproductive rate 1.25 day<sup>1</sup> [19] μNA Non-Wolbachia aquatic death rate 1/7.78 day<sup>1</sup> [47] γ<sup>N</sup> Non-Wolbachia maturation rate 1/6.67 day<sup>1</sup> [48] e<sup>N</sup> The proportion of non-Wolbachia adult male offspring 0.5 Proportion [49] μ<sup>N</sup> Non-Wolbachia adult death rate 1/14 day<sup>1</sup> [47] μWA Wolbachia aquatic death rate 1/7.78 day<sup>1</sup> [47] μ<sup>W</sup> Wolbachia adult death rate 1/7 day<sup>1</sup> [46] r<sup>W</sup> Wolbachia reproductive rate 0.95r<sup>N</sup> day<sup>1</sup> [19] γ<sup>W</sup> Wolbachia maturation rate 1/6.67 day<sup>1</sup> [46] e<sup>W</sup> The proportion of Wolbachia-infected male adults 0.5 N/A Assumed eNW The rate of uninfected males hatched from a Wolbachia-infected mother 0.5 N/A Assumed

K Carrying capacity 300,000 [48]

Figure 1. Numerical simulations of the Model (4). The parameter values used are given in Table 1, but the parameter μ<sup>W</sup>

Table 1. Parameters, description, values and sources for the model of Wolbachia-carrying mosquitoes.

0.9 N/A [7, 46, 50, 51]

α<sup>W</sup> The proportion of Wolbachia-infected offspring from a Wolbachia-infected

mother

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

is 2 μ<sup>N</sup> to reflect the WMelPop Wolbachia strain.

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 (SH), exposed to serotype i (Ei <sup>H</sup>), infected to serotype i (I i <sup>H</sup>), temporary immunity to the serotype i (X<sup>i</sup> H), recovered class (RH), susceptible, exposed and infected to j strain (Sji H, Eji H, Iji <sup>H</sup>, respectively). The superscript ji means individuals that were previously infected by serotype i and currently infected by serotype j. The mosquito population is divided into susceptible (SN and SW),

Figure 3. Numerical simulations of the Model (4). The parameter values used are given in Table 1. The parameter values reflect the WMel Wolbachia strain but different initial conditions. The initial conditions are AN<sup>0</sup> ¼ 0, FN<sup>0</sup> ¼ MN<sup>0</sup> ¼ 7253, AW<sup>0</sup> ¼ 0, and MW<sup>0</sup> ¼ FW<sup>0</sup> ¼ 145.

exposed to serotype i (Ei <sup>N</sup> and <sup>E</sup><sup>i</sup> <sup>W</sup>) and infected to serotype i (I i <sup>N</sup> and I i <sup>W</sup>). The subscript N and W is for non-Wolbachia and Wolbachia-carrying mosquitoes.

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

$$\frac{d\mathbf{S}\_H}{dt} = B\mathbf{N}\_H - \sum\_{i=1}^{2} \lambda\_H^i \mathbf{S}\_H - \mu\_H \mathbf{S}\_{H\nu} \tag{5}$$

dIji H dt <sup>¼</sup> <sup>γ</sup>HEji

Model for non-Wolbachia mosquito population

dSN

Model for Wolbachia-carrying mosquito population

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

dSW

dE<sup>i</sup> W dt <sup>¼</sup> <sup>λ</sup><sup>i</sup>

dt <sup>¼</sup> ατWAW

dIW

<sup>H</sup> <sup>¼</sup> bNT<sup>i</sup>

<sup>N</sup> <sup>¼</sup> bNT<sup>i</sup>

<sup>W</sup> <sup>¼</sup> bWT<sup>i</sup>

λi

λi

λi

dt <sup>¼</sup> <sup>γ</sup>WE<sup>i</sup>

I i N NH

I i H NH

I i H NH

þ bW T<sup>i</sup> HW I i W NH

þ ϕ<sup>i</sup>

þ ϕ<sup>i</sup>

where ϕ<sup>i</sup> 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

bNT<sup>i</sup> I ij H NH

bW T<sup>i</sup> I ij H NH

dAW

where the force of infections are

dt <sup>¼</sup> <sup>r</sup>NF<sup>2</sup>

dt <sup>¼</sup> <sup>τ</sup>NAN

dAN

dRH dt <sup>¼</sup> <sup>X</sup> 2

N 2ð Þ FN þ FW

> dEi N dt <sup>¼</sup> <sup>λ</sup><sup>i</sup>

<sup>H</sup> � σI ji

<sup>j</sup>¼<sup>1</sup>, <sup>j</sup>6¼<sup>i</sup>

<sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>α</sup> <sup>τ</sup><sup>W</sup> AW

dIi N dt <sup>¼</sup> <sup>γ</sup>NEi

<sup>2</sup> <sup>1</sup> � AN <sup>þ</sup> AW

σI ji

<sup>1</sup> � AN <sup>þ</sup> AW K � �

<sup>N</sup>SN � <sup>γ</sup>NE<sup>i</sup>

K � �

> i¼1 λi

> > <sup>W</sup> � μ<sup>W</sup> I i

<sup>W</sup> SW � <sup>γ</sup>WE<sup>i</sup>

<sup>2</sup> �<sup>X</sup> 2

<sup>2</sup> �<sup>X</sup>

<sup>N</sup> � μNI i

2

i¼1 λi

<sup>N</sup> � <sup>μ</sup>NE<sup>i</sup>

<sup>W</sup> � <sup>μ</sup><sup>W</sup> Ei

<sup>H</sup> � <sup>μ</sup><sup>H</sup> <sup>þ</sup> <sup>d</sup> � �<sup>I</sup>

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

ji

<sup>H</sup>, (11)

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<sup>H</sup> � μHRH, (12)

� τNAN � μNAAN, (13)

<sup>N</sup>SN � μNSN, (14)

<sup>N</sup>, (15)

<sup>N</sup>: (16)

� τ<sup>W</sup> AW � μWAAW , (17)

<sup>W</sup>SW � μ<sup>W</sup> SW , (18)

<sup>W</sup> , (19)

<sup>W</sup> , (20)

, (21)

, (22)

, (23)

$$\frac{dE\_H^i}{dt} = \lambda\_H^i S\_H - \gamma\_H E\_H^i - \mu\_H E\_{H'}^i \tag{6}$$

$$\frac{dI\_H^i}{dt} = \gamma\_H E\_H^i - \sigma I\_H^i - \mu\_H I\_{H'}^i \tag{7}$$

$$\frac{dX\_H^i}{dt} = \sigma I\_H^i - \theta^i X\_H^i - \mu\_H X\_{H'}^i \tag{8}$$

$$\frac{dS\_H^{\vec{\mu}}}{dt} = \lambda\_H^\circ S\_H^{\vec{\mu}} - \mu\_H S\_{H'}^{\vec{\mu}} \tag{9}$$

$$\frac{d E\_H^{\ddagger i}}{dt} = \lambda\_H^\circ \mathbf{S}\_H^{\ddagger i} - \gamma\_H \mathbf{E}\_H^{\ddagger i} - \mu\_H \mathbf{E}\_{H\prime}^{\ddagger i} \tag{10}$$

Mathematical Model as a Tool for the Control of Vector-Borne Diseases: *Wolbachia* Example http://dx.doi.org/10.5772/intechopen.79754 121

$$\frac{dI\_H^{\vec{\mu}}}{dt} = \gamma\_H E\_H^{\vec{\mu}} - \sigma I\_H^{\vec{\mu}} - \left(\mu\_H + d\right) I\_{H'}^{\vec{\mu}} \tag{11}$$

$$\frac{d\mathcal{R}\_H}{dt} = \sum\_{j=1, j \neq i}^{2} \sigma I\_H^{\vec{\mu}} - \mu\_H \mathcal{R}\_{H\prime} \tag{12}$$

Model for non-Wolbachia mosquito population

$$\frac{dA\_N}{dt} = \frac{\rho\_N F\_N^2}{2(F\_N + F\_W)} \left( 1 - \frac{A\_N + A\_W}{K} \right) - \tau\_N A\_N - \mu\_{NA} A\_N \tag{13}$$

$$\frac{d\mathbf{S}\_N}{dt} = \frac{\tau\_N A\_N}{2} + \frac{(1 - \alpha)\tau\_W A\_W}{2} - \sum\_{i=1}^2 \lambda\_N^i \mathbf{S}\_N - \mu\_N \mathbf{S}\_{N\prime} \tag{14}$$

$$\frac{dE\_N^i}{dt} = \lambda\_N^i \mathbf{S}\_N - \gamma\_N E\_N^i - \mu\_N E\_{N'}^i \tag{15}$$

$$\frac{dI\_N^i}{dt} = \mathcal{V}\_N E\_N^i - \mu\_N I\_N^i. \tag{16}$$

Model for Wolbachia-carrying mosquito population

$$\frac{dA\_W}{dt} = \frac{\rho\_W F\_W}{2} \left( 1 - \frac{A\_N + A\_W}{K} \right) - \tau\_W A\_W - \mu\_{WA} A\_{W\prime} \tag{17}$$

$$\frac{dS\_W}{dt} = \frac{\alpha \tau\_W A\_W}{2} - \sum\_{i=1}^{2} \lambda\_W^i S\_W - \mu\_W S\_W \tag{18}$$

$$\frac{dE^i\_W}{dt} = \lambda^i\_W S\_W - \gamma\_W E^i\_W - \mu\_W E^i\_{W'} \tag{19}$$

$$\frac{dI\_W}{dt} = \gamma\_W E\_W^i - \mu\_W I\_{W'}^i \tag{20}$$

where the force of infections are

exposed to serotype i (Ei

AW<sup>0</sup> ¼ 0, and MW<sup>0</sup> ¼ FW<sup>0</sup> ¼ 145.

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

<sup>N</sup> and <sup>E</sup><sup>i</sup>

W is for non-Wolbachia and Wolbachia-carrying mosquitoes.

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

dt <sup>¼</sup> BNH �<sup>X</sup>

dSH

dEi H dt <sup>¼</sup> <sup>λ</sup><sup>i</sup>

> dIi H dt <sup>¼</sup> <sup>γ</sup>HE<sup>i</sup>

dX<sup>i</sup> H dt <sup>¼</sup> <sup>σ</sup><sup>I</sup> i <sup>H</sup> � <sup>θ</sup><sup>i</sup> Xi

dEji H dt <sup>¼</sup> <sup>λ</sup><sup>j</sup>

dSji H dt <sup>¼</sup> <sup>λ</sup><sup>j</sup>

<sup>W</sup>) and infected to serotype i (I

Figure 3. Numerical simulations of the Model (4). The parameter values used are given in Table 1. The parameter values reflect the WMel Wolbachia strain but different initial conditions. The initial conditions are AN<sup>0</sup> ¼ 0, FN<sup>0</sup> ¼ MN<sup>0</sup> ¼ 7253,

2

i¼1 λi

<sup>H</sup>SH � <sup>γ</sup>HE<sup>i</sup>

<sup>H</sup> � σI i <sup>H</sup> � μHI i

HSji

<sup>H</sup> � <sup>γ</sup>HEji

HSji

<sup>H</sup> � <sup>μ</sup>HE<sup>i</sup>

<sup>H</sup> � <sup>μ</sup>HX<sup>i</sup>

<sup>H</sup> � <sup>μ</sup>HEji

<sup>H</sup> � <sup>μ</sup>HSji

i <sup>N</sup> and I i

<sup>H</sup>SH � μHSH, (5)

<sup>H</sup>, (6)

<sup>H</sup>, (7)

<sup>H</sup>, (8)

<sup>H</sup>, (9)

<sup>H</sup>, (10)

<sup>W</sup>). The subscript N and

$$
\lambda\_H^i = \frac{b\_N T^i I\_N^i}{N\_H} + \frac{b\_W T\_{HW}^i I\_W^i}{N\_H},
\tag{21}
$$

$$
\lambda\_N^i = \frac{b\_N T^i I\_H^i}{N\_H} + \phi\_i \frac{b\_N T^i I\_H^{ij}}{N\_H} \,\tag{22}
$$

$$
\lambda\_W^i = \frac{b\_W T^i I\_H^i}{N\_H} + \phi\_i \frac{b\_W T^i I\_H^{ij}}{N\_H},
\tag{23}
$$

where ϕ<sup>i</sup> 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

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

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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,

Figures 4 and 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

Symbol Description Value Unit Source α Maternal transmission 0.9 N/A [19, 46, 52]

B Human birth rate 1=ð Þ 70 � 365 day�<sup>1</sup> [3] bN Biting rate of non-W mosquitoes 0.63 day�<sup>1</sup> [53] bW Biting rate of W mosquitoes 0.95 bN day�<sup>1</sup> [54] γ<sup>H</sup> Progression rate from exposed to infectious 1/5.5 day�<sup>1</sup> [1] γ<sup>N</sup> Progression from exposed to infectious class of non-W mosquitoes 1/10 day�<sup>1</sup> [55] γ<sup>W</sup> Progression from exposed to infectious class of W mosquitoes 1/10 day�<sup>1</sup> [55] K Carrying capacity 3 � NH N/A [55]

μ<sup>N</sup> Adult mosquito death rate (non-W) 1/14 day�<sup>1</sup> [47] μ<sup>H</sup> Human death rate 1/ 70 ð Þ � 365 day�<sup>1</sup> [3] μNA Death rate of aquatic non-W mosquitoes 1/14 day�<sup>1</sup> [47] μ<sup>W</sup> Adult aquatic death rate 1.1μ<sup>N</sup> day�<sup>1</sup> [46, 51] μWA Death rate of W mosquitoes 1/14 day�<sup>1</sup> [47] ϕ ADE 1.1 N/A [56] r<sup>N</sup> Reproductive rate of non-W mosquitoes 1.25 day�<sup>1</sup> [19] r<sup>W</sup> Reproductive rate of W-mosquitoes 0:95r<sup>N</sup> day�<sup>1</sup> [46] σ Recovery rate 1/5 day�<sup>1</sup> [1] TN Transmission probability from non-W mosquitoes to human 0.5 N/A [36] THW Transmission probability from W mosquitoes to human 0:5TN N/A [36, 57] θ Progression rate from temporary immunity class to susceptible class 1=ð Þ 0:5 � 365 day�<sup>1</sup> [58] τ<sup>N</sup> Maturation rate of non-W mosquitoes 1/10 day�<sup>1</sup> [47] τ<sup>W</sup> Maturation rate of W mosquitoes 1/10 day�<sup>1</sup> [47]

Table 2. Parameter descriptions, values, and sources. Note that W and N are used to indicate Wolbachia-carrying and

.

non-Wolbachia mosquitoes in the parameter descriptions, respectively. NH <sup>¼</sup> 105

λ Force of infection Eqs. (21)–(23)

dengue serotype to that they are previously infected.

4.2.2. Numerical simulations

respectively.

Figure 4. Numerical simulations of primary and secondary infections in the absence of Wolbachia-carrying mosquitoes. The parameters values used are given in Table 2. Initial conditions are I 1 <sup>H</sup>ð Þ¼ 0 I 2 <sup>H</sup>ð Þ¼ <sup>0</sup> 1 and NH <sup>¼</sup> <sup>10</sup>5. ANð Þ¼ <sup>0</sup> SNð Þ¼ <sup>0</sup> <sup>3</sup> � NH.

Figure 5. Numerical simulations of primary and secondary infections in the presence of Wolbachia-carrying mosquitoes. The parameters values used are given in Table 2. Initial conditions are I 1 <sup>H</sup>ð Þ¼ 0 I 2 <sup>H</sup>ð Þ¼ <sup>0</sup> 1 and NH <sup>¼</sup> 105. ANð Þ¼ <sup>0</sup> SNð Þ¼ <sup>0</sup> AW ð Þ¼ 0 SW ð Þ¼ 0 1:5 � NH.

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.
