Dynamics and optimal control of a Zika model with sexual and vertical transmissions

1.   Introduction

The Zika virus is named after it was first discovered in 1947 in rhesus monkeys in Uganda's Zika forest^[1]. The Zika virus is widely distributed in the tropics and subtropics. It is mainly transmitted to humans through the bites of infected female aedes mosquitoes^[2]. Many people infected with Zika will experience no symptoms or only mild symptoms. Common Zika symptoms are fever, rash, headache, joint pain, red eyes and muscle pain that can last from a few days to a week^[3]. The Zika virus was first detected in Uganda and Tanzania in 1952^[4]. Virus activity had been muted and only sporadic cases of Zika virus infection had been reported in equatorial Africa, the Americas, Asia and the Pacific. Since then, there had been a pandemic on the Western Pacific island of Yap in 2007^[5]. In 2013 to 2014, there was a larger outbreak in French Polynesia in the Pacific region, which infected about 32,000 people ^[6]. Zika spread rapidly in Brazil after the first case in South America was detected there in 2015^[7]. In October of that year, Brazil reported an increase in microcephaly cases. Pregnant women with Zika risk transmitted the virus to their newborns and gave birth to babies with microcephaly or other birth defects ^[8]. In addition to mosquito-borne transmission, Zika is transmitted sexually, mainly from men to women. Studies have shown that the Zika virus persists in men's semen longer than other body fluids, up to six months ^[9]. Zika virus can be transmitted to women several months after a man has recovered. Another mode of transmission that has a high impact is vertical transmission, where the virus is passed from an infected pregnant woman to her newborn. This mode of transmission is likely to result in children being born with microcephaly or other severe fetal brain defects. The virus is also spread in laboratories and through blood transfusions^[10].

All sorts of sophisticated models have been built to study the spread of Zika. The possible transmission of Zika virus through sexual transmission was first identified by Foy et al. ^[11]. Gao et al. ^[12] established a Zika transmission model with mosquito-borne and sexual transmission of the virus, using the Zika epidemic data from Brazil, Colombia and El Salvador for data simulation. It was concluded that sexual transmission was relatively small contributor to Zika virus transmission but that sexual transmission increases the risk of infection and epidemic size and may prolong outbreaks. He et al. ^[13] simulated Zika virus infections reported in French Polynesia, Colombia, and the State of Bahia in Brazil. By comparing the simulation results, we would be able to better understand the likely evolution of Zika virus, control the spread of the outbreak and prevent potential transmission. Baca-Carrasco et al. ^[14] proposed three mathematical models including vector transmission, sexual transmission and population migration. The common conclusion was that the level of endemic disease following Zika virus outbreak was very low and sexual transmission contributed to the extent of the outbreak. Agusto et al. ^[15] established a transmission model of Zika virus in the absence of disease deaths and performed stability analysis. When expanding the model to include mortality due to Zika virus, different model stability analyses were established and numerical simulations were used to assess the importance of sexual transmission to study the dynamics of the disease. Imran et al. ^[16] established a model considering the vertical transmission of human and mosquito as well as analyzing its dynamic behavior in detail. Denes et al. ^[17] established a complex transmission model, which distinguished males and females in the mode of sexual transmission. Data simulations used Zika virus data from Costa Rica and Suriname concluded that mosquito birth and death rates were the most important factors in Zika virus transmission, but sexual transmission also had a significant impact on disease transmission. Ibrahim et al. ^[18] established a model of Zika virus transmission involving sexually transmitted and asymptomatic carriers. The effects of weather periodicity on mosquito-related parameters were studied. Yuan et al. ^[19] proposed three modes of transmission of Zika virus, including vector transmission between humans and mosquitoes, transmission by human sexual contact and vertical transmission within mosquitoes. The contribution of each transmission route to the basic reproduction number was analyzed. Numerical simulations confirm that sexual and vertical transmission had different effects on the early and long-term transmission of Zika virus. Busenberg and Cooke ^[20] earlier discussed the modeling and dynamic analysis of various vertically transmitted diseases. In an $S-I-R$ compartment model, we incorporate vertical transmission into infected category $(I_{f}, I_{m})$ by assuming that a subset of the offspring of infected women $(B_{h}{\frac{\lambda_{h}I_{f}}{N_{f}}})$ are infectious at birth, while the remaining newborns $(B_{h}-B_{h}{\frac{\lambda_{h}I_{f}}{N_{f}}})$ enter the susceptible category. Where $B_{h}$ represents natural birth rate of humans and $\lambda_{h}$ represents proportion of offspring with congenital infection of infected females. Li et al.^[21] used a similar method to consider vertical transmission. Some literatures have conducted detailed studies on the optimal control strategies for the transmission of Zika virus, studied the impact of different control strategy combinations on the transmission of Zika virus in human populations, and obtained some effective strategies for preventing and controlling the transmission of Zika virus disease ^[22,23].

Few previous studies on Zika transmission have dealt with the vertical transmission in humans. However, the people most affected by Zika infection are newborns. It is of great significance to distinguish the sexes and add neonatal infection for analysis. With numerical simulations, it is clear which control measures are in place to rapidly control the spread of Zika disease, and the changes in actual cases in Colombia show that effective control measures can rapidly control the spread of the disease. Motivated by the above discussion, the goal of this paper is to study the joint influence of mosquito-borne transmission, sexual transmission and vertical transmission in human and mosquitoes on the spread of Zika virus. The sections of this paper are as follows. In Section 2, we introduce a Zika transmission model with three transmission modes: mosquito-borne transmission, sexual contact transmission and vertical transmission. In Section 3, we determine the basic reproduction number of the model and study the local asymptotic stability of the disease-free equilibrium as well as the global stability of the disease-free equilibrium in the case of $R_{0} < 1$. In Section 4, we derive and explore the expression of the optimal control policy. In Section 5, we provide parameter estimates using actual transmission cases in Colombia, explaining why the actual transmission cases do not fit well with the fitted curve. The number of infected cases is simulated under different control strategies. The paper ends with a discussion in last section.

2.   The model formulation

2.1.   System description

The total human population $N_{h}(t)$ consists of female human and male human. Female human consists of three compartments: $S_{f}(t)$, $I_{f}(t)$, $R_{f}(t)$. $S_{f}(t)$ represents the number of female susceptible individual, $I_{f}(t)$ represents the number of female infected individual, $R_{f}(t)$ represents the number of female recovered individual. Male human consists of four compartments: $S_{m}(t)$, $I_{m}(t)$, $I_{m}^{r}(t)$, $R_{m}(t)$. $S_{m}(t)$ represents the number of male susceptible individuals, $I_{m}(t)$ represents the number of male infected individuals, $I_{m}^{r}(t)$ represents the number of male convalescent individuals who have recovered from the disease but can still transmit it through sexual contact. $R_{m}(t)$ represents the number of male recovered individual. The total vector population $N_{v}(t)$ consists of two compartments: $S_{v}(t)$, $I_{v}(t)$. $S_{v}(t)$ represents the number of susceptible mosquitos, $I_{v}(t)$ represents the number of infected mosquitos. The transmission diagram of the model is shown in Figure 1.

The total human population and the total mosquito population are given by:

Then the model can be built as follows:

We define the functions $Y_{1}(t), Y_{2}(t), Y_{3}(t), Y_{4}(t), Y_{5}(t)$ by

The model assumes that the population has a constant natural birth rate ($B_{h}$) and that the number of males and females is the same, denoting the natural birth rate of males and females by $\frac{B_{h}}{2}$. When vertical transmission is considered, if a woman is infected with Zika virus during pregnancy, then the individual newborn may also be infected and proportion of congenital infected offspring in infected females is assumed to be $\lambda_{h}$ (see, e.g., ^[21]). $\frac{I_{f}}{N_{f}}$ is the proportion of infected women, so $\frac{B_{h}}{2}\frac{\lambda_{h}I_{f}}{N_{f}}$ represents the number of infected newborns, and $\frac{B_{h}}{2}-\frac{B_{h}}{2}\frac{\lambda_{h}I_{f}}{N_{f}}$ indicates susceptible female newborns. The same is true of male representations. Male susceptible individuals ($S_{m}$) are infected with Zika virus due to the bite of mosquitos infected with Zika virus, the transmission probability of infected mosquitos biting susceptible human is $\alpha_{h}$, thus entering the category of infected individuals ($I_{m}$). Male infections decrease due to natural mortality ($\mu_{h}$) and cure rates ($\gamma$) of infected persons. The male enters the convalescent phase ($I_{m}^{r}$), during which the virus will not be present in the male blood, but will be present in the male semen for a long period of time (even up to six months). Sexual contact with others during this period of time is a risk of transmitting the virus to others. Due to metabolism, the virus will be metabolized out of the body and eventually transformed into male recovered individual ($R_{m})$ at the recovery rate of $\gamma_{r}$. Female susceptible individuals ($S_{f}$) enter the infected category ($I_{f}$) by acquiring the virus through the bite of infected mosquitos with transmission probability ($\alpha_{h}$) and through sexual contact with transmission probability ($\beta$). Female infections decrease due to natural mortality ($\mu_{h}$) and cure rates ($\gamma$) of infected persons. Recovered women were transferred to the recovered category ($R_{f}$).

Similarly, susceptible mosquitoes ($S_{v}$) become infected by biting infected human with transmission probability ($\alpha_{v}$). The proportion of the offspring of infected mosquitoes ($I_{v}$) will be infected and assume that the proportion of infected female offspring is $\theta$. Consider the expression of vertical propagation similar to that of humans. It is assumed that the mosquito will not recover after infection. The parameters description of the Zika model are summarized in Table 1.

By reference [24], Theorems 1.1.8 and 1.1.9 of Wiggins ^[25] for each nonnegative initial condition there is a unique, non-negative solution.

Lemma 1. Let the initial value $W(0)\geq0$, where $W(t) = (S_{f}, I_{f}, R_{f}, S_{m}, I_{m}, I_{m}^{r}, R_{m}, S_{v}, I_{v}).$ Then the solutions $W(t)$ of model (2.1) are non-negative for all time $t\geq0$. Furthermore

Proof. Let $t_{1} = sup\{t > 0:W(t) > 0\},$ by the first equation of model (2.1) that

we get

It can similarly be shown that $W(t) > 0$ for all $t > 0.$ It can be represented as $0 < S_{f}\leq N_{h}(t), 0 < I_{f}\leq N_{h}(t), 0 < R_{f}\leq N_{h}(t), 0 < S_{m}\leq N_{h}(t), 0 < I_{m}\leq N_{h}(t),$ $0 < I_{m}^{r}\leq N_{h}(t), 0 < R_{m}\leq N_{h}(t), 0 < S_{v}\leq N_{v}(t), 0 < I_{v}\leq N_{v}(t).$

Supplementary model (2.1) are given as

Hence,

as required.

Model (2.1) is analyzed in a biologically meaningful feasible domain. Consider the feasible region

Lemma 2. The region $W = W_{h}\times W_{v}\subset\mathbb{R}_{+}^{7}\times \mathbb{R}_{+}^{2}$ is positively invariant and attracts all positive orbits in $W$.

Proof. Following steps to establish positive invariance of $W$. The rate of change in the population is obtained by supplementary model (2.1) to give

Therefore, it can be obtained

Particularly, $N_{h}\leq\frac{B_{h}}{\mu_{h}},$ if $N_{h}(0)\leq\frac{B_{h}}{\mu_{h}}.$ Similarly, $N_{v}\leq\frac{B_{v}}{\mu_{v}},$ if $N_{v}(0)\leq\frac{B_{v}}{\mu_{v}}.$ Thus, the set $W$ is a positive invariant.

In order to prove that the set $W$ is attractive, note that $\frac{{{\rm{d}}N_{h}}}{{{\rm{d}}t}} < 0$ and $\frac{{{\rm{d}}N_{v}}}{{{\rm{d}}t}} < 0$ if $N_{h}(0) > \frac{B_{h}}{\mu_{h}}$ and $N_{v}(0) > \frac{B_{v}}{\mu_{v}}$ respectively. Thus, either the solution enters $W$ in finite time, or $N_{h}(t)$ and $N_{v}(t)$ approach $\frac{B_{h}}{\mu_{h}}$ and $\frac{B_{v}}{\mu_{v}}$ respectively, and therefore the variables of the infection class $I_{f}, I_{m}, I_{m}^{r}$ and $I_{v}$ tend to 0. Thus, the set $W$ is attractive and all solutions in $\mathbb{R}_{+}^{9}$ eventually enter $W$.

3.   Analysis of the model

3.1.   Local stability of disease-free equilibrium

Obviously, the system (2.1) always has a disease-free equilibrium which is given by:

Using the next-generation matrix method ^[24] to prove local stability of disease free equilibrium, we construct the transmission vector $\mathcal{F}$ representing the new infections flowing only into the infected compartments given by

the transition vector $\mathcal{V}$ represent the outflow from the infectious compartments in system (2.1) is given by

Substitute the value of the disease-free equilibrium $N_{h} = \frac{B_{h}}{\mu_{h}}$ and $N_{f} = N_{m} = \frac{B_{h}}{2\mu_{h}}$, we compute the Jacobian $F$ from $\mathcal{F}$ given by

and the Jacobian $V$ from $\mathcal{V}$ given by

Thus, the characteristic polynomial available from of the generation matrix $FV^{-1}$ is

where $k_{1} = \gamma+\mu_{h}, k_{2} = \gamma_{r}+\mu_{h}.$ We can solve for four eigenroots, one of which is 0, two complex roots, one real root, and find the absolute value of the largest root.

Hence, the basic reproduction number of model (2.1) is

where

Using Theorem 2 of ^[24], we obtain the local stability of disease-free equilibrium.

Theorem 3.1.1. The disease-free equilibrium $\varepsilon_{0}$ is locally asymptotically stable for $R_{0} < 1$ and is unstable for $R_{0} > 1$.

3.2.   Global stability of disease-free equilibrium

Using the method which is applied in ^{[26,27,28,29]}, we obtain global stability of disease-free equilibrium. In order to make the population size of Zika extinction independent of the initial value, we establish the global stability of the disease-free equilibrium point, considering a feasible region

where $X = S_{f}, I_{f}, R_{f}, S_{m}, I_{m}, I_{m}^{r}, R_{m}, S_{v}, I_{v}$

Lemma 3. The $W_{1}$ is positively invariant for model (2.1).

Proof. For the the first equation of the model (2.1), where $S_{f}^{*} = \frac{B_{h}}{2\mu_{h}}$

Further,

Hence, if $S_{f}(0)\leq S_{f}^{*}$ for $t\geq0$, then $S_{f}(t)\leq S_{f}^{*}$ for $t\geq0$. Similarly, it can be obtained if $S_{m}(0)\leq S_{m}^{*}$ for $t\geq0$, then $S_{m}(t)\leq S_{m}^{*}$ for $t\geq0$ and if $S_{v}(0)\leq S_{v}^{*}$ for $t\geq0$, then $S_{v}(t)\leq S_{v}^{*}$ for $t\geq0.$ Therefore, in general the field $W_{1}$ is a positive invariant set and attracts all solutions of model (2.1) in $\mathbb{R}_{+}^{9}$.

Remark 3.2.1. The $W_{1}$ is a special field that needs to be satisfied when $R_{0} < 1$, which represents the basin of attraction.

Theorem 3.2.1. The disease-free equilibrium $\varepsilon_{0}$ is globally asymptotically stable for $R_{0} < 1$.

Proof. To prove the global stability of the disease-free equilibrium, let $X = \left(S_{f}, R_{f}, S_{m}, R_{m}, S_{v}\right)$ and $Z = \left(I_{f}, I_{m}, I_{m}^{r}, I_{v}\right).$ Therefore, the grouping system can be expressed as

where, $F(X, 0)$ is the right side of $\dot{S_{f}}, \dot{R_{f}}, \dot{S_{m}}, \dot{R_{m}}, \dot{S_{v}}$ when $I_{f} = I_{m} = I_{m}^{r} = I_{v} = 0$ and $G(X, Z)$ is the right side of $\dot{I_{f}}, \dot{I_{m}}, \dot{I_{m}^{r}}, \dot{I_{v}}.$

Next, we consider simplifying the system $\frac{{\rm{d}}X}{{\rm{d}}t} = F(X, 0):$

It is easy to get an equilibrium for system (3.3),

Showing that $X^{*}$ is a globally stable equilibrium in $W_{1}.$ For this, by the first and second equations of (3.3) we get

Taking the limits of $S_{f}(t)$ and $R_{f}(t)$ at $t\rightarrow \infty,$ we have

Similarly, it can be shown that $\lim\limits_{t\rightarrow \infty}S_{m}(t) = \frac{B_{h}}{2\mu_{h}}, \lim\limits_{t\rightarrow \infty}R_{m}(t) = 0,$ and $\lim\limits_{t\rightarrow \infty}S_{v}(t) = \frac{B_{v}}{\mu_{v}}.$ These asymptotic dynamics are independent of initial in $W.$ Therefore, the convergence of the solution of (3.3) is global in $W_{1}.$ Further, according to ^[28] we require $G(X, Z)$ to satisfy the two stated conditions:

where $(X^{*}, 0) = \left(\frac{B_{h}}{2\mu_{h}}, 0, 0, \frac{B_{h}}{2\mu_{h}}, 0, 0, 0, \frac{B_{v}}{\mu_{v}}, 0\right)$ and $D_{z}G(X^{*}, 0)$ is the Jacobian of the $G(X, Z)$ at $(X^{*}, 0),$ which is an M-matrix(the off-diagonal elements are nonnegative).

Thus,

and

where, $\xi_{1} = \frac{B_{h}\lambda_{h}}{2N_{f}^{*}}(1-\frac{N_{f}^{*}}{N_{f}}),$ $\xi_{2} = \frac{\beta S_{f}^{*}}{N_{f}^{*}}\left(1-\frac{S_{f}N_{f}^{*}}{N_{f}S_{f}^{*}}\right),$ $\xi_{3} = \frac{\beta k_{r}S_{f}^{*}}{N_{f}^{*}}\left(1-\frac{S_{f}N_{f}^{*}}{N_{f}S_{f}^{*}}\right),$ $\xi_{4} = \frac{\alpha_{h}S_{f}^{*}}{N_{h}^{*}}\left(1-\frac{S_{f}N_{h}^{*}}{N_{h}S_{f}^{*}}\right),$ $\xi_{5} = \frac{\alpha_{h}S_{m}^{*}}{N_{h}^{*}}\left(1-\frac{S_{m}N_{h}^{*}}{N_{h}S_{m}^{*}}\right),$ $\xi_{6} = \frac{B_{v}\theta}{N_{v}^{*}}(1-\frac{N_{v}^{*}}{N_{v}}).$ $\xi_{7} = \frac{\alpha_{v}S_{v}^{*}}{N_{h}^{*}}\left(1-\frac{S_{v}N_{h}^{*}}{N_{h}S_{v}^{*}}\right).$

Further, $S_{f}^{*} = \frac{B_{h}}{2\mu_{h}}, S_{m}^{*} = \frac{B_{h}}{2\mu_{h}}, S_{v}^{*} = \frac{B_{v}}{\mu_{v}}, N_{h}^{*} = \frac{B_{h}}{\mu_{h}}.$ We have in that $S_{f}\leq S_{f}^{*}, S_{m}\leq S_{m}^{*}, S_{v}\leq S_{v}^{*}.$ Hence, if the human population is in a state of equilibrium, we can get $(1-\frac{N_{f}^{*}}{N_{f}}) > 0, \left(1-\frac{S_{f}N_{f}^{*}}{N_{f}S_{f}^{*}}\right) > 0, \left(1-\frac{S_{f}N_{h}^{*}}{N_{h}S_{f}^{*}}\right) > 0, \left(1-\frac{S_{m}N_{h}^{*}}{N_{h}S_{m}^{*}}\right) > 0, (1-\frac{N_{v}^{*}}{N_{v}}) > 0$ and $\left(1-\frac{S_{v}N_{h}^{*}}{N_{h}S_{v}^{*}}\right) > 0.$ Therefore, $\hat{G}\geq0.$ Then, according to the theorem in^[28], the global stability of the disease-free equilibrium can be obtained. This would indicate that Zika virus will die out over time and remain stable globally.

4.   The optimal control problems

4.1.   Existence of optimal control

We add four time-dependent control variables to the model corresponding to four mitigation strategies. In order to derive the necessary conditions for the existence of optimal control, we use Pontryagin's maximum principle ^[30]. Abimbade et al. ^[31] and Olaniyi et al. ^[32] refers to the derivation of the optimal control problem.

In model (4.1), four control strategies $u_{1}(t), u_{2}(t), u_{3}(t), u_{4}(t)$ are added to extend model (2.1) to obtain model(4.1). $u_{1}(t)$ denotes the use of mosquito nets and other methods to reduce human exposure to mosquitoes; $u_{2}(t)$ means to improve the media, internet and other publicity efforts to enhance human awareness and reduce the probability of sexual transmission; $u_{3}(t)$ advocates delaying pregnancy and reducing the number of babies born with abnormalities. $u_{4}(t)$ means the use of insecticides and other methods to reduce mosquito populations. Here, we assume that the control set is

The optimal control model is given as

Due to non-negative initial conditions and bounded Lebesgue measurable control, this system has non-negative bounded solutions ^[33]. We consider an optimal control problem to minimize the objective functional

In Eq (4.2), $A_{1}, A_{2}, A_{3}$ and $A_{4}$ represent the weights of human infected and mosquito infected, respectively. The weights $\eta_{1}, \eta_{2}, \eta_{3}$ and $\eta_{4}$ are measures of the costs associated with the control variables $u_{1}, u_{2}, u_{3}$ and $u_{4},$ respectively.

Theorem 4.1.1. We consider the objective functional $J$ given by Eq (4.2) with $(u_{1}, u_{2}, u_{3}, u_{4})\in U$ subject to the control model (4.1) with initial conditions. There exists $u^{*}(t) = \{u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}\}\in U$ such that

Proof. We can use the result of ^[33] to prove the existence of an optimal control problem.

By its definition, we know that the control set $U$ is closed and convex, and the integrand is also convex on $U$. Obviously these state and control variables are non-negative. The control system is bounded, which means the compactness of optimal control. Furthermore, there exists a constant $\zeta > 1$ and positive values $z_{1}$, and $z_{2}$, such that

which completes the existence of the optimal control. A method to prove the existence of optimal control is proposed in ^[34].

4.2.   Characterization of optimal controls

To find an optimal solution we consider the Lagrangian function of the optimal control problem. The Lagrangian function is

For model (4.1), we derive the necessary conditions for optimal control according to the Pontryagin maximum principle. The corresponding Hamiltonian function is

where $\lambda_{i}, i = 1, ..., 9$ are adjoint variables.

Theorem 4.2.1. Given an optimal control $(u_{1}^{*}, u_{2}^{*}, u_{3}^{*}, u_{4}^{*}),$ and let $S_{f}, I_{f}, R_{f}, S_{m}, I_{m}, I_{m}^{r}, R_{m}, S_{v}$ and $I_{v}$ be the state solutions for model (4.1). Thus, there exist adjoint variables $\lambda_{i}, i = 1, ..., 9$ satisfying

The boundary conditions are

Furthermore, the optimal controls $u_{1}^{*}, u_{2}^{*}, u_{3}^{*}$ and $u_{4}^{*}$ are represented by

where

Hence,

where $i = 1, ...4.$

Proof. The result of the adjoint system can be obtained from Pontryagin's principle

where $x = S_{f}, I_{f}, R_{f}, S_{m}, I_{m}, I_{m}^{r}, R_{m}, S_{v}$ and $I_{v}.$ The boundary conditions are $\lambda_{i}^{*}(t_{f}) = 0, i = 1, ..., 9.$

To derive the characterization of the optimal control given by Eq (4.3), we solve the equations on the interior of the control set,

Plugging the bounds for the controls, we obtain the desired characterization.

5.   A case study for colombia

5.1.   Numerical results

In this section, we apply our model to study Zika virus transmission cases in Colombia during 2015 to 2017. Based on the actual data of Zika virus transmission, numerical experiments are carried out using matlab and the parameters of the model is estimated using the Markov Chain Monte Carlo (MCMC) procedure. According to ^[35] data, we know that the average number of births per day in Colombia is about 1826.81 and the mortality rate is 0.0000368. It has been reported that it will take three to seven days for infected individuals with Zika to recover. Therefore, the recovery rate is set to 1/4. We know from ^[36,37] that $\gamma_{r}$ is 0.06. We choose a set of values of parameters in Table 2.

According to Table 2, we know that transmission rate from infected humans to susceptible humans $(\beta)$ is 0.046586, baseline value of transmission rate from mosquitoes to humans $(\alpha_{h})$ is 0.63722 and baseline value of transmission rate from humans to mosquitoes (${\alpha_{v}}$) is 0.6893. About vertical transmission, proportion of offspring congenital infection of infected female mosquitoes (${\theta}$) is 0.0021348 and proportion of offspring with congenital infection of infected females (${\lambda_{h}}$) is 0.0050859. The probability is relatively small, but it has caused serious consequences, not only the financial impact on society but also the impact on life and health. Therefore, we must pay attention to it.

Clearly, fitting results Figure 2, the data simulation does not match the actual data very well. According to ^[7], we found that in June 2016, the World Health Organization developed a plan and implemented measures to address the threat posed by Zika virus, including in Colombia, which also implemented measures to reduce the probability of transmission through sexual contact and human mosquito contact. This is the reason for the rapid decline in the actual number of cases from June 2016. Therefore, we're going to do a piecewise simulation(see Figure 3). Before June 2016, transmission rate from infected humans to susceptible humans $(\beta)$ is 0.046586, baseline value of transmission rate from mosquitoes to humans $(\alpha_{h})$ is 0.63722 and baseline value of transmission rate from humans to mosquitoes $({\alpha_{v}})$ is 0.6893. After June 2016, transmission rate from infected humans to susceptible humans $(\beta)$ is 0.045953, baseline value of transmission rate from mosquitoes to humans $(\alpha_{h})$ is 0.19573 and baseline value of transmission rate from humans to mosquitoes $(\alpha_{v})$ is 0.30891. The values of other parameters are shown in Table 2. In conclusion, control measures can quickly bring the disease under control. In the next section, we will discuss the impact of several control measures on the transmission of Zika virus.

5.2.   Analysis of control measures

In this section, we examine the impact of several control measures on the prevalence of the disease and the change in the number of infected people under optimal control. We take the initial value $S_{f} = 30000, I_{f} = 200, R_{f} = 500, S_{m} = 30000, I_{m} = 150, I_{m}^{r} = 100,$ $R_{m} = 400, S_{v} = 2000, I_{v} = 100.$ We choose the parameter values as follows, $B_{h} = 1826.81, B_{v} = 4506, \mu_{h} = 3.65*10^{-5}, \mu_{v} = \frac{1}{8}, \beta = 0.046586,$ $\alpha_{h} = 0.63722, \alpha_{v} = 0.6893, k_{r} = 0.4249, \gamma = \frac{1}{4}, \gamma_{r} = 0.06, \lambda_{h} = 0.0054773, \theta = 0.0021348.$ In order to compare the effectiveness of control measures, we conduct simulations as follows.

To study the impact of different control measures on Zika virus outbreak, we will simulate cases over time in female susceptible individual, male susceptible individual and male convalescent individuals under different control measures. Figure 4 shows the change of the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ over time when only the control measure $u_{1}$ is added and other control measures are zero. As can be seen from the figure, when no control measures are taken, the peak of infected people will be brought to a high degree and then the number will decrease. Reduction of human mosquito exposure rate $(u_{1} = 0.25)$ will greatly reduce the peak of infection with slight control, but its impact on Zika virus will be reduced in later stages. Under moderate control of $u_{1}$ the disease will not break out or become extinct and will remain at a very low incidence. Figure 5 shows the change of the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ over time when control measure $u_{4}$ is added and other control measures are zero. As can be seen from the figure, only mild implementation of control measures $(u_{4} = 0.2)$ is required to prevent the outbreak of disease and gradually eliminate the disease over time. Figure 6 shows the change in the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ when only $u_{2}$ is added and other control measures are zero. As can be seen from the figure, when $u_{2}$ control measures are moderate, the peak value of infected females $(I_{f})$ is slightly reduced and the effect on infected females $(I_{m})$ and convalescent men $(I_{m}^{r})$ is not significant. Comparing Figures 4–6, it is clear that controlling mosquito transmission to humans is more effective than controlling sexual transmission.

Figure 7 shows the change in the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ when only control measure $u3$ is added and other control measures are zero. As can be seen from the figure, due to the large number of infected people in the outbreak of the disease, only implementing the control strategy of delaying pregnancy $(u_{3} = 0.5)$ has little impact on the number of infected people. Figure 8 shows the change in the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ over time when all four control measures were implemented. As can be seen from the figure, only minor control measures are needed to quickly control the outbreak and wipe out the disease in a short time. Figure 9 shows the change of the number of infected persons $(I_{f}, I_{m}, I_{m}^{r})$ over time under the optimal control measures. We know that under optimal control, the disease rarely breaks out and is most effectively controlled.

Reducing the mosquito population is the most effective way to control the spread of the disease by comparing several control measures. In conclusion, rapid control of a Zika outbreak requires a combination of mosquito-borne control strategies and internal human control strategies. The strategy of delaying pregnancy may be effective because of the greater adverse effects of Zika virus infection in newborns and the possibility that delaying pregnancy may reduce the number of infections in newborns.

6.   Discussion and conclusions

We develop a Zika virus transmission model with mosquito-borne transmission, sexual transmission and vertical transmission. Because sexual transmission of Zika is mainly male to female, we have differentiated the genders to more accurately represent the method of sexual transmission. Vertical transmission distinguishes between vertical transmission by mosquitoes and vertical transmission by humans. The basic reproduction number is obtained by the next generation matrix method. The global stability of the disease-free equilibrium is derived. The existence and mathematical expression of optimal control are obtained by using Pontryagin's maximum principle.

Based on the data of Zika virus in Colombia from 2015 to 2017, the unknown parameters of the model are estimated by MCMC algorithm. The reason why the fitted curve was not consistent with the actual cases was that the corresponding control measures adopted in Columbia in June 2016 led to a rapid decline in the number of infections after June 2016. A comparison of the four control strategies revealed that a combination of mosquito vector control and internal human control is necessary to bring the disease under control quickly. So in order to quickly control the spread of the Zika virus, we need to not only control the mosquito population and avoid contact with mosquitoes, but also increase awareness of disease transmission in humans and reduce sexual transmission. The Zika virus can cause severe brain defects in newborns and control strategies to delay pregnancy may reduce the number of infections in newborns. However, the change in the number of neonatal infected persons is not clearly indicated in this paper, which will likely be related to follow-up work in this respect.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11861044), the NSF of Gansu of China (21JR7RA212 and 21JR7RA535).

Conflict of interest

The authors declare that they have no conflict of interest.