Preventions and Controls on Congenital Transmissions of Zika: Mathematical Analysis

Vector-borne diseases threat lives of millions of people in many countries of the world. Zika is one of the vector-borne diseases which initially spread by the bite of an infected Aedes species mosquito (Ae. aegypti and Ae. albopictus) and then it transmits vertically from a pregnant woman to her fetus or from an infected human to their sexual partners. The congenital transmission of Zika virus (ZIKV) results in new born with microcephaly and other neurological abnormalities. The control of infected mosquitos is the best efficient way to control spread of ZIKV. Spraying insecticide is the safest and easiest way to control mosquitos, but sometimes it is cost worthy for long period of spraying. Controlled prevention from the vector bites can also help to control disease spread. To control congenital transmission and sexual transmission of ZIKV, preventions should be taken to reduce/stop pregnancy rate and safe heterosexual transmission among adults. Also, there is no specific treatment available for Zika disease. Treatment is aimed at relieving symptoms with rest, fluids and medications. Controlled combinations of rest, fluids and medications will help to recover early. As costs are incorporated with spraying, preventions and treatment, our aim is to minimise the total cost associated by controlling spraying, preventions and treatment. To fulfil this purpose a mathematical model is developed with disease dynamics in nine compartments namely Susceptible human child, Susceptible human male, Susceptible human female, Infected human child, Infected human male, Infected human female, Recovered human, Susceptible vector and Infected vector including vertical transmission of Zika disease. Numerical simulations have been carried out to optimise controls, and basic reproduction number and stability are calculated.


Introduction
Zika virus (ZIKV) is a Flavivirus. It is initially transmitted to humans by the bites of infected female mosquitoes from the Aedes genus. The pathogen responsible for spread of Zika disease is known as Zika virus (ZIKV).In past two years remarkable changes has been seen in the epidemiology of Zika virus (ZIKV).
The transmission of ZIKV has been first reported from continental America and the Caribbean. Also, recent reports indicate an increase in detected cases of congenital malformations and neurological complications associated with ZIKV infection. To treat, prevent, or diagnose ZIKV infection there is no specific treatment, vaccine, or fast diagnostic test available at this time.
Dick et al. [1], Rodrigez et al. [2] and Macnamara et al. [3] observed that in the Zika forest of Uganda the Zika virus was initially isolated from a rhesus monkey in 1947. In 1954 same virus was isolated from humans in Nigeria. There after only sporadic infected human cases were reported from Africa and Southeast Asia. Duffy et al. [4] observed that first largest outbreak of ZIKV infection was reported in 2007 on Yap Island of the North Pacific. Musso et al. [5] observed that, in French Polynesia during October 2013, 28000 ZIKV infection cases were reported. Pan American Health Organization [6] noted that the recent outbreak began in April 2015 from Brazil, has covered many southern, central American countries and the Caribbean with spread of disease, and more than 140,000 suspected and confirmed cases are reported by the end of February 2016.
Foy et al. [7] reported that infection of ZIKV spread from an infected male to a female during their sexual intercourse. Thereafter, in February 2016 cases of sexually transmitted ZIKV were reported from Dallas [8] County, U.S. and France by CDC [9], Hills et al. [10] and Mansuy et al. [11] respectively. According to the Toronto Star [12] spread of Zika infection because of sexual activities has been reported in Argentina, Canada, Chile, France, Italy, New Zealand, Peru, Portugal, and the USA from 2015. Hills et al. [10] recorded that the disease has minor impact on sexual activity as ZIKV infections have mild symptoms for two to seven days. As ZIKV is sexually transmissible, CDC [9] issued related guidelines for preventions should be taken for safe sex during a Zika outbreak. Didier et al. [13] noted that ZIKV has been found in semen samples which point out the possibility of transmission of ZIKV through sexual activity.
Gatherer et al. [14] found that like other flaviviruses ZIKV could also be transmitted by blood transfusion.
Mlakar et al. [15] and Cauchemez et al. [16] indicated that ZIKV increases the chances of microcephaly in new-born babies of infected mothers. Cao-Lormeau et al. [17] [19] observed that in February 2015, abnormal brain development in the foetuses of pregnant and ZIKV infected women were found.
Dalia et al. [20] suggested for prevent pregnancy and effective strategies against the vector will control disease spread. Ebenezer et al. [21] suggested that to control disease the best strategy is to combine all preventive, treatment and Insecticides controls simultaneously. Daozhou et al. [22] noted that to control spread of ZIKV preventive steps are necessary during sexual activities in ZIKV outbreak area.
Together all these facts lead to the increase of the people susceptible to the disease. As there is no vaccination available for the ZIKV disease, it is consider as a severe problem. Mathematical models are essential tools to study the dynamics of the spread of infectious diseases like ZIKV. Basic reproduction number provides information about how infection will be sustained. In this study, a new model with optimal control on spraying insecticides, preventions and treatment is examined. Pontryagin's maximum principle, established by Pontrayagin et al. [23] is used to determine the optimal control. Result proves that optimal control gives significant reduction in ZIKV spread.
The paper is organised as follow. In Section 2, mathematical model by a system of ordinary differential equations with notations, assumptions and the flow of populations between compartments are described. For autonomous model having fixed rates of controls basic reproduction number of the whole system (human-mosquito combined) is calculated at disease free equilibrium and endemic equilibrium points. In Section 3, Stability of model has been discussed. In Section 4, cost control function for controls on spraying, preventions and treatment is formed and validated for the system of equations obtained in Section 2.
In Section 5, numerical simulations are carried out, for both autonomous and control model. In Section 6, conclusions suggest that how to control disease spread with minimal cost.

Mathematical Model
The mathematical model is developed with following notations.
A system of non-linear differential equations is formulated to investigate  To investigate effects of congenital transmission of disease and its control by spraying on mosquitoes, preventions and treatment, human population is distributed in human child and human male (adult) and human female (adult) classes.
For model formulation increase in total population of vectors (mosquitoes) is considered, as subpopulation of vectors (mosquitoes) survive from the spraying, get matures and reproduces.
To prepare model, following possibilities of disease spread are considered: 1) Vertical Transmission of infection to new-borns from infected mothers; 2) Horizontal transmission to human child/adult (male and female) from infected vectors (mosquitoes) when they bite to human child/adult (male and female); 3) Heterosexual transmissions amongst human adults (male and female); 4) Horizontal transmission to vectors (mosquitoes) from infected human child/adult (male and female) when vectors (mosquitoes) bite to human child/ adult (male and female).

Disease Dynamics amongst Human Population
After birth from an infected human female, infected human children will join infected human child class at rate ( )

Disease Dynamics amongst Vector (Mosquito) Population
Vector ( tors (mosquitoes) will be eliminated due to either natural death rate in vectors (mosquitoes) V µ or spraying at rate 5 u r . Disease induced death rate V α will also affect the infected vector (mosquito) population.
With above discussion the dynamics of Zika disease can be represented by the system of non-linear differential equations as Adding Equations (1) to (7) (8) and (9), So, the feasible region of the system is
Using next generation method [24], let     u  I  u k I  I  I  u k I  I  I  u k I  I  I  I I u rI      Table 1, 0 0.4757 R = .

Stability at Disease Free Equilibrium
If all eigenvalues of Jacobian matrix of the system of differential Equations (1)       With the parametric values given in Table 1, This implies that disease free equilibrium is locally asymptotically stable if 0 1 R < and unstable, if 0 1 R > .

Stability at Endemic Equilibrium Point
For the endemic equilibrium, using           With the parametric values given in Table 1, This implies that endemic equilibrium is locally asymptotically stable if 0 1 R < and unstable, if 0 1 R > .

Optimal Control
For Zika disease an optimal control model is formulated, to derive optimal prevention from mosquito bite 1 u , optimal prevention to stop pregnancy 2 u and sexual transmission 3 u , optimal treatment 4 u and optimal spraying 5 u with minimal implementation cost, in order to minimise the number of infected individuals for model described by Equations (1) to (9) in the time interval [ ] 0,T with the feasible region same as given by Ω in Section 2.
Considering the cost-functional as, Using Lagrangian techniques for a problem along with Hamiltonian, the adjoint variable is needed to construct for the optimal control problem given by (1) to (9).
Introducing the Lagrangian to derive the optimality conditions, To get optimality Pontrayagin's maximum (minimum) principle, for the model is defined as follow.

Numerical Simulations and Observations
For parametric values given in Table 1, the basic reproduction number  (1) to (9) is asymptotically stable at disease free equilibrium point as well as endemic equilibrium point.
In Figure 2, global stability of human population for 0 1 R < is shown. It can be seen that in the beginning susceptible human population increases but after 15 days it moves toward stability. Also, infected human population decreases initially but no prevention from vector bite, treatment, sexual activity, pregnancy, and no spraying increase infected population after 10 days. High disease induced death rate controls the infected human population.
In Figure 3, the effects of increase in prevention from mosquito bites on various infected human classes are analysed. It can be observed that prevention from vector bites in human population is an essential tool to control disease spread.
From Figure 4, it can be analysed that how the infected human population can be controlled by giving proper treatment on time. As there is no specific vaccination or treatment available for Zika disease cure, the treatment is considered as cure of symptoms of disease. Figure 5 indicates that proper preventive steps to stop sexual transmission of disease taken at the time of disease out breaks will help to control disease spread.
From Figure 6, it can be observed that the preventions to stop pregnancy amongst infected female and effected area by ZIKV, will reduce the new child births having microcephaly and hence infected child population. Figure 7 indicates that how the effective spraying can control infected mosquito population and hence finally spraying on mosquito will control disease spread indirectly in human population.
Thus to investigate effects of congenital transmission of ZIKV and its control   by spraying on mosquitoes, preventions and treatment on human population, it is required to make the total cost associated with above controls minimise using optimal policy for all controls. (Figure 8)   To minimise the total effective cost to control disease spray, the policy is to be designed in a way that during first week of disease outbreaks, 26% preventions on treatment and 40% prevention to stop pregnancy, 15% prevention on sexual activities amongst human, 8% spraying is required to have for 1% of prevention from vector bite.

Conclusions
In this paper, the spread of ZIKV considered initially as vector-borne infection and then after its spread amongst sexual partners and from mother to child is analysed. The control over spraying on vectors, prevention from vector bite, prevention during sexual activity, prevention to stop pregnancy and prevention for treatment with time incorporated. It is observed that all controls disease effectively in each compartment.
In future, the model can be studied with addition of transmission of ZIKV by blood transmission in human population.