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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.

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. [

Foy et al. [

Gatherer et al. [

Mlakar et al. [

Dalia et al. [

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. [

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.

The mathematical model is developed with following notations.

A system of non-linear differential equations is formulated to investigate spread of congenital transmission of ZIKV with control spraying, preventions and treatment. Dynamics of human population N H ( t ) and Vector population N V ( t ) are developed. Human population is classified amongst eight compartments viz. number of susceptible human child S H C ( t ) , number of susceptible human female S H F ( t ) , number of susceptible human male S H M ( t ) , number of infected human child I H C ( t ) , number of infected human female I H F ( t ) , number of infected human male I H M ( t ) , number of recovered human adult R H ( t ) , while, vector population N V ( t ) is divided in to two compartments viz. number of susceptible vectors S V ( t ) , number of infected vectors I V ( t ) . The population dynamics of above compartments is shown in

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).

After birth from an infected human female, infected human children will join infected human child class at rate ( 1 − u 3 ) δ I H F . Susceptible human children will either join infected human child class at rate ( 1 − u 1 ) b β H C N H I V due to bite of an

infected vector (mosquito) or susceptible human male (adult) or female (adult) classes at growth rates g M and g F respectively. Infected human child will join recovered human class at recovery/treatment rate u 4 k H C . Disease induced death rate α H C will make very high effect on infected child population as possibility of microcephaly and Guillain-Barre syndrome due to ZIKV transmission occurred during pregnancy.

Susceptible human male (adult) and female (adult) will either join infected human male (adult) and female (adult) class at rates

( 1 − u 1 ) b β H M N H I V + ( 1 − u 2 ) c β S N H I H F and ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M

respectively due to vector bite and sexual intercourse with infected human female (adult) and male (adult) respectively. Infected human male (adult) and female (adult) will join recovered human class at recovery/treatment rate u 4 k H M and u 4 k H F respectively. Disease induced death rate α H will effect on infected human male and female (adult) population. At each stage, the natural death rate in human child/adult classes μ H is taken into account.

Vector (mosquito) population is divided in to two compartments susceptible vectors and infected vectors. Susceptible vector gets infection from infected human child/adult, when vectors (mosquitoes) bite them, and joins infected vector

class at rate b β V N V ( I H C + I H F + I H M ) . A portion of susceptible and infected vec-

tors (mosquitoes) will be eliminated due to either natural death rate in vectors (mosquitoes) μ V or spraying at rate u r 5 . Disease induced death rate α V 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

d S H C ( t ) d t = B H C − ( 1 − u 1 ) b β H C N H I V S H C − μ H S H C − ( g F + g M ) S H C (1)

d S H F ( t ) d t = B H F − ( ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M ) S H F + g F S H C − μ H S H F (2)

#Math_33# (3)

#Math_34# (4)

d I H F ( t ) d t = ( ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M ) S H F − u 4 k H F I H F − ( μ H + α H ) I H F (5)

d I H M ( t ) d t = ( ( 1 − u 1 ) b β H M N H I V + ( 1 − u 2 ) c β S N H I H F ) S H M − u 4 k H M I H M − ( μ H + α H ) I H M (6)

d R H ( t ) d t = u 4 ( k H C I H C + k H F I H F + k H M I H M ) − μ H R H (7)

d S V ( t ) d t = B V − μ V S V − b β V N V ( I H C + I H F + I H M ) S V − u 5 r S V (8)

d I V ( t ) d t = b β V N V ( I H C + I H F + I H M ) S V − ( μ V + α V ) I V − u 5 r I V (9)

Adding Equations (1) to (7),

d N H d t = B H C + B H F + B H M − μ H ( S H C + S H F + S H M + I H C + I H F + I H M + R H ) + ( 1 − u 3 ) δ I H F − α H C I H C − α H ( I H F + I H M )

And, adding Equations (8) and (9),

d N V d t = B V − μ V ( S V + I V ) − u 5 r ( S V + I V ) − α V I V

Hence,

d N H d t ≤ B H C + B H F + B H M − μ H N H ⇒ lim t → ∞ Sup N H ≤ B H C + B H F + B H M μ H

and,

d N V d t ≤ B V − μ V N V ⇒ lim t → ∞ Sup N V ≤ B V μ V .

So, the feasible region of the system is

Ω = { ( S H C , S H F , S H M , I H C , I H F , I H M , R H , S V , I V ) : S T H C + S U H C + S T H A + S U H A + I T H C + I U H C + I T H A + I U H A ≤ B H C + B H F + B H M μ H , S V + I V ≤ B V μ V , S H C > 0 , S H F > 0 , S H M > 0 , I H C > 0 , I H F ≥ 0 , I H M ≥ 0 , R H ≥ 0 , S V > 0 , I V ≥ 0 }

As at disease free equilibrium, I H C = 0 , I H F = 0 , I H M = 0 , I V = 0 and R H = 0 .

Hence, let X 0 = ( B H C μ H + g F + g M , B H F μ H − g F , B H M μ H − g M , 0 , 0 , 0 , 0 , B V μ V + u 5 r , 0 ) be disease free equilibrium point of system.

Using next generation method [

X ′ = ( I H C , I H F , I H M , I V , S H C , S H F , S H M , S V , R H ) ′

where dash denotes derivative

∴ X ′ = d X d t = ℑ ( x ) − v ( x )

where,

ℑ ( X ) = ( ( 1 − u 1 ) b β H C N H I V S H C ( 1 − u 1 ) b β H F N H I V S H F + ( 1 − u 2 ) c β S N H I H M S H F ( 1 − u 1 ) b β H M N H I V S H M + ( 1 − u 2 ) c β S N H I H F S H M b β V N V ( I H C + I H F + I H M ) S V 0 0 0 0 0 )

and

υ ( X ) = ( − ( 1 − u 3 ) δ I H F + u 4 k H C I H C + μ H I H C + α H C I H C u 4 k H F I H F + μ H I H F + α H I H F u 4 k H M I H M + μ H I H M + α H I H M μ V I V + α V I V + u 5 r I V − B H C + ( 1 − u 1 ) b β H C N H I V S H C + μ H S H C + ( g F + g M ) S H C − B H F + ( 1 − u 1 ) b β H F N H I V S H F + ( 1 − u 2 ) c β S N H I H M S H F + μ H S H F − g F S H C − B H M + ( 1 − u 1 ) b β H M N H I V S H M + ( 1 − u 2 ) c β S N H I H F S H F + μ H S H M − g M S H C − B V + b β V N V ( I H C + I H F + I H M ) S V + u 5 r S V + μ V S V α V S V + μ V I V + u 5 r I V )

Using, F = [ ∂ ℑ i ( X 0 ) ∂ X j ] and V = [ ∂ υ i ( X 0 ) ∂ X j ] for i , j = 1 , 2 , 3 , ⋯ , 9

Therefore,

F = ( 0 0 0 b 14 0 0 0 0 0 0 0 b 23 b 24 0 0 0 0 0 0 b 32 0 b 34 0 0 0 0 0 b 41 b 42 b 43 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 )

where,

b 14 = ( 1 − u 1 ) b β H C B H C ( μ H + g F + g M ) N H , b 23 = ( 1 − u 2 ) c β S B H F ( μ H − g F ) N H , b 24 = ( 1 − u 1 ) b β H F B H F ( μ H − g F ) N H , b 32 = ( 1 − u 2 ) c β S B H M ( μ H − g M ) N H , b 34 = ( 1 − u 1 ) b β H M B H M ( μ H − g M ) N H , b 41 = b β V B V ( μ V + u 5 r ) N V , b 42 = b β V B V ( μ V + u 5 r ) N V , b 43 = b β V B V ( μ V + u 5 r ) N V

and

V = ( a 11 a 12 0 0 0 0 0 0 0 0 a 22 0 0 0 0 0 0 0 0 0 a 33 0 0 0 0 0 0 0 0 0 a 44 0 0 0 0 0 0 0 0 a 54 a 55 0 0 0 0 0 0 a 63 a 64 0 a 66 0 0 0 0 a 72 0 a 74 0 0 a 77 0 0 a 81 a 82 a 83 0 0 0 0 a 88 0 a 91 a 92 a 93 0 0 0 0 0 a 99 )

where,

a 11 = u 4 k H C + μ H + α H C , a 12 = − ( 1 − u 3 ) δ , a 22 = u 4 k H F + μ H + α H , a 33 = u 4 k H M + μ H + α H a 33 = u 4 k H M + μ H + α H , a 44 = μ V + α V + u 5 r , a 54 = ( 1 − u 1 ) b β H C B H C ( μ H + g F + g M ) N H , a 55 = μ H + g F + g M , a 63 = ( 1 − u 2 ) c β S B H F ( μ H − g F ) N H , a 64 = ( 1 − u 1 ) b β H F B H F ( μ H − g F ) N H , a 66 = μ H − g F , a 72 = ( 1 − u 2 ) c β S B H M ( μ H − g M ) N H , a 74 = ( 1 − u 1 ) b β H M B H M ( μ H − g M ) N H , a 77 = μ H − g M , a 81 = b β V B V ( μ V + u 5 r ) N V , a 82 = b β V B V ( μ V + u 5 r ) N V , a 83 = b β V B V ( μ V + u 5 r ) N V , a 88 = μ V + u 5 r , a 91 = − u 4 k H C , a 92 = − u 4 k H F , a 93 = − u 4 k H M , a 99 = μ H

The basic reproduction number R 0 is spectral radius of matrix F V − 1 .

With the parametric values given in

If all eigenvalues of Jacobian matrix of the system of differential Equations (1) to (10) have negative real parts at

X 0 = ( B H C μ H + g F + g M , B H F μ H − g F , B H M μ H − g M , 0 , 0 , 0 , 0 , B V μ V + u 5 r , 0 )

then disease free equilibrium point becomes stable. Jacobian matrix of the system at DFE is,

J = ( c 11 0 0 0 0 0 0 0 c 19 0 c 22 0 0 0 c 26 0 0 c 29 0 0 c 33 0 c 35 0 0 0 c 39 0 0 0 c 44 c 45 0 0 0 c 49 0 0 0 0 c 55 c 56 0 0 c 59 0 0 0 0 c 65 c 66 0 0 c 69 0 0 0 c 74 c 75 c 76 c 77 0 0 0 0 0 c 84 c 85 c 86 0 c 88 0 0 0 0 c 94 c 95 c 96 0 0 c 99 )

where,

c 11 = − μ H − g F − g M , c 19 = − ( 1 − u 1 ) b β H C B H C ( μ H + g F + g M ) N H , c 21 = g F , c 22 = − μ H , c 26 = − ( 1 − u 2 ) c β S B H F ( μ H − g F ) N H , c 29 = − ( 1 − u 1 ) b β H F B H F ( μ H − g F ) N H , c 31 = g M , c 33 = − μ H , c 35 = − ( 1 − u 2 ) c β S B H M ( μ H − g M ) N H , c 39 = − ( 1 − u 1 ) b β H M B H M ( μ H − g M ) N H , c 44 = − u 4 k H C − μ H − α H C , c 45 = ( 1 − u 3 ) δ , c 49 = − c 19 , c 55 = − u 4 k H F − μ H − α H , c 56 = − c 26 , c 59 = − c 29 , c 65 = − c 35 , c 66 = − u 4 k H M − μ H − α H , c 69 = − c 39 , c 74 = u 4 k H C , c 75 = u 4 k H F , c 76 = u 4 k H M , c 77 = − μ H , c 84 = − b β V B V ( μ V + u 5 r ) N V , c 85 = c 84 , c 86 = c 84 , c 88 = − μ V − u 5 r , c 94 = − c 84 , c 95 = − c 84 , c 96 = − c 84 , c 99 = c 88 − α V .

With the parametric values given in

This implies that disease free equilibrium is locally asymptotically stable if R 0 < 1 and unstable, if R 0 > 1 .

For the endemic equilibrium, using

d S T H C d t = 0 , d S U H C d t = 0 , d S T H A d t = 0 , d S U H A d t = 0 , d I T H C d t = 0 , d I U H C d t = 0 , d I T H A d t = 0 , d I U H A d t = 0 , d S V d t = 0 , d I V d t = 0

These give, X ∗ = ( S T H C ∗ , S U H C ∗ , S T H A ∗ , S U H A ∗ , I T H C ∗ , I U H C ∗ , I T H A ∗ , I U H A ∗ , S V ∗ , I V ∗ ) as an endemic equilibrium point.

J ∗ = ( d 11 0 0 0 0 0 0 0 d 19 d 21 d 22 0 0 0 d 26 0 0 d 29 d 31 0 d 33 0 d 35 0 0 0 d 39 d 41 0 0 d 44 d 45 0 0 0 d 49 0 d 52 0 0 d 55 d 56 0 0 d 59 0 0 d 63 0 d 65 d 66 0 0 d 69 0 0 0 d 74 d 75 d 76 d 77 0 0 0 0 0 d 84 d 85 d 86 0 d 88 0 0 0 0 d 94 d 95 d 96 0 d 98 d 99 )

where,

Notation | Description | Parametric values |
---|---|---|

Human Population at time | 1600 | |

Vector Population at time | 20000 | |

Number of susceptible Human Child at time | 20 | |

Number of susceptible Human female at time | 40 | |

Number of susceptible Human male at time | 30 | |

Number of Infected Human Child at time | 10 | |

Number of Infected Human female at time | 20 | |

Number of Infected Human male at time | 12 | |

Number of Recovered Human at time | 80 | |

Number of susceptible vector at time | 9000 | |

Number of Infected vector at time | 5000 | |

New recruitments in susceptible human child class | 40 | |

New recruitments in susceptible human female class | 80 | |

New recruitments in susceptible human male class | 50 | |

New recruitments in susceptible vector class | 100 | |

Vector biting rate | 50 | |

Recovery/Treatment Rate in human child (Rate at which infected human child get recovered and moves towards recovered human class) | 0.15 | |

Recovery/Treatment Rate in human female (Rate at which infected human female get recovered and moves towards recovered human class) | 0.35 | |

Recovery/Treatment Rate in human male (Rate at which infected human male get recovered and moves towards recovered human class) | 0.35 | |

Disease transmission rate in human child from vector bite | 0.5 | |

Disease transmission rate in human female from vector bite | 0.25 | |

Disease transmission rate in human male from vector bite | 0.25 | |

Disease transmission rate in human from sexual contacts | 0.2 | |

Sexual contact rate in human | 25 | |

Disease transmission rate in vector | 0.4 | |

Growth rate of human from child to female | 0.35 | |

Growth rate of human from child to male | 0.35 | |

Rate of Spraying to kill vectors(insecticides) | 0.9 | |

Birth rate coefficient of infected new borns | 0.15 |

Mortality rate for human | 0.14 | |
---|---|---|

Mortality rate for vectors | 0.5 | |

Disease induced death rate amongst human in Infection classes | 0.7 | |

Disease induced death rate amongst vectors | 0.5 | |

Prevention control from vector bites | Are controls and taken initially zero, will be optimized to control disease spread. | |

Prevention control from sexual activities amongst infected humans | ||

Prevention control for infected human female from conceiving child | ||

Treatment control amongst human | ||

Spraying (insecticide) control amongst vector |

d 11 = − ( 1 − u 1 ) b β H C N H I V ∗ − μ H − g F − g M , d 19 = − ( 1 − u 1 ) b β H C N H S H C ∗ , d 21 = g F , d 22 = − ( 1 − u 1 ) b β H F N H I V ∗ − ( 1 − u 2 ) c β S N H I H M ∗ − μ H , d 26 = − ( 1 − u 2 ) c β S N H S H F ∗ , d 29 = − ( 1 − u 1 ) b β H F N H S H F ∗ , d 31 = g M , d 33 = − ( 1 − u 1 ) b β H M N H I V ∗ − ( 1 − u 2 ) c β S N H I H F ∗ − μ H , d 35 = − ( 1 − u 2 ) c β S N H S H M ∗ , d 39 = − ( 1 − u 1 ) b β H M N H I V ∗ , d 41 = − d 39 , d 44 = − u 4 k H C − μ H − α H C , d 45 = ( 1 − u 3 ) δ , d 49 = − d 19 , d 52 = ( 1 − u 1 ) b β H F N H I V ∗ + ( 1 − u 2 ) c β S N H I H M ∗ , d 55 = − u 4 k H F − μ H − α H , d 56 = − d 26 , d 59 = − d 29 , d 63 = ( 1 − u 1 ) b β H M N H I V ∗ + ( 1 − u 2 ) c β S N H I H F ∗ , d 65 = − d 35 , d 66 = − u 4 k H M − μ H − α H , d 69 = − d 39 , d 74 = u 4 k H C , d 75 = u 4 k H F , d 76 = u 4 k H M , d 77 = − μ H , d 84 = − b β V N V S V ∗ , d 85 = d 84 , d 86 = d 84 , d 88 = − μ V − b β V N V （ I H c * + I H F * + I H M ） * − u 5 r , d 94 = − d 84 , d 95 = − d 84 , d 96 = − d 84 , d 98 = b β V N V （ I H c * + I H F * + I H M ） * , d 99 = − μ V − α V − u 5 r .

With the parametric values given in

For Zika disease an optimal control model is formulated, to derive optimal prevention from mosquito bite u 1 , optimal prevention to stop pregnancy u 2 and sexual transmission u 3 , optimal treatment u 4 and optimal spraying u 5 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,

J ( u 1 , u 2 , u 3 , u 4 , u 5 ) = ∫ 0 T ( w 21 ( S H C ) 2 + w 22 ( S H F ) 2 + w 23 ( S H M ) 2 + w 24 ( I H C ) 2 + w 25 ( I H F ) 2 + w 26 ( I H M ) 2 + w 27 ( R H ) 2 + w 28 ( S V ) 2 + w 29 ( I V ) 2 + w 11 u 1 2 + w 12 u 2 2 + w 13 u 3 2 + w 14 u 4 2 + w 15 u 5 2 ) d t

where, x = ( S H C , S H F , S H M , I H C , I H F , I H M , R H , S V , I V ) , u = ( u 1 , u 2 , u 3 , u 4 , u 5 ) and w i j are weights to regularise the optimal control. In order minimise the disease spread using, w i j > 0 , for i = 2 , j = 1 , 2 , ⋯ , 9 . Also to minimise the cost associated with strategies applied for preventions, treatment and spraying in a way that spread of infection can be controlled by choosing weights w k m > 0 , for k = 1 , m = 1 , 2 , ⋯ , 5 . So, the optimal control task reads as min u J ( u 1 , u 2 , u 3 , u 4 , u 5 ) such that P ( x , u ) = 0 . Where P ( x , u ) = 0 denotes the system of equations de-

fined in (1) to (9), i.e. x ˙ = M ( t , x , u ) , x ( 0 ) = x 0 . The optimal control u ∗ can be obtained from J ( u 1 ∗ , u 2 ∗ , u 3 ∗ , u 4 ∗ , u 5 ∗ ) = min u J ( u 1 , u 2 , u 3 , u 4 , u 5 ) such that P ( x , u ) = 0 .

The control set is

Γ = { ( u 1 , u 2 , u 3 , u 4 , u 5 ) / u i ( t ) ispiecewisecontinouson [ 0 , T ] , a i ≤ u i ( t ) ≤ b i , i = 1 , 2 , 3 , 4 , 5 , a i , b i areconstantsin [ 0 , 1 ] }

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,

L ( x , u , λ ) = ( w 21 ( S H C ) 2 + w 22 ( S H F ) 2 + w 23 ( S H M ) 2 + w 24 ( I H C ) 2 + w 25 ( I H F ) 2 + w 26 ( I H M ) 2 + w 27 ( R H ) 2 + w 28 ( S V ) 2 + w 29 ( I V ) 2 + w 11 u 1 2 + w 12 u 2 2 + w 13 u 3 2 + w 14 u 4 2 + w 15 u 5 2 )

To obtain the minimal value of the Lagrangian, defining the Hamiltonian H for the control problem as,

H ( x , u , λ ) = w 21 ( S H C ) 2 + w 22 ( S H F ) 2 + w 23 ( S H M ) 2 + w 24 ( I H C ) 2 + w 25 ( I H F ) 2 + w 26 ( I H M ) 2 + w 27 ( R H ) 2 + w 28 ( S V ) 2 + w 29 ( I V ) 2 + w 11 u 1 2 + w 12 u 2 2 + w 13 u 3 2 + w 14 u 4 2 + w 15 u 5 2 + λ 1 ( B H C − ( 1 − u 1 ) b β H C N H I V S H C − μ H S H C − ( g F + g M ) S H C ) + λ 2 ( B H F − ( ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M ) S H F + g F S H C − μ H S H F ) + λ 3 ( B H M − ( ( 1 − u 1 ) b β H M N H I V + ( 1 − u 2 ) c β S N H I H F ) S H M + g M S H C − μ H S H M ) + λ 4 ( ( 1 − u 1 ) b β H C N H I V S H C + ( 1 − u 3 ) δ I H F − u 4 k H C I H C − ( μ H + α H C ) I H C ) + λ 5 ( ( ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M ) S H F − u 4 k H F I H F − ( μ H + α H ) I H F ) + λ 6 ( ( ( 1 − u 1 ) b β H M N H I V + ( 1 − u 2 ) c β S N H I H F ) S H M − u 4 k H M I H M − ( μ H + α H ) I H M ) + λ 7 ( u 4 ( k H C I H C + k H F I H F + k H M I H M ) − μ H R H ) + λ 8 ( B V − μ V S V − b β V N V ( I H C + I H F + I H M ) S V − u 5 r S V ) + λ 9 ( b β V N V ( I H C + I H F + I H M ) S V − ( μ V + α V ) I V − u 5 r I V )

To get optimality Pontrayagin’s maximum (minimum) principle, for the model is defined as follow.

If ( u 1 ∗ , u 2 ∗ , u 3 ∗ , u 4 ∗ , u 5 ∗ ) is optimal solution of an optimal control problem then there exists a nontrivial vector function λ ( t ) = ( λ 1 ( t ) , λ 2 ( t ) , λ 3 ( t ) , λ 4 ( t ) , λ 5 ( t ) , λ 6 ( t ) , λ 7 ( t ) , λ 8 ( t ) , λ 9 ( t ) ) satisfying following equations:

1) The state equation d x d t = ∂ H ( t , u 1 ∗ , u 2 ∗ , u 3 ∗ , u 4 ∗ , u 5 ∗ , λ ( t ) ) ∂ λ

2) The optimality condition 0 = ∂ H ( t , u 1 ∗ , u 2 ∗ , u 3 ∗ , u 4 ∗ , u 5 ∗ , λ ( t ) ) ∂ u

3) The adjoint equation d λ d t = ∂ H ( t , u 1 ∗ , u 2 ∗ , u 3 ∗ , u 4 ∗ , u 5 ∗ , λ ( t ) ) ∂ x .

Using equations stated above for the Hamiltonian defined by (13), using state equations as given by (1) to (9), adjoint equations as

λ ˙ 1 = − 2 w 21 S H C + ( λ 1 − λ 4 ) ( ( 1 − u 1 ) b β H C N H I V ) + λ 1 ( μ H + g F + g M )

λ ˙ 2 = − 2 w 22 S H F + ( λ 2 − λ 5 ) ( ( 1 − u 1 ) b β H F N H I V + ( 1 − u 2 ) c β S N H I H M ) + λ 2 ( μ H − g F )

λ ˙ 3 = − 2 w 23 S H M + ( λ 3 − λ 6 ) ( ( 1 − u 1 ) b β H M N H I V + ( 1 − u 2 ) c β S N H I H F ) + λ 3 ( μ H − g M )

λ ˙ 4 = − 2 w 24 I H C + ( λ 8 − λ 9 ) ( b β V N V S V ) + ( λ 4 − λ 7 ) u 4 k H C + λ 4 ( μ H + α H C )

λ ˙ 5 = − 2 w 25 I H F + ( λ 3 − λ 6 ) ( ( 1 − u 2 ) c β S N H S H M ) + ( λ 8 − λ 9 ) ( b β V N V S V ) + ( λ 5 − λ 7 ) u 4 k H F + λ 5 ( μ H + α H ) − λ 5 ( 1 − u 3 ) δ

λ ˙ 6 = − 2 w 26 I H F + ( λ 2 − λ 5 ) ( ( 1 − u 2 ) c β S N H S H F ) + ( λ 8 − λ 9 ) ( b β V N V S V ) + ( λ 6 − λ 7 ) u 4 k H M + λ 6 ( μ H + α H )

λ ˙ 7 = − 2 w 27 R H + μ H λ 7

λ ˙ 8 = − 2 w 28 S V + ( λ 8 − λ 9 ) ( b β V N V ( I H C + I H F + I H M ) ) + λ 8 ( μ V + u 5 r )

λ ˙ 9 = − 2 w 29 I V + ( λ 1 − λ 4 ) ( ( 1 − u 1 ) b β H C N H S H C ) + ( λ 2 − λ 5 ) ( ( 1 − u 1 ) b β H F N H S H F ) + ( λ 3 − λ 6 ) ( ( 1 − u 1 ) b β H M N H S H M ) + λ 9 ( μ V + α V + u 5 r )

andoptimality conditions are as

2 w 11 u 1 + ( λ 1 − λ 4 ) ( b β H C N H I V S H C ) + ( λ 2 − λ 5 ) ( b 1 β H F N H I V S H F ) + ( λ 3 − λ 6 ) ( b β H M N H I V S H M ) = 0

2 w 12 u 2 + ( λ 2 − λ 5 ) c β S N H S H F I H M + ( λ 3 − λ 6 ) c β S N H S H M I H F = 0

2 w 13 u 3 − λ 4 δ I H F = 0

2 w 14 u 4 + ( λ 7 − λ 4 ) k H C I H C + ( λ 7 − λ 5 ) k H F I H F + ( λ 7 − λ 6 ) k H M I H M = 0

2 w 15 u 5 − λ 8 r S V − λ 9 r I V = 0

Solving optimality conditions for optimal control and the property of control space u give,

u 1 ∗ ( t ) = max { min { 1 , 1 2 w 11 [ ( λ 4 − λ 1 ) ( b β H C N H I V S H C ) + ( λ 5 − λ 2 ) ( b 1 β H F N H I V S H F ) + ( λ 6 − λ 3 ) ( b β H M N H I V S H M ) ] } , 0 } (10)

u 2 ∗ ( t ) = max { min { 1 , 1 2 w 12 ( λ 5 − λ 2 ) c β S N H S H F I H M + ( λ 6 − λ 3 ) c β S N H S H M I H F } , 0 } (11)

u 3 ∗ ( t ) = max { min { 1 , 1 2 w 13 λ 4 δ I H F } , 0 } (12)

#Math_177# (13)

u 5 ∗ ( t ) = max { min { 1 , 1 2 w 15 ( λ 8 r S V + λ 9 r I V ) } , 0 } (14)

For parametric values given in

In

In

From

From

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. (

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.

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.

The authors thank DST-FIST file # MS1-097 for support to the department of Mathematics.

Shah, N.H., Patel, Z.A. and Yeolekar, B.M. (2017) Preventions and Controls on Congenital Transmissions of Zika: Mathematical Analysis. Applied Mathematics, 8, 500-519. https://doi.org/10.4236/am.2017.84040