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In this paper, we studied the transmission dynamics of ZIKV in the presence of a vector under the combined effects of treatment and vaccination in a hypothetical population. The disease-free
ε
_{o} and endemic
ε
_{1} equilibria were established with local stability on
ε
_{o}. We established the basic reproduction number
R
_{o} which served as a threshold for measuring the spread of the infection in the population using the next-generation matrix and computed its numerical value to be
R
_{o} = 0.0185903201 using the parameter values. It was established that the disease-free equilibrium
ε
_{o} is locally asymptotically stable since
R
_{o} < 1; meaning ZIKV infection would be eradicated from the population. The computational results of the study revealed that combining the two interventions of vaccination and treatment concomitantly proffers an optimal control strategy in taming the transmission of the virus than a single intervention strategy.

In 1948, scientists (from Yellow Fever Research Institute) caged a Rhesus macaque monkey in the Zika Forest of Uganda for epidemic study. After some time, the monkey developed fever, the scientists took a sample of the monkey’s serum and isolated a ‘‘filterable transmission agent’’ which was named Zika Virus [

ZIKV, a flavivirus responsible for an unprecedented epidemic in Brazil and the Americas, has been causally associated with fetal microcephaly, intrauterine growth restriction, and other birth defects in both humans and mice [

Mathematical modeling in epidemiology provides understanding of the mechanisms that influence the spread of diseases, and suggests control strategies [

A general SEIR model with vertical transmission for the dynamics of an infectious disease was studied by [

In this work, we proposed a mathematical model for the dynamics of Zika virus under the combined effects of vaccine and oral treatment. In developing the model we treated Zika virus infection as asymptomatic and symptomatic infections and partition the infected compartment into asymptomatic and symptomatic infected compartments.

The model was developed on the assumptions that the birth and death per capita mosquito rates are constant and distinct, the infectious compartment was partitioned into asymptomatic and symptomatic compartments such that asymptomatic and symptomatic infected individuals are equally infectious, the recruitment rate into the human population is by birth and is at a constant rate, the deaths in the human population are either natural or disease-induced and are constants, proportion of vaccinated susceptible per unit time are regarded removed with immunity response against Zika virus infection for a particular period of time, recovered individuals may become susceptible to infection again during a single outbreak or in the future outbreaks, recovery rates for asymptomatic and symptomatic infectious individuals are assumed the same and are constant. The vector (Mosquito) dos no recover from the infection. We targeted 50% vaccine coverage on the susceptible population, this leads us to the estimation of the proportion of susceptible vaccinated per unit time as α h = 0.5 , we also assume that lost of temporary immunity from recovery λ h , and that of vaccination φ h , to become susceptible again is very small due to the strength of treatment and the vaccine, this leads to the estimation of this parameters as λ h = 0.09 and φ h = 0.02 respectively. For the start up population for the simulation, we used hypothetical total human population of 5120 distributed across the compartments of the model in the human host as indicated in

The susceptible human patch S h is recharged by birth. A susceptible human from S h maybe vaccinated and move to the vaccinated patch V h , or gets into contact with an infected vector from I v and become exposed (if not vaccinated) to move to E h . After the incubation period, the exposed human may become infected and move to I h , a or I h , s . When treatment is administered on individuals in I h , a or I h , s , they may recover and move to R h or die naturally or as a result of the disease. A proportion of the vaccinated human in V h may lose the temporary immunity conferred by the vaccine after some time due to the deterioration of the efficacy of the vaccine and become susceptible again, thus moving back to S h . A vector from S v may become exposed after coming into contact with an infected human from I h , a or I h , s and move to E v . After the incubation period, the exposed vector becomes infected and moves to I v and since there is no recovery for the vector, they die naturally. The cycle continues in this manner. The schematic diagram of the dynamics can be found in

Based on the transmission dynamics described above, we obtained the following model equations:

S ˙ h = b h − b β v h S h I v N h − α h S h + λ h R h + φ h V h − d h S h V ˙ h = α h S h − ( φ h + d h ) V h E ˙ h = b β v h S h I v N h − v h E h − d h E h I ˙ h , s = ( 1 − q ) v h E h − δ h I h , s − ( d h + ϕ h ) I h , s I ˙ h , a = q v h E h − δ h I h , a − ( d h + ϕ h ) I h , a R ˙ h = δ h ( I h , a + I h , s ) − λ h R h − d h R h S ˙ v = μ v − b β h v S v I h , a + I h , s N h − d v S v E ˙ v = b β h v S v I h , a + I h , s N h − ( d v + v v ) E v I ˙ v = v v E v − d v I v } (1)

Using standard approaches, we obtained the disease-free equilibrium ε ∘ and the endemic equilibrium ε 1 and are respectively given to be:

ε ∘ = ( S h , V h , E h , I h , s , I h , a , R h , S v , E v , I v ) = ( b h ( φ h + d h ) d h ( α h + φ h + d h ) , b h α h d h ( α h + φ h + d h ) , 0 , 0 , 0 , 0 , μ v d v , 0 , 0 ) (2)

ε 1 = ( S h ∗ , V h ∗ , E h ∗ , I h , s ∗ , I h , a ∗ , R h ∗ , S v ∗ , E h ∗ , I v ∗ ) (3)

where:

S h ∗ = d v ( d h + v h ) ( d v + v v ) ( δ h + d h + ϕ h ) ( b h − ϕ h I h ) 2 [ b d h β h v I h + d v ( b h − ϕ h I h ) ] b 2 d h 2 v h v v μ v β h v β v h ( b h − ϕ h I h )

V h ∗ = α h d v ( d h + v h ) ( d v + v v ) ( b h − ϕ h I h ) 2 ( δ h + d h + ϕ h ) [ b d h β h v I h + d v ( b h − ϕ h I h ) ] b 2 d h 2 v h v v μ v β h v β v h ( b h − ϕ h I h ) ( φ h + d h )

E h ∗ = ( δ h + d h + ϕ h ) v h I h , I h , s ∗ = ( 1 − q ) I h ,

I h , a ∗ = q I h , R h ∗ = δ h λ h + d h I h

S v ∗ = μ v ( b h − ϕ h I h ) b d h β h v I h + ( b h − ϕ h I h )

E v ∗ = b d h μ v β h v ( b h − ϕ h I h ) I h ( d v + v v ) ( b h − ϕ h I h ) [ b d h β h v I h + d v ( b h − ϕ h I h ) ]

I v ∗ = b d h v v μ v β h v ( b h − ϕ h I h ) I h d v ( d v + v v ) ( b h − ϕ h I h ) [ b d h β h v I h + d v ( b h − ϕ h I h ) ]

such that I h , a + I h , s = I h .

Since all parameters in this model are nonnegative, the solution of the model variables in the disease-free equilibrium and the endemic equilibrium is obviously positive. The total populations of the humans and the vectors in the model were defined by the following demographic equations:

N h = S h + V h + E h + I h , a + I h , s + R h (4)

N v = S v + E v + I v (5)

Therefore, under the dynamics described by Equations (1)-(3), the region:

Ω = { ( S h , V h , E h , I h , a , I h , s , R h ) ∈ ℜ + 6 : S h ≥ 0 , V h ≥ 0 , E h ≥ 0 , I h , a ≥ 0 , I h , s ≥ 0 , R h ≥ 0 , S h + V h + E h + I h , a + I h , s + R h ≤ N h ≤ b h d h }

is positively invariant. Hence, the system is mathematically and epidemiologically well-posed. Therefore, for initial starting point, x ∈ ℜ + 6 ; the trajectory lies in Ω . Thus, we can restrict our analysis to the region Ω .

The basic reproduction number of the model was obtained as;

R ∘ = b 2 d h v h v v μ v β h v β v h ( φ h + d h ) b h d v 2 ( α h + φ h + d h ) ( d h + v h ) ( d v + v v ) ( d h + δ h + ϕ h ) (6)

The Jacobian matrix of the model was obtained as:

J = [ − ( s + α h + d h ) φ h 0 0 0 λ h 0 0 − w 1 α h − p 0 0 0 0 0 0 0 s 0 − c 1 0 0 0 0 0 0 0 0 c 2 − x 0 0 0 0 0 0 0 q v h 0 − x 0 0 0 0 0 0 0 δ h δ h − c 3 0 0 0 0 0 0 − m 1 − m 1 0 − ( n 1 + d v ) 0 0 0 0 0 m 1 m 1 0 n 1 − c 4 0 0 0 0 0 0 0 0 v v − d v ] (7)

where: s = b β v h I v ∗ N h ∗ , x = ( δ h + d h + ϕ h ) , p = ( φ h + d h ) , w 1 = b β v h S h ∗ N h ∗ , n 1 = b β h v I h , a ∗ + I h , s ∗ N h ∗ , m 1 = ( b β h v S v ∗ N h ∗ ) , c 1 = ( v h + d h ) , c 2 = ( 1 − q ) v h , c 3 = ( λ h + d h ) , c 4 = ( v v + d v ) .

Evaluating J at the disease-free equilibrium ε ∘ , we obtained:

J = [ − ( α h + d h ) φ h 0 0 0 λ h 0 0 − w 2 α h − p 0 0 0 0 0 0 0 0 0 − c 1 0 0 0 0 0 0 0 0 c 2 − x 0 0 0 0 0 0 0 q v h 0 − x 0 0 0 0 0 0 0 δ h δ h − c 3 0 0 0 0 0 0 − m 2 − m 2 0 − d v 0 0 0 0 0 m 2 m 2 0 0 − c 4 0 0 0 0 0 0 0 0 v v − d v ] (8)

where: w 2 = − b β v h ( φ h + d h ) ( α h + φ h + d h ) , m 2 = b d h β h v μ v b h d v .

Next, we used elementary row-operations to row-reduce the Jacobian matrix evaluated at disease free equilibrium as used by [

#Math_70# (9)

where c 5 = b α h β v h ( φ h + d h ) ( α h + φ h + d h ) , c 6 = d h ( α h + φ h + d h ) and c 7 = ( α h + d h ) .

Therefore, since (4) is an upper triangular matrix, we obtained the following nine eigenvalues from the reduced Jacobian matrix which are the entries on the leading diagonal:

λ 1 = − c 7 , λ 2 = − c 6 , λ 3 = c 3 , λ 4 = − c 3 x , λ 5 = − c 3 x , λ 6 = c 3 2 x , λ 7 = − d v c 3 x , λ 8 = c 4 c 3 x , λ 9 = d v c 4 c 3 x

After substitutions, the eigenvalues became:

λ 1 = − ( α h + d h ) (10)

λ 2 = − d h ( α h + φ h + d h ) (11)

λ 3 = − ( λ h + d h ) (12)

λ 4 = − ( λ h + d h ) ( δ h + d h + ϕ h ) (13)

λ 5 = − ( λ h + d h ) ( δ h + d h + ϕ h ) (14)

λ 6 = − ( λ h + d h ) 2 ( δ h + d h + ϕ h ) (15)

λ 7 = − d v ( λ h + d h ) ( δ h + d h + ϕ h ) (16)

λ 8 = − ( v v + d v ) ( λ h + d h ) ( δ h + d h + ϕ h ) (17)

λ 9 = − d v ( v v + d v ) ( λ h + d h ) ( δ h + d h + ϕ h ) (18)

Since λ i < 0 , for i = 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , therefore, the disease-free equilibrium is locally asymptotically stable from the following theorem:

Theorem 1: The disease-free equilibrium is locally asymptotically stable if R ∘ < 1 and unstable if R ∘ > 1 .

The basic reproduction number was then evaluated with Scientific Work Place using parameter values in

We carried out Numerical Simulation using the values in

The result of the numerical simulations is given in Figures 2-9 under the following subheads:

・ Graphical simulation results with combined intervention (Vaccine and treatment);

・ Graphical simulation results with treatment only;

・ Graphical simulation results with Vaccine only at vaccine coverage rates:

α h = 0.5 , α h = 0.6 , α h = 0.7

Parameter | Description | Value | Source | |
---|---|---|---|---|

β_{vh} | Rate of infectiousness of human to mosquitoes | 0.01 | [ | |

Β_{hv} | Rate of infectiousness of mosquitoes to humans | 0.01 | [ | |

b | Biting rate | 0.3 | [ | |

b_{h} | Recruitment rate into the human population | 0.0391389432 | [ | |

d_{h} | Natural death rate in the human population | 0.00003913894325 | [ | |

d_{v} | Natural death rate in the mosquito’s population | 0.0476190476 | [ | |

ϕ h | Disease-induced death rate in human population | 0.001 | [ | |

ν_{h} | Human’s incubation rate | 0.2 | [ | |

ν_{v} | Vector incubation rate | 0.1 | [ | |

q | Proportion of latent that become asymptomatic and infectious | 0.8 | [ | |

δ_{h} | Human recovery rate | 0.1666666667 | [ | |

µ_{v} | Recruitment rate into the vector population | 476.19047619 | [ | |

α_{h} | Proportion of susceptible vaccinated per unit time | 0.5 | Assumed | |

λ_{h} | Proportion of the recovered that loses temporary immunity and become susceptible again | 0.09 | Assumed | |

φ_{h} | Proportion of vaccinated individuals that lose immunity and become susceptible again | 0.02 | Assumed |

Variable | Description | Value | Source |
---|---|---|---|

S_{h} | Susceptible humans | 1430 | Assumed |

S_{v} | Susceptible vectors | 9500 | Assumed |

E_{h} | Exposed/Latent humans | 220 | Assumed |

E_{v} | Exposed/Latent vectors | 600 | Assumed |

I_{h,a} | Asymptomatic infectious humans | 80 | Assumed |

I_{h,s} | Symptomatic infectious humans | 20 | Assumed |

I_{v} | Infected vector | 170 | Assumed |

R_{h} | Recovered humans | 50 | Assumed |

V_{h} | Vaccinated humans | 3320 | Assumed |

N_{h} | Total human population | 5120 | Assumed |

N_{v} | Total vector population | 10270 | Assumed |

populations decreases gradually within the first hundred 100 days of the intervention and remained asymptotic to zero, while

From the results of the numerical simulations above, we see that: 1) Treatment intervention alone or vaccine intervention alone have impact on reducing the spread of the current ongoing Zika virus outbreak; 2) Vaccine intervention alone reduces the susceptible human population that may become exposed and later infected; 3) Combined vaccination and treatment interventions therapy have greater impact in reducing the spread of the virus in fewer days of the joint interventions with vaccine administered at about 50% coverage or greater on the susceptible human population, hence combined therapy gives a better result in a shorter period than treatment or vaccination alone. Therefore, we recommend combining vaccine and treatment in a population cohort intervention program to control the spread of Zika virus infection.

Usman, S., Adamu, I.I. and Babando, H.A. (2017) Mathematical Model for the Transmission Dynamics of Zika Virus Infection with Combined Vaccination and Treatment Interventions. Journal of Applied Mathematics and Physics, 5, 1964-1978. https://doi.org/10.4236/jamp.2017.510166

1) Description of Model State Variables

Variable Description

S h : Susceptible human population

S v : Susceptible vector population

E h : Expose/Latent human population

E v : Expose/Latent vectorpopulation

I h , a : Asymptomatic infectious humanpopulation

I h , s : Symptomatic infectious humanpopulation

I v : Infectious vectorpopulation

R h : Recovered humanpopulation

V h : Vaccinated human population

N h : Total human population

N v Total vector population

2) Description of the Model Parameters

Parameter Description

β v h : Rate of infectiousness of human to mosquito

β h v : Rate of infectiousness of mosquito to human

b : Biting rate

b h : Recruitment rate into the human population

d h : Natural death rate in the human population

d v : Natural death rate in the vector population

ϕ h : Disease-induced death rate in the human population

v h : Disease incubation rate in human

q : Proportion of latent that becomes asymptomatic & infectious

δ h : Recovery rate for human

μ v : Recruitment rate into the vector population

v v : Disease incubation rate in mosquito (vector)

α h : Proportion of susceptible vaccinated per unit time

λ h : Proportion of treated individuals that becomes susceptible again

φ h : Proportion of vaccinated individuals that lose immunity and become susceptible again

3) Noun

ZIKV: Zika Virus

CDC: Center for Disease Prevention and Control

DNA: Deoxyribonucleic acid

SEIR: Susceptible, Exposed, Infected and Recovered