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A delayed mathematical model of Dengue dynamical transmission between vector mosquitoes and human, incorporating a control strategy of perfect pediatric vaccination is proposed in this paper. By some analytical skills, we obtain the existence of disease-free equilibria and endemic equilibrium, the necessary conditions of global asymptotical stability about two disease-free equilibria. Further, by Pontryagin’s maximum principle, we obtain the optimal control of the disease. Finally, numerical simulations are carried out to verify the correctness of the theoretical results and feasibility of the control measure.

Dengue fever and dengue hemorrhagic fever are the vector-borne diseases which transcend international borders as the most important arboviral diseases currently threatening human populations. The research found that more than approximately 50 million people are affected by dengue disease each year [

In 1760, the Swiss mathematician Daniel Bernoulli published an investigation on the impact of immunization with cowpox. From then on, the means of protecting figures from infection through immunization begin to be widely used; in addition, the method has already successfully decreased both mortality and morbidity [

Since the 1940s, dengue vaccines have been under development. But industry interest languished throughout the 20th century owing to the limited appreciation of global disease burden and the potential markets for it. In recent years, however, with the increase in dengue infections, the development of dengue vaccines has amazingly accelerated, as well as the prevalence of all four circulating serotypes. It became a serious concern for faster development of a vaccine [

On the other hand, there are three successive aquatic juvenile phases (egg, larva and pupa) and one adult pupa for the life cycle of the mosquitoes. It is large compared the duration from the egg to the adult (1 - 2 weeks) with the average life span (about 3 weeks) of an adult mosquito. The size of the mosquito population is strongly affected by temperature. The number of female mosquitoes changes accordingly due to seasonal variations. When the size of the mosquito population increases during the favorable periods, the dengue virus infection among individuals also increases, therefore the incidence for humans’ increases. Then it is vital to consider the maturation time of mosquitoes, the length of the larval phase from egg to adult mosquitoes, and the impact on the transmission of dengue virus.

Based on above-mentioned conditions, a dengue dynamical model with maturation delay and pediatric vaccination is proposed to consider the effects of maturation delay and pediatric vaccination for the transmission of dengue between mosquitoes and human. The remaining parts of this paper are organized as follows. A form of vaccination model is formulated: a perfect pediatric vaccination model, in the next Section. And the stability of equilibria of the model is analysed in Section 3. In Section 4, the optimal control strategy of the disease is discussed. Finally, the numerical simulation is performed in Section 5.

Dengue can be a serious candidate for a type of vaccination which is much focus on vaccinating newborns or very young infants. It parallels many potentially human infections, such as measles, rubella, polio. In this section, we propose a SVIR model in which a continuous vaccination strategy is considered, and a proportion of the newborn

The mathematical model can be described as:

where

The initial condition of model (1) is given as

where

Firstly, it follows from model (1) that the total number of adult female mosquitoes satisfies the following equation

With initial condition

Letting

Param. | Description | Value | Source |
---|---|---|---|

Susceptible: individuals who can contract the disease | − | − | |

Vaccinated: individuals who were vaccinated and are now immune | − | − | |

Infected: individuals who are capable of transmitting the disease | − | − | |

Resistant: individuals who have acquired immunity | − | − | |

Susceptible: mosquitoes able to contract the disease | − | − | |

Infected: mosquitoes capable of transmitting the disease to humans | − | − | |

Average number of bites by mosquito infected with virus (day) | 0.8 | [ | |

Transmission probability from | 0.375 | [ | |

Transmission probability from | 0.375 | [ | |

Average host life expectancy (year) | [ | ||

Dengue recovery rate in human population (day) | [ | ||

Natural death rate of adult female mosquitoes | [ | ||

Size of the mosquito population at which egg laying is maximized without delay | 10000 | [ | |

Maximum per capita daily mosquito egg production rate | [ | ||

Maturation time of the mosquito | [ | ||

Death rate of juvenile mosquitoes | [ | ||

Vertical transmission probability of the virus in the mosquito population | [ |

it follows that

The following theorem describes the global asymptotic behavior of equation (3).

Theorem 1. For model (3) with the initial condition (4), the solution

(i) If

(ii) If

The process of proofing is absolutely same as Theorem 1 in Reference [

Now, define two threshold values

Assuming that the vaccine is perfect, which means that it confers life-long protection. For model (1), we can get two nontrivial disease-free equilibria and a endemic equilibrium. That is, the disease-free equilibrium without mosquitoes

Firstly, on the globally asymptotical stability of disease-free equilibrium without mosquito

Theorem 2. If

Proof. It obvious that

For the first equation of model (1) we have

Obviously, it is easy to obtain that system (6) has a unique positive equilibrium

By comparison principle,

Due to

From the third and the forth equation of model (1) we get

Consider the comparison differential equation

It is easy to obtain that

By comparison principle,

On the stability of disease-free equilibrium with mosquitoes, linearizing model (1) about

Similarly, as for the endemic equilibrium

Obviously, the study of solving these transcendental equations of (8) and (9) is out of the scope of this one. Therefore, we give the stability of

Theorem 3. Supposing that

then model (1) has a unique disease-free equilibrium with mosquitoes

Proof. From the sixth equation of model (1) we get

Consider another auxiliary system

it is obvious that the equilibrium

where

To obtain two negative solutions about (12), require that

Then only requires to satisfies that

According to above discussion and comparison principle we know that

As for the stability about other variations of model (1), they are absolutely same as Theorem 2, omitted.

Remark 1. In fact, q is small enough since the vertical transmission of Dengue virus in mosquitoes is rare. Therefore,

In this section, the vaccination of model (1) is seen as a control variable to reduce or even eradicate the disease. Let p be the control variable:

where

We have the following theorem on the existence of optimal vaccination.

Theorem 4. The problem (1) and (13) with the initial condition (2), admits a unique optimal solution

and the transversality conditions

Proof. The existence of optimal solution

and

where the Hamiltonian H is defined by

Together with the minimality condition

Satisfied almost everywhere on

Now, some numerical simulations are performed to illustrate the main theoretical results above for stability of equilibria using the Runge-Kutta method in the software MATLAB. The values of parameters for model (1) are listed in

In

To illustrate the asymptotic behaviors of infectious classes (individuals and mosquitoes) and susceptible mosquitoes when the parameter conditions satisfying Theorem 3, set

decreasing to zero eventually, whereas the number of susceptible mosquitoes are not decreasing to zero but having a positive stable state from

In order to further investigate the dynamic behavior of model (1), setting

are both having positive stable states (see

To better visualize the impact of maturation delay of

pace with increasing of the value of

increasing with the value q.

This work was supported in part by the Natural Science Foundation of Xinjiang (Grant No. 2016D01C046).

Xue, Y.N. and Nie, L.F. (2016) A Model of Perfect Pediatric Vac- cination of Dengue with Delay and Optimal Control. International Journal of Modern Nonlinear Theory and Application, 5, 133- 146. http://dx.doi.org/10.4236/ijmnta.2016.54014