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The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.

Mathematical modeling has become an important tool for analyzing the spread as well as control of infectious diseases. It is also a useful tool for the measurement of the effect of different strategies for controlling the spread of infectious diseases within a population. In recent years epidemiological modeling of infectious disease transmission has had an increasing influence on the theory and practice of disease management and control [

Nipah virus, a member of the genus Henipavirus, a new class of virus in the Paramyxoviridae family, has drawn attention as an emerging zoonotic virus in south-east and south-Asian region [

This paper deals with application of optimal control to a dynamic model of Nipah Virus (NiV) infections and its possible control and preventive strategy with the help of optimal control technique. Our aim is to minimize the total number of infectious individuals and the cost which is related for creating awareness and treatment.

Nipah virus infection is a zoonotic virus and transmitted first from animal to human. Once it has been transmitted to human, then it continues to be transmitted through human to human (H2H) by the close contact of infected individuals due to its highly infectivity [

We consider the following system of non-linear differential equation, is a type of standard SIR disease model, to describe the dynamics of Nipah Virus (NiV) infections in the community.

with initial conditions,

and where, the parameter

Since there is no proper vaccination program or appropriate drugs for NiV infections, so in the model we introduce two control strategies, namely, creating awareness (u_{1}) among the community about the risky areas before outbreak of the disease and the treatment (u_{2}). Here the control

Now the NiV model with two control strategies is given below:

with initial conditions,

Here our main objective is to minimize the total number of infected individuals and to reduce the cost which is needed for creating awareness and treatment on a specified time interval. For the fulfillment of our purpose, we work with the following objective function which is similar as [

where, B_{1} and B_{2} are weight parameters that help to balance the corresponding costs. We define the control set as follows:

In the objective function, A_{1}I represents the total number of infected individuals,

Adding first three equations of the system (3) we get,

On integrating we get,

So we have

From the fourth equation of (3) we have

and then

So, finally we get

Since

By proving the following theorem we can establish the existence of the objective functional:

Theorem 1. Consider the control problem with system (3). Then there exists optimal controls

Proof: To use an existence result in [

1) The set of controls and corresponding state variables is non-empty.

2) The control set U is convex and closed.

3) The right-hand side of the state system is bounded by a linear function in the state and control variables.

4) The integrand of the objective functional is convex on U and is bounded below by

To prove the above theorem we need to use the following theorem 2 and 3.

Theorem 2. Let each of the functions

be continuous in a region

that also satisfies the initial conditions

Theorem 3. Let

Now with the help of the above two theorems we prove the four conditions of theorem (1).

Proof of theorem 1: To use an existence result in [

1) The set of controls and corresponding state variables is non-empty.

2) The control set U is convex and closed.

3) The right-hand side of the state system is bounded by a linear function in the state and control variables.

4) The integrand of the objective functional is convex on U and is bounded below by

Proof of 1): Let us consider,

where, F_{1}, F_{2}, F_{3} and F_{4} buildup the right hand side of the system (3). Let _{1}, C_{2}. F_{1}, F_{2}, F_{3} and F_{4} must be linear and they are also continuous everywhere. Moreover, the partial derivatives of F_{1}, F_{2}, F_{3} and F_{4} with respect to all state are constants and they are continuous everywhere.

So by following the theorem 3, we can say that there exists an unique solution

Proof of 2): By definition, U is closed. Take any controls

Additionally, observe that

Hence,

Proof of 3):

We consider,

The state system is given below:

Now we rewrite the system in matrix form:

where,

and

which gives a linear function of the controls u_{1} and u_{2} defined by time and state variables. Then we can find out the bound of the right hand side. It is noted that all parameters are constant and greater than or equal to zero. Therefore we can write,

where

Proof of 4):

For the proof of the condition 4) we use the result in [_{1} is satisfied. The control variables

Now we have to prove that

Here,

where,

In order to derive the necessary condition for the optimal control, we use pontryagin’s maximum principle [_{1} and u_{2}. In the objective function, the value A is the balancing parameter, B_{1} and B_{2} are the weight parameters balancing the cost. Here we can see from the system (3) that R appears only in the recovered class. So, when we build up the optimality system, we will ignore the recovered class.

By using pantraygin’s Maximum principle we first derive the Hamiltonian which is given below

where, λ_{S}, λ_{I}, λ_{N} are the associated adjoints for the state S, I, N respectively. By differentiating the Hamiltonian (H) with respect to each state variable, we find the differential equation for the associated adjoint. Hence, the adjoint system is,

with the final conditions,

So by differentiating the Hamiltonian with respect to two controls u_{1} and u_{2} we obtain:

and

State equations:

with initial conditions,

Adjoint equations:

Transversality equations:

Characterization of the optimal control

and

In compact notion we can write,

and

Numerical solutions to the optimal system are executed using MATLAB. The considered two controls (u_{1}, u_{2}) depend on the adjoints λ_{S}, λ_{I} and λ_{N} of the state variables S, I and N respectively. We simulate the model without control and with control and then we compare the results. We considered the numerical value of the controls u_{1} and u_{2} in between zero(0) and one(1) as they are not 100 percent effective. We also monitored the effectiveness of the weight parameter to see how the control is related to weight function. In this simulation we assumed the initial values of S, I and N as proportions instead of whole numbers.

The parameter values used in the simulations are presented in the following

On the other hand for the higher rate of a (where awareness does not work, u_{1} = 0 and the treatment u_{2} works for a short period of time) there is a sharp decrease of infection due to death resulting the existence of fewer recovered people.

On the other hand, for the very high contact rate (b = 2), which resulted a severe disease burden, the controls work for a longer period of time to reduce the disease burden.

Variable | Description | Initial values |
---|---|---|

S_{0} | Initial susceptible individuals | 0.90 [assumed] |

I_{0} | Initial infected individuals | 0.05 [assumed] |

R_{0} | Initial recovered individuals | 0.05 [assumed] |

Parameters | Description | Initial values |

Birth rate | 0.03 [assumed] | |

Mortality rate | 0.002 [assumed] | |

Contact rate | 0.75 [ | |

Recovery rate | 0.005 [assumed] | |

Disease induced death rate | 0.01 [assumed] | |

A_{1} | Weight parameter | 10 [ |

B_{1} | Weight parameter | 1 [ |

B_{2} | Weight parameter | 2 [ |

T | Number of years | 6 [assumed] |

weight parameters (B_{1} = 0.2, B_{2} = 0.3) the infectious individuals decrease sharply for first few years (as the controls work at maximum level). It is also noticed that the infected individuals start to increase when the effectiveness of the controls start to decrease.

In the case of high weight parameter values (B_{1} = 2, B_{2} = 3) the high effectiveness of the controls are monitored and as a result there is a sharp reduction of infection during that effective level.

The important findings are given below:

・ A comparison between with and without control strategy is monitored. The effect of control parameters is very much notable for reducing the infected individuals to control the disease dynamics.

・ The controls need to be effective for longer period of time in case of high incidence.

・ The optimal control is much more effective to minimize the infected individuals (as a result recovered individuals will be maximized) and also to minimize the cost of the two control measures.

・ For low weight parameter values, the controls show their effectiveness at a maximum level.

・ From the simulations it is monitored that the optimal combination of treatment and creating awareness is very prominent for disease elimination.

The author, JS acknowledge, with thanks, the support in part of the National Science and Technology (NST), Dhaka. The authors are grateful to the reviewers for their constructive comments.

Jakia Sultana,Chandra N. Podder, (2016) Mathematical Analysis of Nipah Virus Infections Using Optimal Control Theory. Journal of Applied Mathematics and Physics,04,1099-1111. doi: 10.4236/jamp.2016.46114