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In this paper, we build an epidemiological model to investigate the dynamics of th e spread of dengue fever in human population. We apply optimal control theory via the Pontryagins Minimum Principle together with the Runge-Kutta solution technique to a “ simple ” SEIRS disease model. Controls representing education and drug therapy treatment are incorporated to reduce the latently infected and actively infected individual populations. The overall thrust is the minimization of the spread of the disease in a population by adopting an optimization technique as a guideline.

Dengue fever is a painful, debilitating mosquito-borne disease caused by one of four closely related dengue viruses (Noorani [

A very important aspect of the strategy related to dengue fever spreading is quick and effective action (Rodriguez and Monteiro [

The discovery of antibiotics and vaccines heralded a new hope in disease control. Despite this, new challenges resulting from factors such as drug-resistance have also emerged. Sometimes this led to the emergence of more virulent forms of previously eradicated diseases. For example resistances to such diseases as malaria, tuberculosis, dengue and yellow fever have emerged and, as a result of climate changes, they have been spreading into new regions (Helena &Teresa [

Quantitative methods are often applied to achieve optimization of investments in the control of a disease. This is necessary in order to obtain maximum benefits from a fixed amount of financial resources. In this case, our efforts will be directed towards the dynamics of the Aedes mosquito vector as well as some management protocols aimed at controlling or alleviating the spread of the disease. Such management principles involving the termination of the reproduction cycle of mosquitoes by avoiding the accumulation of still water in pot-holes and ditches especially after a heavy downpour, are of vital importance as well as educating the local population on issues related to basic hygiene through the television (TV) and radio.

Model Assumptions and Mathematical Formulation1) The population is uniform and mixes homogeneously. The total population size, N ( t ) = S ( t ) + E ( t ) + I ( t ) + R ( t ) at any time t > 0, where N stands for the total population, E for exposed I for infected, S for susceptible and R for recovered.

2) The natural birth rate b and death rates μ_{n} are assumed to be different.

3) Each individual in the population is considered as having an equal probability of contacting the disease with a contact rate β.

4) An infected individual makes contact and is able to transmit the disease with βN per unit time, that is, the contact rate is proportional to the total population size.

5) The fraction of contacts by an infected with a susceptible is S/N. Therefore the number of new infections in unit time per infective becomes (βN)(S/N). This rate is called an infection rate. This gives the rate of new infections or those leaving the susceptible category as (βN(S/N)I = βSI, which is called an incidence of the disease. This type of incidence is called bilinear incidence i.e., proportional to the product of the number of infective individuals and the number of susceptible individuals.

6) The number of infected individuals move from the exposed compartment per unit time is δE at time t.

7) The exposed E move from their compartment to I-compartment at a constant rate δ, so that 1/δ is the mean latent period.

8) The infectious I move from their compartment to R-compartment at a constant rate γ, so that 1/γ is the mean infectious period.

9) The rate of susceptible, exposed, infected and recovered individual removed from each compartments through natural death and disease induced death are μ_{n}S, μ_{n}E, μ_{n}I, μ_{n}R and μ_{d}I respectively.

10) The recovered individual R move from their compartment to susceptible(S)-compartment at a constant rate α,

11) The differential equations from these assumptions can be represented by a system of ordinary differential equations:

d S d t = b − β I S − μ n S − u S + α R d E d t = β I S − μ n E − δ E d I d t = δ E − ( μ n + μ d + γ ) I d R d t = γ I − ( μ n + α ) R + u S (1)

An optimal control strategy aimed at minimizing the objective (cost) functional J of the cost of education for a susceptible population is described by the following differential equations:

d S d t = b − β I S − μ n S − u S + α R d E d t = β I S − μ n E − δ E d I d t = δ E − ( μ n + μ d + γ ) I d R d t = γ I − ( μ n + α ) ) R + u S (2)

S ( 0 ) = S 0 ≥ 0 , E ( 0 ) = E 0 ≥ 0 , I ( 0 ) = I 0 ≥ 0 , R ( 0 ) = R 0 ≥ 0

Subject to: min J ( u ) = ∫ 0 T ( A I + 1 2 B u 2 ) d t

where, A is balancing cost factor due to the infective and B is the weight on the cost of education.

Based on the above assumptions, an optimal control problem is formulated by incorporating one of the intervention strategies into our basic mathematical model (see Equations (1) and (2)).

・ u(t) is the control which represents the education ratio of susceptible individuals being educated per unit of time with bounds between 0 and 1.

・ The inflow of population to the susceptible class is obtained, by combining assumptions 2, 5, 9, 10 and control (education).

・ A number of individuals leaves S and enter E, at the same time, a fraction of exposed E moves to infectious group I with a latent rate δ. δE represents an individual’s move from exposed to infectious. Some of the exposed group die through natural death rate μ_{n}, μ_{n}E represents movement from exposed to death.

・ Some individuals leave E and enter into the infected individuals I with latent rate δ.

・ A part of the population leaves I and enter the recovered group with recovery rate γ. Combination of assumptions 2, 5, 9, 10 in addition to the control u, gives the rate of recovered.

Antiviral drugs are known to be very helpful in decreasing or preventing disease symptoms at the first sign of a dengue outbreak even when there is no evidence of fever. Before we incorporate drug therapy as part of our treatment protocol and control measures, we will deal with how the application of drug therapy affects some of the model compartments.

・ Consider control variables u_{1}, u_{1}E as representing an individual’s move from exposed to recovered. The exposed populations change per unit of time becomes,

d E d t = β I S − μ n E − δ E − u 1 E (3)

・ In addition, a number of individuals leaves the infected group I and enter the recovered group with recovery rate γ. A number of individuals also leaves the susceptible and exposed groups S and E to enter the recovered group with controls u and u_{1} respectively. This gives rate of recovered as:

d R d t = γ I − ( μ n + α ) R + u S + u 1 E (4)

The differential equation of the diagram for t ≥ 0 is given in a system of ordinary differential equation. Introducing the controls representing the education and drug therapy treatment the model of Equation (1) becomes

d S d t = b − β I S − μ n S − u S + α R d E d t = β I S − μ n E − δ E − u 1 E d I d t = δ E − ( μ n + μ d + γ ) I d R d t = γ I − ( μ n + α ) R + u S + u 1 E (5)

where, S(0), E(0), I(0), R(0) are the initial conditions. The definitions of above model parameters are listed in _{1}(t) are bounded, Lebesgue integrable functions(van den Driessche and Watmough [_{1}(t), represents the effort on drug therapy treatment of latently infected individuals to reduce the number of individuals that may be infectious. While the control u(t) is the effort on education of susceptible individuals to increase the number of recovered individuals.

A is balancing cost factor due to the infective, B and B_{1} are the weight on the cost of education and drug respectively.

The control problem involves a number of individuals with latent and active dengue fever infections. The cost of applying education and drug therapy treatment controls u(t) and u_{1}(t) are minimized subject to the differential equations (6). The performance specification involves the numbers of individuals with latent and susceptible components respectively, as well as the cost for applying education control (u) and drug therapy treatment control (u_{1}). The objective functional is defined as:

J ( u , u 1 ) = min [ u , u 1 ] ∫ 0 T ( A I + 1 2 ( B u 2 + B 1 u 1 2 ) ) d t (6)

where T is the final time and the coefficients, A, B, B_{1} are balancing cost factors reflecting the importance of the three parts of the objective function. We need to

Symbols | Description | Value | Reference |
---|---|---|---|

μ_{n} β b | Natural death rate Contact rate Average birth rate | 1/(71 * 365) per year 0.375 per year 1/(71 * 365) per year | Helena [ |

μ_{d} | Disease related death rate | 1/11 per year | Assumption |

δ γ | Exposed rate Recovery rate | 1/4 per year 1/3 per year | Helena [ |

α | Recovering rate of remove disease to Susceptible | 0.00008 per year | Assumption |

A B | Balancing cost factor due to the infective The weight on the cost of education | 100 0.04 | Esayas [ |

B_{1} | The weight on the cost of treatment | 0.06 | Assumption |

find an optimal control pair, u and u_{1}, such that

J ( u , u 1 ) = min J ( u , u 1 ) | u , u 1 ∈ U (7)

where, U = ( u ( t ) , u 1 ( t ) ) | ( u ( t ) , u 1 ( t ) ) measurable, a i ≤ ( u ( t ) , u 1 ( t ) ) ≤ b i , i = 1 , 2 , t ∈ [ 0 , T ] is the control set.

The necessary conditions that an optimal pair must satisfy come from the Pontryagins Maximum Principle (Helena [_{1}). First we formulate the Hamiltonian from the cost functional (6) and the governing dynamics (5) to obtain the optimality conditions. Pontryagin introduced the adjoint function to relate the differential equation to the objective functional. The necessary conditions needed to solve this OC problem, can be followed stepwise:

Step 1: Formulate the Hamiltonian for the problem and by applying Pontryagin’s principle to the Hamiltonian and find optimal controls u^{*}, u 1 ∗ with the corresponding solution S^{*}, E^{*}, I^{*} and R^{*} of equation (5).

Step 2: Write the adjoint differential equation, the optimality condition and transversality boundary condition (if necessary). Using the Hamiltonian to find the differential equation of the adjoint λ, and obtain the adjoint variables λ_{1}, λ_{2}, λ_{3} and λ_{4} that satisfy adjoint condition.

λ ′ i ( t ) = ∂ H ∂ x i , where i = 1 , 2 , 3 , 4

Adjoint Functionsλ ′ i = − ∂ H ∂ x ⇒ λ ′ = − ( F x + λ g x ) , adjoint condition (8)

λ ′ 1 = − ∂ H ∂ S ⇒ λ ′ 1 = λ 1 ( β I + μ n + u ) − λ 2 β I − λ 4 u

λ ′ 2 = − ∂ H ∂ E ⇒ λ ′ 2 = λ 2 ( μ n + δ ) − λ 3 δ

λ ′ 3 = − ∂ H ∂ I ⇒ λ ′ 3 = − A + λ 1 ( β S ) − λ 2 β S + λ 3 ( μ n + m u d + γ ) − λ 4 γ

λ ′ 4 = − ∂ H ∂ E ⇒ λ ′ 4 = − λ 1 α + λ 4 ( μ n + α )

with transversality conditions λ i ( T ) = 0 , i = 1 , 2 , 3 , 4 .

The optimality condition is given by,

δ H δ u = 0 at u = u * ⇒ F u + λ g u = 0 δ H δ u 1 = 0 at u 1 = u 1 ∗ ⇒ F u 1 + λ g u 1 = 0

Step 3: Solve for u^{*} and u 1 ∗ in terms of S^{*}, E^{*}, I^{*}, R^{*} and λ

δ H δ u = B u + S ( λ 4 − λ 1 ) = 0 (9)

In this way we obtain an expression for the OC:

δ H δ u = 0 at u = u * ⇒ F u + λ g u = 0

u * = S ( λ 1 − λ 4 ) B

δ H δ u 1 = B 1 u 1 + E ( λ 4 − λ 2 ) = 0 (10)

u 1 ∗ = E ( λ 2 − λ 4 ) B 1

Step 4: Solve the four differential equations for S^{*}, E^{*}, I^{*}, R^{*} and λ with boundary conditions, substituting u^{*} and u 1 ∗ in the differential equations with the expression for the optimal control from the previous step.

Step 5: After finding the optimal state and adjoint, solve for the optimal control.

We solve that system of differential equations for the optimal state and adjoint. The solution of the optimal control in problem terms of S^{*}, E^{*}, I^{*}, R^{*} and λ, represents the characterization of the optimal control (u^{*}). The state equations and the adjoint equations together with the characterization of the optimal control and the boundary conditions constitute the optimality system.

Remark 1: If the Hamiltonian is linear in the control variable u, it can be difficult to calculate u^{*} from the optimality equation, since δ H δ u would not contain u. Specific ways of solving these kind of problems can be found in Lenhart and John [

Backward-forward Sweep Method

From the model the optimal control problem becomes:

min J ( u ) = ∫ 0 T ( A I + 1 2 B u 2 ) d t

Subject to:

d S d t = b − β I S − μ n S − u S + α R d E d t = β I S − μ n E − δ E d I d t = δ E − ( μ n + μ d + γ ) I d R d t = γ I − ( μ n + α ) R + u S (11)

With initial value,

S ( 0 ) = S 0 ≥ 0 , E ( 0 ) = E 0 ≥ 0 , I ( 0 ) = I 0 ≥ 0 , R ( 0 ) = R 0 ≥ 0 and

min J ( u ) = ∫ 0 T ( A I + 1 2 ( B u 2 + B 1 u 1 2 ) ) d t

Subject to:

d S d t = b − β I S − μ n S − u S + α R d E d t = β I S − μ n E − δ E − u 1 E d I d t = δ E − ( μ n + μ d + γ ) I d R d t = γ I − ( μ n + α ) R + u S + u 1 E (12)

As previously indicated, any solution to the above optimal control problem must also satisfy

λ ′ i ( t ) = − ∂ H ∂ x i (13)

where, i = 1, 2, 3, 4, x_{1} = S, x_{2} = E, x_{3} = I, x_{4} = R

δ H δ u = 0 at u * (14)

δ H δ u 1 = 0 at u 1 ∗ (15)

The optimal controls are,

u * = { 0 if δ H δ u < 0 S ( λ 1 − λ 4 ) B if δ H δ u = 0 0.9 if δ H δ u > 0 (16)

u 1 ∗ = { 0 if δ H δ u 1 < 0 E ( λ 2 − λ 4 ) B 1 if δ H δ u 1 = 0 0.9 if δ H δ u 1 > 0 (17)

The optimality condition can usually be manipulated to find a representation of u^{∗} in terms of t, state variables and λ. If this representation is substituted back into the ODEs for the state variables and λ then the Equations (11) and (12) form a two-point boundary value problem. The Runge-Kutta method is then applied to solve initial value problems, and resolve the optimality system of the optimal control problem. This approach is generally referred to as the Forward-Backward Sweep method. Information about convergence and stability of this method can be found in (Lenhart & John [

Numerical solutions to the optimality system comprising the state Equation (5)and adjoint equations are carried out using MATLAB and using parameters in _{1} = 0.06, S(0) = 86.46%, E(0) =4.5%, I(0) = 9.042%, R(0) = 0%. The algorithm is the forward-backward scheme; starting with an initial guess for the optimal controls u and u_{1}, the state variables are then solved forward in time from the dynamics (5) using a Runge-Kutta method of the fourth order. Then those state variables and initial guess u and u_{1} are used to solve the adjoint equations backward in time with given final conditions (16) and (17) by employing a fourth order Runge-Kutta method. The controls u and u_{1} are updated and used to solve the state and then the adjoint system. This iterative process terminates when current state, adjoint, and control values converge sufficiently (Helena &Teresa [

With this strategy, education (u) is utilized in the disease control while the control on drug therapy treatment (u_{1}) is set to zero, with weight factors B_{1} = 0, A = 100, B = 0.04. For this strategy, we observed that the number of susceptible individuals is higher when education and drug therapy treatment are absent (

The control (u_{1}) on drug therapy treatment is utilized while the control on education(u) is set to zero, with weight factors A = 100, B = 0.04, B_{1} = 0.06. For this strategy, it can be observed in

education and with education and treatment during the first 5 weeks. It is obvious that the impact of education takes time to be felt or manifested in the dynamics. However there is a dramatic change in the dynamics after this period as the percentage of the exposed with education and treatment becomes significantly lower than for those with education alone. For the infected individuals in

With this strategy, the controls on education (u) and drug therapy treatment (u_{1}) are utilized, with weight factors A = 100, B = 0.04, B_{1} = 0.06.

exposed to treatment and education the more they are likely not to get infected.

The results displayed herein not only confirm the validity of the mathematical formulation derived but also illustrate how to optimally apply control measures involving treatment and education for the control of dengue fever. Utilizing education and drug therapy treatment lead to better disease control in the population than utilizing drug therapy treatment only. In addition, the application of only one form of control measure though it results in a delayed peak in the percentage of exposed and infected, is not as effective as using both controls. Thus control programs that specialize in an optimal application of multi-control measures can effectively reduce or alleviate the effects of dengue fever spread.

Further work should include other control variables like the effect of bio-immunology on the spread of dengue fever, the use of medicated mosquito nets, development and application of vaccines, creation of sterile mosquito males for the control of mosquito population etc.

The authors declare no conflicts of interest regarding the publication of this paper.

Onyejekwe, O.O., Tigabie, A., Ambachew, B. and Alemu, A. (2019) Application of Optimal Control to the Epidemiology of Dengue Fever Transmission. Journal of Applied Mathematics and Physics, 7, 148-165. https://doi.org/10.4236/jamp.2019.71013