The Solution of a Mathematical Model for Dengue Fever Transmission Using Differential Transformation Method

Differential Transformation Method (DTM) is a very effective tool for solving linear and non-linear ordinary differential equations. This paper uses DTM to solve the mathematical model for the dynamics of Dengue fever in a population. The graphical profiles for human population are obtained using Maple software. The solution profiles give the long term behavior of Dengue fever model which shows that treatment plays a vital role in reducing the disease burden in a population.


Introduction
Dengue fever is an infectious vector borne disease spreading in tropical and subtropical countries with more than 50 million dengue fever cases per year. It is transmitted to humans by the bite of infected aedes mosquitoes. The major vector, Aedes aegypti, thrives in tropical regions, mainly in urban areas, closely linked to human populations providing artificial water-holding containers as breeding sites. A second potential vector, Aedes Albopictus, resides in temperate regions (North America and Europe), where it may give rise to occasional dengue outbreaks. [1,2].
As part of awareness campaigns, different kinds of precautions have been suggested towards preventing mosquito's bite. Some of the precautions that can be taken are: to keep home, environment and surrounding clean, to remove all stagnant water and containers, to cover all containers properly to prevent dengue mosquito breeding there, to use mosquito repellents to avoid mosquito bite, to use mosquitoes net around bed while sleeping etc. [1]. Different studies have shown the importance of mathematical approaches in understanding dengue disease transmission and evaluating the effectiveness and/or costeffectiveness of control strategies [1].
In recent years, the field of public health has benefited tremendously from the use of mathematical models to study the spread of infectious disease. Most epidemiological models are represented using systems of non-linear ordinary differential equations [3]. Differential Transformation Method is a semi-analytical method of solving both linear and nonlinear system of ordinary differential equations (ODE) to obtain approximate series solutions. This method which was derived from the Taylor's series expansion has been used to solve problems in Mathematics and Physics [4], fractional differential-algebraic equations [5], fourth-order parabolic partial differential equations [6], fractional-order integrodifferential equations [7], differential equation [8] and problems in epidemic models [9].

Model Formulation
Two populations consisting of human and vector population will be considered in this work. The model sub-divides these populations into a number of mutually-exclusive compartments, as given below. The total population of human and vectors is divided into the following mutually exclusive epidemiological classes, namely, susceptible humans ( S H (t) ), humans with dengue in latent stage ( E 1 (t)), humans with dengue ( I 1 (t) ), humans treated of dengue ( R 1 (t) ), susceptible vectors ( S V (t) ), vectors with latent dengue ( E V (t) ), vectors with dengue ( I V (t) ).

Derivation of Model Equations
Let N H (t) and N V (t) denote the total number of humans and vectors at time t, respectively. Hence, we have that, Susceptible humans are recruited at a rate H while the susceptible vectors are recruited at a rate V.
Susceptible humans contract dengue at a rate where η v < 1 , this accounts for the relative infectiousness of vectors with latent dengue I v compared to vectors in the I v class. Susceptible vectors acquire dengue infection from infected humans at a rate where η A < η B this accounts for the relative infectiousness of humans with latent dengue E 1 compared to humans in the I 1 class [10]. The model equations for dengue disease transmission incorporating treatment as a control measure are given below: An arbitrary function f (t) can be expanded in Taylor series about a point t = 0 as The differential transformation of f (t) is defined as Then the inverse differential transform is If y (t) and g (t) are two uncorrelated functions with t where Y (k) and G (k) are the transformed functions corresponding to y (t) and g (t) then, the fundamental mathematical operations performed by differential transform can be proved easily and are listed as follows Table 1: The Fundamental Mathematical Operations by Differential Transformation Method (DTM). Source: [11,12] Transformed Function Original Function Analytical solution of the model equations using differential transformation method (DTM) In this section, the Differential Transformation Method (DTM) is employed to solve the system of non-linear differential equations which describe our model for Dengue fever.
Let the model equation be a function q (t), q (t) can be expanded in Taylor series about a point t = 0 as where, The differential transformation of q (t) is defined as Then the inverse differential transform is Using the fundamental operations of differential transformation method in Table 1, we obtain the following recurrence relation of equation (1) as With the initial conditions Then the closed form of the solution where k = 0, 1, 2, 3 can be written as Then the closed form of the solution where k = 0, 1, 2, 3 can be written as Then the closed form of the solution where k = 0, 1, 2, 3 can be written as

Numerical Simulation and Graphical Representation of the Solutions of the Model Equations
The numerical simulation which illustrates the analytical solution of the Model is demonstrated using Maple software. This is achieved by using some set of parameter values given in the Table 2

Discussion of results
The Figures 1 to 6 give the numerical profiles of the solutions (17), (19) and (21) using DTM. Figure 1 shows increase in the population of susceptible individuals while Figure 2 indicates a decreasing population of the exposed owing to the progression out of exposed class to class of human with Dengue. Figure 3 implies a decrease in the population of human infected with dengue which later increases. Figure 4 implies the treated  Figure 5 indicates that increasing the treatment rate of individuals in the infected population leads to a reduction in the number of infected individuals which is due to progression into the treated population while Figure 6 shows that increasing the treatment rate of individuals in the infected population leads to a corresponding increase in the treated population.

Conclusion
We formulated a compartmental model to investigate the dynamics of dengue fever in a population with treatment as a control measure. Differential Transform Method (DTM) was employed to obtain the series solution of the model. Numerical simulations were carried out to determine the long term behavior of Dengue fever model which shows that treatment plays a vital role in reducing the disease burden in a population. The results of the simulations were displayed graphically.