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Complexity
Volume 2018, Article ID 4902142, 18 pages
https://doi.org/10.1155/2018/4902142
Research Article

Different Solution Strategies for Solving Epidemic Model in Imprecise Environment

1Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India
2Department of Mathematics, Netaji Subhash Engineering College, Techno City, Garia, Kolkata, West Bengal 700152, India
3Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore, West Bengal 721101, India
4Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
5Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran

Correspondence should be addressed to Ali Ahmadian; moc.liamg@iniessoh.naidamha

Received 6 February 2017; Revised 6 June 2017; Accepted 15 January 2018; Published 13 May 2018

Academic Editor: Carla Pinto

Copyright © 2018 Animesh Mahata et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the different solution strategy for solving epidemic model in different imprecise environment, that is, a Susceptible-Infected-Susceptible (SIS) model in imprecise environment. The imprecise parameter is also taken as fuzzy and interval environment. Three different solution procedures for solving governing fuzzy differential equation, that is, fuzzy differential inclusion method, extension principle method, and fuzzy derivative approaches, are considered. The interval differential equation is also solved. The numerical results are discussed for all approaches in different imprecise environment.

1. Introduction

1.1. Modeling with Impreciseness

The aim of mathematical modeling is to imitate some real world problems as far as possible. The presence of imprecise variable and parameters in practical problems in the field of biomathematical modeling became a new area of research in uncertainty modeling. So, the solution procedure of such problems is very important. If the solution of said problems with uncertainty is developed, then, many real life models in different fields with imprecise variable can be formulated and solved easily and accurately.

1.2. Fuzzy Set Theory and Differential Equation

Differential equations may arise in the mathematical modeling of real world problems. But when the impreciseness comes to it, the behavior of the differential equation is altered. The solution procedures are taken in different way. In this paper we take two types of imprecise environments, fuzzy and interval, and find their exact solution. In 1965, Zadeh [1] published the first of his papers on the new theory of Fuzzy Sets and Systems. After that Chang and Zadeh [2] introduced the concept of fuzzy numbers. In the last few years researchers have been giving their great contribution on the topic of fuzzy number research [35]. As for the application of the fuzzy set theory applied in fuzzy equation [6], fuzzy differential equation [7], fuzzy integrodifferential equation [810], fuzzy integral equation [11], and so on were developed.

1.3. Different Approaches for Solving Fuzzy Differential Equation

The application of differential equations has been widely explored in various fields like engineering, economics, biology, and physics. For constructing different types of problems in real life situation the fuzzy set theory plays an important role. The applicability of nonsharp or imprecise concept is very useful for exploring different sectors for its applicability. A differential equation can be called fuzzy differential equation if only the coefficient (or coefficients) of the differential equation is fuzzy valued number, only the initial value (or values) or boundary value (or values) is fuzzy valued number, the forcing term is fuzzy valued function, and all the conditions , , and or their combination is present on the differential equation.

There exist two types of strategies for solving the FDEs, which are as follows:(a)Zadeh’s extension principle method.(b)Differential inclusion method.(c)Approach using derivative of fuzzy valued functions.(d)Approach using fuzzy bunch of real valued functions instead of fuzzy valued functions.

Now we look on some different procedure and concepts of derivation in Table 1.

Table 1

There exist different numerical techniques [3436] for solving the fuzzy differential equation. The techniques are not fully similar to any differential equation solving techniques.

In this paper we only study the first three approaches.

1.4. Interval Differential Equation

An interval number is itself an imprecise parameter. Because the value is not a crisps number, the value lies between two crisp numbers. When we take any parameter, may be coefficients or initial condition or both, of a differential equation then the interval differential equation comes. The basic behaviors of that number are different from a crisp number. Hence, the calculus of those types numbers valued functions is different. So we need to study the differential equation in these environments. From the time that Moore [37] and Markov [38] as the pioneers introduced the interval analysis and related notions, several monographs and papers were devoted to connect the fuzzy analysis and interval analysis [39], but, the later one was not well-realized and applicable to model dynamical systems. After introducing generalized Hukuhara differentiability, different perspectives, which leaded to nice schemes and strategies to achieve the solutions, were discussed in the literature [4043]. Lupulescu in [44] developed the notions of RL- and Caputo-types derivatives for interval-valued functions. Salahshour et al. [45, 46] proposed a nonsingular kernel and conformable fractional derivative for interval differential equations of fractional order. Recently interval differential equation is studied by da Costa et al. [47] and Gasilov and Emrah Amrahov [48].

1.5. Work Done Using Fuzzy Differential Equation and Interval Differential Equation on Biomathematical Problem

Fuzzy differential equation and biomathematics are not new topics. A lot of research was done in this field. For instance, check [4968]. Many authors consider interval parameter with differential equation in biomathematical model. For presence of interval parameter the equation becomes interval differential equation. Using the property of interval number they solve the concerned model and discuss the behavior. Pal and Mahapatra [62] consider a bioeconomic modeling of two-prey and one-predator fishery model with optimal harvesting policy through hybridization approach in interval environment. Similarly, optimal harvesting of prey–predator system with interval biological parameters is discussed in [63]. Sharma and Samanta consider optimal harvesting of a two species competition model with imprecise biological parameters in [69]. Although Barros et al. [70] studied SIS model in fuzzy environment using fuzzy differential inclusion still we can study the model in different environments by different approaches.

1.6. Motivation

Impreciseness comes in every model for biological system. The necessity for taking some parameter as imprecise in a model is an important topic today. There are so many works done on biological model with imprecise data. Sometimes parameters are taken as fuzzy and sometimes it is an interval. Our main aim is modeled as a biological problem associated with differential equation with some imprecise parameters. Thus fuzzy differential equation and imprecise differential equation are important. Now we can concentrate some previous works on biological modeling in imprecise environments:

1.7. Novelties

Although some developments are done, some new and interesting research works have been done by ourselves, which are mentioned as follows:(i)SIS model is studied in imprecise environment.(ii)The fuzzy and interval environments are taken for analyses in the model.(iii)The governing fuzzy differential equation is solved by three approaches: fuzzy differential inclusion, extension principal, and fuzzy derivative approaches.(iv)The SIS model is solved by reducing the dimension of the model for fuzzy cases. For these reasons we use completely correlated fuzzy number.(v)Numerical examples are taken for showing the comparative view of different approaches.

Moreover, we can say all developments can help for the researchers who are engaged with uncertainty modeling, differential equation, and biological system if fuzzy parameters are assumed in the models. One can model and find the solution on any biological model with fuzzy and differential equation by the same approaches.

2. Basic Definitions

2.1. Definition

Definition 1 (fuzzy set). Let be a fuzzy set which is defined by a pair , where is a nonempty universal set andFor each , is the grade of membership function of .

Definition 2 (fuzzy number in trapezoidal form). A fuzzy number in trapezoidal form represented by three points like as and the presentation can be illustrated as membership function as

Definition 3 (-cut of a fuzzy set). The -cut of is given by

Definition 4 (correlated fuzzy number: [71]). Let and are two fuzzy sets whose membership function is written as follows: and . Then there exist with such that their joint possibility distribution is given bywhere is the characteristic function of the line .
In this case we have , where , , for any . Moreover if ,

Definition 5 (correlated trapezoidal fuzzy number). Two trapezoidal fuzzy numbers and are said to be correlated if , where is arbitrary operation and is constant.

Example 6. Let a trapezoidal fuzzy number be like . Now we have to find another trapezoidal fuzzy number that is correlated to .
Let be of the form .
So clearly we have
So, , , , .
So we can write , but here .
We can write it in modified form as .

Note 7 (use of correlated fuzzy number). There can be a basic question arising here, which is why we take correlated fuzzy variables. Fuzzy number can be employed and applied in various fields for various models. Sometimes for simplification of a model, we give a transformation so that the operation between two variables becomes unit. Now, if the initial condition or the solution is defined as a fuzzy parameter then the certain operation on this quantity is obviously a unit number. Otherwise, the importance of using a correlated fuzzy number is to take the data in fewer amounts, which can be very helpful for calculation.

Definition 8 (strong and weak solution of fuzzy differential equation). Consider the first order fuzzy differential equation with . Here or (and) is fuzzy number(s).
Let the solution (the solution comes from any method) of the above FDE be and its -cut be .
If then is called strong solution; otherwise is called weak solution and in that case the -cut of the solution is given by

Definition 9 (interval number). An interval number is represented by closed interval and defined by , where is the set of real numbers and and are the left and right boundary of the interval number, respectively.

3. Method for Solving Fuzzy Differential Equation

Let us consider the differential equationwhere is constant, is initial condition, and is the function which may be linear or nonlinear.

The differential equation (7) can be fuzzy differential equation if(i), that is, initial condition, is fuzzy number.(ii), that is, coefficient, is a fuzzy number.(iii) and , that is, initial condition and coefficient, are both fuzzy numbers.

3.1. Differential Inclusion Method for Solving Fuzzy Differential Equation

There are the papers where the concept of fuzzy differential equations is understood as the family of differential inclusions. For details see Agarwal et al. [72, 73], Diamond [74, 75], Laksmikantham et al. [76], and Lakshmikantham and Tolstonogov [77]. This new approach allowed considering some interesting aspects of fuzzy differential equations such as periodicity, Lyapunov stability, regularity of solution sets, attraction, and variation of constants formula (see [74, 75, 78, 79]). Also the numerical methods for FDEs have been developed in Hüllermeier [13, 80] and Ma et al. [81].

Let us assume the following differential inclusion is of the form with .

is a set valued function and (here is space of fuzzy numbered valued functions). We have to solve of (8) with provided:(a)The function is absolutely continuous in .(b)The function satisfies (8) for .

Now we denote the attainable set at time which is subset of associated with the problem (8) defined by which is solution of (8).

In fuzzy environment dynamical system the problem (8) can be formed as where is a fuzzy set valued function and .

According to Hüllermeier [13] the fuzzy initial value problem can be formed as family of differential inclusion given aswhere are the α-cuts of fuzzy set

Here the attainable sets related to the problem (10) can be defined by which is a solution of (10) in .

Hence there is fuzzy interval which is a fuzzy solution of (10) via differential inclusion if for all the collection of α-cuts satisfies the condition of the following theorem.

Theorem 10 (see [71]). Let be family of sets satisfying the following:(i) is a compact and convex interval, for all ;(ii)  for ;(iii)  for any nondecreasing sequence in .

Then there is a unique fuzzy interval such that . Conversely, the α-cuts sets for any satisfy these conditions.

Therefore, we have the solution of (9) if it is a solution of (10).

Theorem 11 (see [71]). Suppose are completely correlated fuzzy numbers; let G be their joint possibility distribution and be a continuous function; then ).

Theorem 12 (see [71]). For all ( there exists a unique solution to (10) in . Then the solution of the problem (7) via extension principle when and are noninteractive, and when and are completely correlated satisfies the following relation , for all , where , meaning is the joint possibility distribution of the noninteractive fuzzy numbers .

3.2. Extension Principle for Solving Fuzzy Differential Equation

Extension principle is a method by which we can easily find the solution of a fuzzy differential equation. Some researchers considered this method to find the solution of fuzzy differential equations [8284].

Definition 13 (extension principle on fuzzy sets). Suppose that we have some usual sets and choose some fuzzy sets .
The extension principle for fuzzy sets states that if such that ,and for every , is defined in the following wayfor every .

Example 14. Let be a fuzzy set where membership function is written as Let us choose a function .

Now by Zadeh’s extension principle, can be determined and its membership function is written as

Method 15 (solution of fuzzy differential equation using extension principle). Let us consider the fuzzy initial value problem (FIVP)If we denoteBy using the extension principle we have the membership functionThe result is a fuzzy function.

And

3.3. Fuzzy Derivative and Solution of Fuzzy Differential Equation by Fuzzy Derivative Approach

Bede and Gal [85] presented a concept of generalized Hukuhara differentiability of fuzzy valued map-pings, which permits them to obtain the solutions of FDEs with a diminishing diameter of solutions values. This was followed up in the literature [8591]. This comprehensive definition allows us to resolve the disadvantages of the previous fuzzy derivatives. Indeed, the strongly generalized derivative is defined for a larger class of fuzzy number valued functions in the case of the Hukuhara derivative.

Before going to the fuzzy differential equation approach we first know the following definition.

Definition 16 (generalized Hukuhara difference). The generalized Hukuhara difference of two fuzzy numbers is defined as follows:Consider ; then and .

Here the parametric representation of a fuzzy valued function is expressed by

Definition 17 (generalized Hukuhara derivative on a fuzzy function). The generalized Hukuhara derivative of a fuzzy valued function at is defined as If satisfying (21) exists, we say that is generalized Hukuhara differentiable at .
Also we say that is (i)-gH differentiable at if and is (ii)-gH differentiable at if

Method 18 (solution of fuzzy differential equation using fuzzy differential equation approach). Consider the fuzzy differential equation taking in (15).
We have the following two cases.

Case 1. If we consider in the first from (i), then we have to solve the following system of ODEs:

Case 2. If we consider in the first from (ii), then we have to solve the following system of ODEs:In both cases, we should ensure that the solution is valid level sets of a fuzzy number valued function and are valid level sets of a fuzzy valued function.

4. Model Formulation on Epidemic

There are so many mathematical models in biology; SIS model is an important model of them. In a given species population at time , let be the number of susceptible, which means the number of those who can be infected, and be the number of infected persons in the species population. In this model, a susceptible species can become infected at a rate proportional to and an infected species can recover and become susceptible again at a rate of so that the model can be formulated as follows:where   at is the initial condition.

Here a susceptible can become infected at rate proportional of SI and on infected can recover and become susceptible again at a rate proportional to .

(total number of population).

Now taking , , the model can be written aswhere with initial condition .

Note 19 (dimension less of a model). Sometimes for a mathematical model, it is critical to find the dynamical behavior. However, the dependent variables in the model are connected with another dependent variable, which makes the finding of the behavior complicated. In this regard, there is some criterion in which we can eliminate the conditions and make the model more simple and which is very easy to solve. According to these circumstances, we reduce the dimension of the above model.

The crisp solution of the above system of equations is written in two different cases.

Case 1 (when ). In this case the solution can be written as

Case 2 (when ). In this case the solution can be written as

Note 20. May be someone will ask why do we take SIS model for comparing different solution strategy for solving in uncertain environment? Basically we take the particular SIS model and apply the different techniques in uncertain environment. Once one can be familiar with it, anyone can take one of the strategies which best fits their model.

5. Solution of the above SIS Model in Fuzzy Environment by Different Strategy

5.1. Solution of Fuzzy SIS Model via Differential Inclusion

The solution of the problem (30) using differential inclusion is obtained from the solution of the auxiliary where .

The attainable sets of the problem of (31) can be written as

, solution of (31).

Case 1 (when ).

Case 2 (when ).

5.2. Solution of Fuzzy SIS Model by Extension Principle Method

Let and be the solution by extension principle method.

Now different cases arise.

Case 1 (when ). In this case the solution can be written asThe solution depends on the function whether it is increasing or decreasing. The solution can be written as in Table 2.
Here, So, it depends upon , whether it is negative or positive. If we take thenand also The solution depends on the function whether it is increasing or decreasing. The solution can be written as in Table 3.
Here,So, the solution is given by

Table 2
Table 3

Case 2 (when ). In this case the solution can be written asThe solution depends on the function whether it is increasing or decreasing. The solution can be written as in Table 4.
Here,Hence the solution is and also The solution depends on the function whether it is increasing or decreasing. The solution can be written as in Table 5.
Here,Hence the solution is

Table 4
Table 5
5.3. Solution of Fuzzy SIS Model by Fuzzy Differential Equation Approach

Let and be the solution using generalized Hukuhara derivative approach.

Now different cases can be found as follows.

Case 1 ( and is (i)-gH differentiable). In this case the differential equation transforms to with initial conditions , , , and .

Case 2 ( is (i)-gH and is (ii)-gH differentiable). In this case the differential equation transforms to with initial conditions , , , and .

Case 3 ( is (ii)-gH and is (i)-gH differentiable). In this case the differential equation transforms to with initial conditions , , , and .

Case 4 ( and is (ii)-gH differentiable). In this case the differential equation transforms to with initial condition , , , and

6. Modeling SIS in Interval Environment

The problem in interval environment iswhere with initial condition at , .

We get the solution for two cases as follows.

Case 1 (when ). The solution is written as

Case 2 (when ). The solution is written as

7. Numerical Examples

7.1. Numerical Example on Fuzzy Cases

Find the solution after when and , when and .

Solution by differential inclusion and extension principle and fuzzy differential equation is given by

7.1.1. Solution by Differential Inclusion

Case 1 (when ). The boundary of the solution is given by

Remarks 21. From Figure 1 and Table 6 it shows that is decreasing and is increasing whereas is increasing and is decreasing. The figure demonstrates the boundary of the solution. The solution for gives the natural weak solution but gives the natural strong solution.

Table 6: Solution boundary for .
Figure 1: Solution boundary for .

Case 2 (when ). The boundary of the solutions is

Remarks 22. From Figure 2 and Table 7 it shows that is decreasing and is increasing whereas is increasing and is decreasing. The figure demonstrates the boundary of the solution. The solution for gives the natural weak solution but gives the natural strong solution.

Table 7: Solutions boundary for .
Figure 2: Solutions boundary for .
7.2. Solution by Extension Principle

Case 1 (when ). Here the solutions are given by

Remarks 23. From Figure 3 and Table 8 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural strong solution but gives the natural strong solution.

Table 8: Solution for .
Figure 3: Solution for .

Case 2 (when ). The solutions are given by

Remarks 24. From Figure 4 and Table 9 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural strong solution and gives the natural strong solution.

Table 9: Solution for .
Figure 4: Solution for .
7.3. Solution by Fuzzy Differential Equation Approach

Now the solutions for different cases are given by the following.

Case 1 ( and is (i)-gH differentiable).

Remarks 25. From Figure 5 and Table 10 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural strong solution and gives the natural strong solution.

Table 10: Solutions for .
Figure 5: Figure for .

Case 2 ( is (i)-gH and is (ii)-gH differentiable).

Remarks 26. From Figure 6 and Table 11 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural weak solution but gives the natural strong solution.

Table 11: Solution for .
Figure 6: Figure for .

Case 3 ( is (ii)-gH and is (i)-gH differentiable).

Remarks 27. From Figure 7 and Table 12 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural weak solution but gives the natural strong solution.

Table 12: Solution for .
Figure 7: Solution for .

Case 4 ( and is (ii)-gH differentiable).

Remarks 28. From Figure 8 and Table 13 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural strong solution and gives the natural strong solution.

Table 13: Solution for .
Figure 8: Figure for .
7.4. Numerical Example on Interval Cases

Find the solution after when and , when and .

Case 1 (when ).

Case 2 (when ).

Remarks 29. From Figures 9 and 10 and Tables 14 and 15 it shows that is increasing and is decreasing whereas is increasing and is decreasing. The figure demonstrates the solution of the problem. The solution for gives the natural strong solution but gives the natural strong solution.