Different Solution Strategies for Solving Epidemic Model in Imprecise Environment

. 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.


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.

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 [3][4][5].As for the application of the fuzzy set theory applied in fuzzy equation [6], fuzzy differential equation [7], fuzzy integrodifferential equation [8][9][10], fuzzy integral equation [11], and so on were developed.

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 (1) only the coefficient (or coefficients) of the differential equation is fuzzy valued number, (2) only the initial value (or values) or boundary value (or values) is fuzzy valued number, (3) the forcing term is fuzzy valued function, and (4) all the conditions (1), (2), and (3) 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.
(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.
There exist different numerical techniques [34][35][36] 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.

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 wellrealized 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 [40][41][42][43].Lupulescu in [44] developed the notions of RLand 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].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.

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

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 ( 0 ) =  0 .Here  or (and)  0 is fuzzy number(s).

Method for Solving Fuzzy Differential Equation
Let us consider the differential equation where  is constant,  0 is initial condition, and (, , ()) is the function which may be linear or nonlinear.The differential equation ( 7) can be fuzzy differential equation if (iii)  0 and , that is, initial condition and coefficient, are both fuzzy numbers.

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 (0) =  0 ∈  0 . : [0, ] ×   →   is a set valued function and  0 ∈   (here   is space of fuzzy numbered valued functions).We have to solve (⋅,  0 ) of ( 8) with  0 ∈  0 provided: In fuzzy environment dynamical system the problem (8) can be formed as where g : [0, ] ×   →   is a fuzzy set valued function and  0 ∈   .According to Hüllermeier [13] the fuzzy initial value problem can be formed as family of differential inclusion given as where (,   ()) are the -cuts of fuzzy set g(, ()).

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 [82][83][84].
Definition 13 (extension principle on fuzzy sets).Suppose that we have some usual sets   and choose some fuzzy sets Ã ∈ (  ).
The extension principle for fuzzy sets states that if ( Ã) ∈ (  ) such that  ∈   , and for every B ∈ (),  −1 ( B) is defined in the following way for every  ∈   .
Example 14.Let Ã be a fuzzy set where membership function is written as Let us choose a function () = 2 + 3.
Now by Zadeh's extension principle, ( Ã) 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 denote By using the extension principle we have the membership function The result (, ()) is a fuzzy function. And

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 [85][86][87][88][89][90][91].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 ,  ∈ R F is defined as follows: Consider Here the parametric representation of a fuzzy valued function  : [, ] → R F is expressed by Definition 17 (generalized Hukuhara derivative on a fuzzy function).The generalized Hukuhara derivative of a fuzzy valued function  : (, ) → R F at  0 is defined as If   ( 0 ) ∈ R F satisfying (21) exists, we say that  is generalized Hukuhara differentiable at  0 .Also we say that () is (i)-gH differentiable at  0 if and 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 [  1 (),   2 ()] is valid level sets of a fuzzy number valued function and [(/)(  1 ()), (/)(  2 ())] are valid level sets of a fuzzy valued function.

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 () =  0 () =  0 at  = 0 is the 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  =  −  ̸ = 0).In this case the solution can be written as Case 2 (when  =  −  = 0).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.
We get the solution for two cases as follows.
Remarks 25.From Figure 5 and Table 10 it shows that  1 (, ) is increasing and  2 (, ) is decreasing whereas  1 (, ) is  the natural weak solution but ĩ() gives the natural strong solution.
Remarks 27.From Figure 7 and

Conclusion
In this paper we study the different solution strategies for analyzing fuzzy differential equation and application in mathematical biology model, namely, SIS model, which is considered to be an important area of research in biological research.The approaches regarding fuzzy differential inclusion, extension principle, and fuzzy differential equation were applied to find the fuzzy solutions of the given model.The whole paper is concluded as follows: (i) Demonstrating SIS model with fuzzy numbers which enabled meeting the uncertain parameters as well, which is appreciatively helpful for the decision makers to investigate the situation in a more precise manner.
(ii) The different approaches having significant place in fuzzy calculus efficiently made it possible to obtain the fuzzy solution of the governing model by different methods.
(iii) The use of correlated fuzzy number in the said model is for finding the fuzzy solution.
Thus in the future we seek to apply these concepts to different types of differential equation models in fuzzy environments.
Definition 9 (interval number).An interval number  is represented by closed interval [  ,   ] and defined by  = [  ,   ] = { :   ≤  ≤   ,  ∈ R}, where R is the set of real numbers and   and   are the left and right boundary of the interval number, respectively.
Table 6 it shows that  1 (, ) is decreasing and  2 (, ) is increasingwhereas  1 (, ) is increasing and  2 (, ) is decreasing.The figure demonstrates the boundary of the solution.The solution for s() gives the natural weak solution but () gives the natural strong solution.Remarks 22. From Figure 2 and Table 7 it shows that  1 (, ) is decreasing and  2 (, ) is increasing whereas  1 (, ) is increasing and  2 (, ) is decreasing.The figure demonstrates the boundary of the solution.The solution for s() gives the natural weak solution but () gives the natural strong solution.Remarks 23.From Figure3and Table8it shows that  1 (, ) is increasing and  2 (, ) is decreasing whereas  1 (, ) is increasing and  2 (, ) is decreasing.The figure demonstrates the solution of the problem.The solution for s() gives the natural strong solution but () gives the natural strong solution.

Table 12
it shows that  1 (, ) is increasing and  2 (, ) is decreasing whereas  1 (, ) is increasing and  2 (, ) is decreasing.The figure demonstrates the solution of the problem.The solution for s() gives the natural weak solution but ĩ() gives the natural strong solution.

Table 12 :
Solution for  = 10.Remarks 28.From Figure8and Table13it shows that  1 (, ) is increasing and  2 (, ) is decreasing whereas  1 (, ) is increasing and  2 (, ) is decreasing.The figure demonstrates the solution of the problem.The solution for s() gives the natural strong solution and () gives the natural strong solution.