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.
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.
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 [
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
There exist two types of strategies for solving the FDEs, which are as follows: Zadeh’s extension principle method. Differential inclusion method. Approach using derivative of fuzzy valued functions. 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
Name of the theory | Some references | |
---|---|---|
Fuzzy differential equation | Fuzzy differential inclusion | Baidosov [ |
Hüllermeier [ | ||
Zadeh’s Extension principle | Oberguggenberger and Pittschmann [ | |
Approach using derivative of fuzzy valued functions | ||
Dubois-Prade derivative | Dubois and Prade [ | |
Puri-Ralescu derivative | Puri and Ralescu [ | |
Goetschel-Voxman derivative | Goetschel Jr. and Voxman [ | |
Friedman-Ming-Kandel derivative | Friedman et al. [ | |
Seikkala derivative | Seikkala [ | |
SGH derivative | Bede and Gal [ | |
Same-order and reverse-order derivative | Zhang and Wang [ | |
| Chalco-Cano et al. [ | |
gH-derivative | Stefanini and Bede [ | |
g-derivative | Bede and Stefanini [ | |
H2-derivative | Mazandarani and Najariyan [ | |
Interactive derivative | de Barros and Santo Pedro [ | |
gr-derivative | Mazandarani et al. [ | |
Approach using fuzzy bunch of real valued functions instead of fuzzy valued functions | Gasilov et al. [ |
There exist different numerical techniques [
In this paper we only study the first three approaches.
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 [
Fuzzy differential equation and biomathematics are not new topics. A lot of research was done in this field. For instance, check [
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:
Although some developments are done, some new and interesting research works have been done by ourselves, which are mentioned as follows: SIS model is studied in imprecise environment. The fuzzy and interval environments are taken for analyses in the model. The governing fuzzy differential equation is solved by three approaches: fuzzy differential inclusion, extension principal, and fuzzy derivative approaches. The SIS model is solved by reducing the dimension of the model for fuzzy cases. For these reasons we use completely correlated fuzzy number. 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.
Let
A fuzzy number in trapezoidal form represented by three points like as
The
Let
In this case we have
Two trapezoidal fuzzy numbers
Let a trapezoidal fuzzy number be like
Let
So clearly we have
So,
So we can write
We can write it in modified form as
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.
Consider the first order fuzzy differential equation
Let the solution (the solution comes from any method) of the above FDE be
If
An interval number
Let us consider the differential equation
The 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. [
Let us assume the following differential inclusion is of the form
The function The function
Now we denote the attainable set at time
In fuzzy environment dynamical system the problem (
According to Hüllermeier [
Here the attainable sets related to the problem (
Hence there is fuzzy interval
Let
Then there is a unique fuzzy interval
Therefore, we have the
Suppose
For all (
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 [
Suppose that we have some usual sets
The extension principle for fuzzy sets states that if
Let
Now by Zadeh’s extension principle,
Let us consider the fuzzy initial value problem (FIVP)
And
Bede and Gal [
Before going to the fuzzy differential equation approach we first know the following definition.
The generalized Hukuhara difference of two fuzzy numbers
Here the parametric representation of a fuzzy valued function
The generalized Hukuhara derivative of a fuzzy valued function
Also we say that
Consider the fuzzy differential equation taking in (
We have the following two cases.
If we consider
If we consider
There are so many mathematical models in biology; SIS model is an important model of them. In a given species population at time
Here a susceptible
Now taking
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.
In this case the solution can be written as
In this case the solution can be written as
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.
The attainable sets of the problem of (
Let
Now different cases arise.
In this case the solution can be written as
Here,
Here,
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In this case the solution can be written as
Here,
Here,
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Let
Now different cases can be found as follows.
In this case the differential equation transforms to
In this case the differential equation transforms to
In this case the differential equation transforms to
In this case the differential equation transforms to
The problem in interval environment is
We get the solution for two cases as follows.
The solution is written as
The solution is written as
Find the solution after
Solution by differential inclusion and extension principle and fuzzy differential equation is given by
From Figure
Solution boundary for
| | | | |
---|---|---|---|---|
0 | 0.5022 | 0.1832 | 0.4974 | 0.8160 |
0.2 | 0.4549 | 0.1923 | 0.5446 | 0.8070 |
0.4 | 0.4152 | 0.2022 | 0.5843 | 0.7971 |
0.6 | 0.3814 | 0.2129 | 0.6181 | 0.7864 |
0.8 | 0.3523 | 0.2246 | 0.6471 | 0.7747 |
1 | 0.3270 | 0.2375 | 0.6724 | 0.7618 |
Solution boundary for
The boundary of the solutions is
From Figure
Solutions boundary for
| | | | |
---|---|---|---|---|
0 | 0.9565 | 0.8750 | 0.0435 | 0.1250 |
0.2 | 0.9492 | 0.8790 | 0.0508 | 0.1210 |
0.4 | 0.9421 | 0.8831 | 0.0579 | 0.1169 |
0.6 | 0.9355 | 0.8874 | 0.0645 | 0.1126 |
0.8 | 0.9291 | 0.8919 | 0.0709 | 0.1081 |
1 | 0.9231 | 0.8966 | 0.0769 | 0.1034 |
Solutions boundary for
Here the solutions are given by
From Figure
Solution for
| | | | |
---|---|---|---|---|
0 | 0.1832 | 0.5022 | 0.4974 | 0.8160 |
0.2 | 0.1923 | 0.4549 | 0.5446 | 0.8070 |
0.4 | 0.2022 | 0.4152 | 0.5843 | 0.7971 |
0.6 | 0.2129 | 0.3814 | 0.6181 | 0.7864 |
0.8 | 0.2246 | 0.3523 | 0.6471 | 0.7747 |
1 | 0.2375 | 0.3270 | 0.6724 | 0.7618 |
Solution for
The solutions are given by
From Figure
Solution for
| | | | |
---|---|---|---|---|
0 | 0.8750 | 0.9565 | 0.0435 | 0.1250 |
0.2 | 0.8790 | 0.9492 | 0.0508 | 0.1210 |
0.4 | 0.8831 | 0.9421 | 0.0579 | 0.1169 |
0.6 | 0.8874 | 0.9355 | 0.0645 | 0.1126 |
0.8 | 0.8919 | 0.9291 | 0.0709 | 0.1081 |
1 | 0.8966 | 0.9231 | 0.0769 | 0.1034 |
Solution for
Now the solutions for different cases are given by the following.
From Figure
Solutions for
| | | | |
---|---|---|---|---|
0 | 0.4471 | 0.9046 | 0.0954 | 0.5529 |
0.2 | 0.4845 | 0.8810 | 0.1190 | 0.5155 |
0.4 | 0.5209 | 0.8564 | 0.1436 | 0.4791 |
0.6 | 0.5564 | 0.8308 | 0.1692 | 0.4436 |
0.8 | 0.5908 | 0.8043 | 0.1957 | 0.4092 |
1 | 0.6243 | 0.7768 | 0.2232 | 0.3757 |
Figure for
From Figure
Solution for
| | | | |
---|---|---|---|---|
0 | 0.7403 | 0.6601 | 0.1097 | 0.4899 |
0.2 | 0.7359 | 0.6662 | 0.1341 | 0.4638 |
0.4 | 0.7313 | 0.6722 | 0.1587 | 0.4378 |
0.6 | 0.7266 | 0.6782 | 0.1834 | 0.4118 |
0.8 | 0.7218 | 0.6840 | 0.2082 | 0.3860 |
1 | 0.7168 | 0.6898 | 0.2332 | 0.3602 |
Figure for
From Figure
Solution for
| | | | |
---|---|---|---|---|
0 | 0.7403 | 0.6601 | 0.1097 | 0.4899 |
0.2 | 0.7359 | 0.6662 | 0.1341 | 0.4638 |
0.4 | 0.7313 | 0.6722 | 0.1587 | 0.4378 |
0.6 | 0.7266 | 0.6782 | 0.1834 | 0.4118 |
0.8 | 0.7218 | 0.6840 | 0.2082 | 0.3860 |
1 | 0.7168 | 0.6898 | 0.2332 | 0.3602 |
Solution for
From Figure
Solution for
| | | | |
---|---|---|---|---|
0 | 0.6855 | 0.7376 | 0.2624 | 0.3145 |
0.2 | 0.6872 | 0.7317 | 0.2683 | 0.3128 |
0.4 | 0.6883 | 0.7254 | 0.2746 | 0.3117 |
0.6 | 0.6875 | 0.7171 | 0.2829 | 0.3125 |
0.8 | 0.6914 | 0.7143 | 0.2857 | 0.3086 |
1 | 0.6950 | 0.7114 | 0.2886 | 0.3050 |
Figure for
Find the solution after
From Figures
Solution for
| | |
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0 | 0.5026 | 0.4974 |
0.2 | 0.4309 | 0.5691 |
0.4 | 0.3610 | 0.6390 |
0.6 | 0.2955 | 0.7045 |
0.8 | 0.2361 | 0.7639 |
1 | 0.1840 | 0.8160 |
Solution at
| | |
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0 | 0.9565 | 0.0435 |
0.2 | 0.9449 | 0.0551 |
0.4 | 0.9310 | 0.0690 |
0.6 | 0.9146 | 0.0854 |
0.8 | 0.8958 | 0.1042 |
1 | 0.8750 | 0.1250 |
Figure at
Figure at
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: 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. 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. 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.
The authors declare that they have no conflicts of interest.
The second author of the article wishes to convey his heartiest thanks to Miss. Gullu for inspiring him to write the article.