In this work, the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus (COVID-19). The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics, namely, susceptible (S), infected (I), treatment (T), and recovered (R). The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points. To indicate the usefulness of this method, we employ it in some cases. For error analysis of the method, the residual of the solutions is reviewed. The reported examples show that the method is reasonably efficient and accurate.

The world has recently contracted the dangerous and deadly disease coronavirus 2019 (COVID-19), which is an almost uncontrollable respiratory infection. The new coronavirus was reported in Wuhan, China in December 2019. The World Health Organization (WHO) announced the global pandemic of the infection in March 2020. Its disease pattern is called COVID-19 [

One way to study the effective growth of this deadly outbreak is to use computer simulation block models. In different researches, different numerical and analytical approaches have been considered to perform the outputs of SITR models [

Differential equations have a remarkable role in several scientific and engineering phenomena that have always been considered at physical and technical applications and they are appeared in various areas as mathematics, physics and engineering sciences [

In recent years, Yüzbaşi et al. have applied the collocation method based on exponential approximation to solve some problems like pantograph equation, the linear neutral delay differential, Fredholm integro-differential difference equations and so on [

The aim of the present work is to find a numerical solution for a nonlinear SITR model using the new dynamic parameters of COVID-19, along with numerical analysis to better understand the expansion using numerical approaches through exponential basis.

The organization of this article is structured as follows: In

Mathematical modelling of the dynamical system is an interesting field of study that has attracted most researchers’ attention. Dynamical systems have a wide range of applications, including population growth models, biomedical research, biological systems, and engineering.

In this approach, for the SITR model is designed for the new COVID-19 dynamics, the population is allocated to various compartments with particular labels, susceptible

The SITR model of the novel COVID-19 dynamics is given by the following nonlinear 1st order differential equations and the description of each compartment is given in

Compartment | Brief definition |
---|---|

Non-infected individuals | |

Non-infected older or major diseased individuals | |

Infected community rate | |

Recovery community rate | |

Treatment |

Each equation is describing the transmission behavior of individuals in the respective compartments. By this transmission number of individuals can vary in each of the five compartments [

Parameter | Interpretation |
---|---|

Contact rate | |

Rate of natural birth | |

Reduce infection from treatment | |

Fever, tiredness and dry cough rate | |

Recovery rate of population | |

Death rate | |

Rate of infection from treatment | |

Healthy food rate | |

Sleep rate | |

Initial conditions |

We try to find the approximate solution of system of differential

To begin with, we assume that the unique solutions of system

After solving these systems, we merge the functions as multivariate function.

In this section, we outline operational matrices of the exponential method we will use in order to solve system

In the first step, we create the differentiation matrices which are the basic tools of the current approach. Differentiation matrices make this method more suitable for managing high-order differential equations. By constructing an operational matrix, it is easy to derive high-order derivatives of the unknown in terms of values at collocation points.

Firstly, we inscribe the approximated solution

Taking advantage of the linearity of expansion

The derivative of the approximate solution can also be expressed as a product of matrices. Namely,

Note that

By using of the matrix relations

After replacing the collocation points

In this section, we explain how to use the exponential collocation method for problem

To acquire an exponential series solution of

The relationship between

Now let’s create the

The system

To determine the unknown coefficients, we use the collocation points

Now, let us find the relations between the matrix

After the substitution of the above relations, we obtain to the following matrix equation:

Briefly,

Here,

By placing the colocation points at the algebraic equations and considering initial conditions, the system be solved and the coefficients be determined.

Since most systems do not have exact solutions and most similar programs usually do not have the ability to generate numerical solutions, we must verify the accuracy of the numerical results using a method. We can confirm the accuracy and effectiveness of the numerical results by considering the residual error functions. The residual method is a general class of methods developed to obtain the approximate solution of differential equations. In the residual method, an approximate solution based on the overall behavior of the dependent variable is considered. The assumed solution is often chosen to satisfy the boundary conditions. This assumed solution is then replaced by the differential equation. Since the assumed solution is only approximate, it generally does not satisfy the differential equation and therefore leads to an error or what we call the residual. The residual then disappears in the mean sense throughout the solution domain to produce a system of algebraic equations.

If

The rate of increase in the number of infections depends on the product of the number of infected and susceptible persons. The system

In this section, we present some applications of the method that are presented in different values for the parameters

These results show the effectiveness of the present method in achieving good accuracy with fewer collocation points and less computational time. All the calculations have been performed using MAPLE. By applying the suggested method for

Parameter | Case (I) | Case (II) |
---|---|---|

0.25 | 0.30 | |

0.30 | 0.30 | |

0.30 | 0.30 | |

0.005 | 0.005 | |

0.08 | 0.10 | |

0.20 | 0.25 | |

0.30 | 0.30 | |

0.20 | 0.20 | |

0.10 | 0.10 |

The residual of solutions has been used to show that this method is efficient and reasonably accurate.

The coefficients of the approximate numerical outcomes of the method for

Function | Coefficients |
---|---|

N | t = 0.1 | t = 0.3 | t = 0.5 | t = 0.7 | t = 0.9 | |
---|---|---|---|---|---|---|

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 |

Function | Coefficients |
---|---|

N | t = 0.1 | t = 0.3 | t = 0.5 | t = 0.7 | t = 0.9 | |
---|---|---|---|---|---|---|

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 | ||||||

N = 4 | ||||||

N = 6 |

In

These graphs show that contact rates are primarily a source of large increase in the number of susceptible individuals, but begin to decrease over time. This is because higher contact rates cause more people to become infected and move to an infected class; therefore, the number of individuals in the susceptible class decreases. About

The figures also show that with a high death rate, the number of recovered individuals suddenly decreases. Because when the death rate is high, so many people in the affected and recovered classes lose their lives, resulting in a decrease in people in all the classes. Since the death rate is so high, the infected and recovered people are almost gone.

In this paper, the exponential approximation is used to solve the numerical investigation of a nonlinear SITR model that represents the dynamics of the new disease of COVID-19. The method is based on exponential functions and the collocation method as an operational matrix. As seen, there is no concern about approximating higher-order derivatives of the unknowns. Also, to show the accuracy and efficiency of the method, two cases with different values have been examined. Through the examples provided, we realize that the obtained numerical results have very good residual errors. Moreover, it is realized that errors decrease when

The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.

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