Stability analysis of fractional order model on corona transmission dynamics.

In this paper a fractional order mathematical model is constructed to study the dynamics of corona virus in Oman. The model consists of a system of eight non-linear fractional order differential equations in Caputo sense. Existence and uniqueness as well as the stability analysis of the solution of the model are given. The stability analysis is in the frame of Ulam-Hyers and generalized Ulam-Hyers criteria. Numerical simulations are given to support the theoretical results. Many informations on the dynamics of COVID -19 in Oman were obtained using this model. Also many informations on the qualitative behaviour of the model were obtained.

Introduction

Some epidemic diseases are capable of producing large number of infections starting from a fewer ones, an example of such diseases is COVID-19 [1]. Coronavirus disease is a respirational and zoonotic disease, caused by a virus of the coronaviridae family which originated in the city of Wuhan China on December 01, 2019 [2]. The Virus strain is severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), resulting in fever, coughing, breathing difficulties, fatigue, and myalgia. It may transform into pneumonia of high intensity.

Many scientists put their heads together in trying to find answers about the spread and infection of corona virus by examining virus samples [3].Although the disease strain is known, but, the vaccine is still not available. Hence it is necessary to ensure strict mitigation actions in order to contain the virus. Many countries are taking different measures to cope with the virus.

Taking the China and South Korea cases into consideration, a lot of countries were able to contain the spread of the virus. It is proven that social distancing and testing are some of the key control measures. Another measure responsible for spreading the disease worldwide was regional and global travellers. Although, air travel is almost suspended now, but, this initial shock and countries (and people) not taking it too seriously has taken many countries in Europe and North America to a real bad situation.

Oman has taken various measures to avoid the spread of COVID-19. From mid of March, Schools and Colleges/Universities were closed, consequently other non essential offices and services were closed. The situation is been monitored on daily basis by the government. However, there are many suspected cases which are not been tested so far. By the time of this report, most part of the country is partially locked down. Although, the spread of the disease is not that much as of today (June 17, 2020), but, the new cases are appearing continuously.

We need modelling approach to understand the exact dynamics of the disease [23], [24], [25]. This is what motivates this research.It is important to note that, in the classical order model, the state of epidemic does not depend on its history. However, in real life memory plays a vital role in studying the pattern of spread of any epidemic disease. It was found that the waiting times between doctor visits for a patient follow a power law model [4]. It is worth to know that Caputo fractional time derivative is a consequence of power law [5]. When dealing with real world problem Caputo fractional-order derivatives allows traditional initial and boundary conditions [26], [27]. Furthermore, due to its non-local behaviour and its ability to change at every instant of time, Caputo fractional-order gives better result than the integer order [28], [29], [30], [31].

In recent studies, Khan et al. studied a fractional-order model that describes the interaction among bats and unknown hosts, then among people and seafood market [6]. To predict the trend of the Corona-Virus Yu et al. constructed a fractional time delay dynamic system that studied the local outbreak of COVID-19 [7]. Also, to predict the possible outbreak of infectious diseases like COVID-19 and other diseases in the future Xu et al. proposed a generalized fractional-order SEIQRD model [8]. Shaikh et al. used Bats-Hosts-Reservoir - People transmission fractional-order COVID-19 model to estimate the effectiveness of preventive measures and various mitigations, predicting future outbreaks and potential control strategies [9].

The aim of this research is to study fractional - order epidemic model that investigates the dynamics of COVID-19 in Oman. Based on the memorability nature of Caputo fractional-order derivatives, this model can be fitted with data reasonably well. Then, based on the official data given by the Federal Ministry of Health Oman daily, numerical examples will be carried out.

This paper is organized as follows. In Section 2, Preliminary definitions are given. In Section 3, the fractional order model for COVID-19 in Caputo sense is formulated. In Section 4, existence and uniqueness of the solution of the model is established. In chapter 5, stability analysis of the solution of the model in the frame of Ulam-Hyers and generalized Ulam-Hyers is given. Chapter 6, contains the numerical scheme and numerical simulations to illustrate the theoretical results. Finally, conclusion is given in chapter 7.

Preliminaries

[10]

The fractional integral of order with the lower limit for a function is defined asprovided the right side is point-wise defined on where denotes the gamma function.

The Caputo fractional derivative of order with the lower limit for a function is defined asprovided the function differentiable on where denotes the gamma function.

Let . Then

Formulation of the model

Let the total population be The population is divided into eight compartments, namely; Susceptible population Exposed population Asymptomatic Infective population Symptomatic Infective Population Isolated Infective Population Hospitalised Infective Population , Recovered Population and then Dead Individuals The dynamics of this population is represented by the following system of fractional order differential equations (FODE), and the meaning of parameters is given in Table 1 .

Table 1

Description of the parameters.

Parameter	Description
β	effective contact rate
α	reduction of infectiousness in asymptomatic people
ξ	reduction of infectiousness in isolated people
k	progression from exposed to infectious class
p	proportion of asymptomatic cases
q	progression from asymptomatic unaware to self - isolated
Ψ	proportion of hospitalised people
γ_A,γ_S,γ_H	recovery rate for asymptomatic, symptomatic and hospitalised individuals respectively
μ_S,μ_H	mortality rate in isolated and hospitalised classes

Existence and uniqueness results

The theory of existence and uniqueness of solutions is one of the most dominant fields in the theory of fractional-order differential equations. The theory has recently attracted the attention of many researchers, we are referring to [11], [12] and the references therein for some of the recent growth. In this section, we discuss the existence and uniqueness of solution of the proposed model using fixed point theorems. Let us reformulate the proposed model in the subsequent formwhereThus, the proposed model (2) takes the formon condition thatwhere represents the transpose operation. In view of Theorem in [10], problem is given byLet denotes the Banach space of all continuous functions from to R endowed with the norm defined bywhereand

Suppose that the functionand maps bounded subset ofinto relatively compact subsets of. In addition, there exists constantsuch that

for alland each. Then the integral equationwhich is equivalent with the proposed modelhas a unique solution provided thatwhere

Consider the operator defined byObviously, the operator is well defined and the unique solution of model is just the fixed point of . Indeed, let us take and . Thus, it is enough to show that where the set is closed and convex. Now, for any yieldsHence, the results follows. Also, given any we getwhich implies that . Therefore, as a consequence of Banach contraction principle, proposed model possess a unique solution. □

Next, we prove the existence of solutions of the proposed model by employing the concept of well-known Krasnoselskii’s fixed point theorem.

[13]

Letbe a closed, bounded and convex subset of a Banach SpaceLetbe two operators that obey the given relations

whenever

is compact and continuous;

is a contraction mapping.

Then there existssuch that

Suppose that the functionis continuous and satisfies conditionIn addition, assume that

.

Then the proposed modelhas at least one solution provided

Setting and we consider Consider the operators on defined byandThus, for any yieldsHence,

Next, we prove the contraction of the operator

Obviously, given any and givesSince the function is continuous, implies that the operator is continuous. Moreover, for any and implies that is uniformly bounded. Finally, we show that the operator is compact. Define givesThus, is equicontinuous and so is relatively compact on Hence, as consequences of Arzelá Ascoli theorem, is compact on Since all the hypotheses of Theorem [10] are true, proposed model has at least one solution. □

Stability results

In this section, we drive the stability of the proposed model in the frame of Ulam-Hyers and generalized Ulam-Hyers stability. The concept of Ulam stability was introduced by Ulam [14], [15]. Then, in several research papers on classical fractional derivatives, the aforementioned stability was investigated, see for example, [16], [17], [18], [19]. Moreover, since stability is fundamental for approximate solution, we strive to use nonlinear functional analysis on Ulam-Hyers and generalized stability of the proposed model . Thus the following definitions are needed. Let and consider the inequality given belowwhere

The proposed problem is Ulam-Hyers stable if there exist such that for every and a solution satisfying there exists a unique solution of equation withwhere

Problem is referred to generalized Ulam-Hyers stable if there exist a continuous function with such that for every solution of the equation there a solution of equation such thatwhere

A function satisfy the inequality if and only if there exist a function with the property below:

Assume thatsatisfies inequalitythensatisfies the integral inequality describe by

Thanks to of Remark 5.1,and Theorem [10], givesUsing of Remark 5.1 and we getHence, the desired results. □

Suppose thatis continuous for everyand hypotheseshold with. Thus, problemis Ulam-Hyers and consequently, generalized Ulam-Hyers stable.

Suppose that satisfies the inequality and be a unique solution of . Thus, for any and Lemma 5.1, givesSo,whereSo, setting such that . We conclude that the proposed problem is both Ulam-Hyers and generalized Ulam-Hyers stable. □

Numerical Simulations and Discussion

Here we use COVID-19 data obtained from Federal Ministry of Health Oman on 18th June 2020 for the Numerical simulations. The parameter values are given in Table 2 . We can observe that the number of infection get to zero with time. This is true as per as any epidemic disease is concern. Herein, the fractional variant of the model under consideration via Caputo fractional operator is numerically simulated via first order convergent numerical techniques as proposed in [20], [21], [22]. These numerical techniques are accurate, conditionally stable, and convergent for solving fractional-order both linear and nonlinear system of ordinary differential equations.

Table 2

Parameter values.

Parameter	Values
β	calibration
α	0.5
ξ	calibration
k	0.2174
p	0.5
q	0.3448
Ψ	data
γ_A,γ_S,γ_H	0.1961
μ_S,μ_H	data

Now we discuss the obtained numerical outcomes of the governing model in respect of the approximate solutions. To this aim, we employed the effective Euler method under the Caputo fractional operator to do the job. The initial conditions are taken as and the parameters values are as in Table 2 below:

Considering the values in the above table, we carry out the following numerical simulations with the fractional order value .

In Fig. 1 , dynamics of all the populations involved in the model are presented. It can be seen that with time, all the populations will tend to zero except the recovered and Dead populations. This requires more effort to be put in place in order to reduce the number of death and increase the recovery of the infected individuals. In Fig. 2 , Asymptomatic and Symptomatic cases are shown. It can be seen that there are more asymptomatic cases as compared to the symptomatic cases. This means, there are many positive individuals without any sign of the disease. Infection through this lane could only be stopped through contact trace. In Fig. 3 , Recovered case were depicted against Death cases. It can clearly be seen that there are more recovery cases than death cases. This is a good news and various means should be put in place to maintain the trend. Fig. 4 presents the total Infected cases against total Death cases. It can be seen that indeed many infected individuals recovered. The death cases is very low, which shows that the disease is not fatal in Oman. Fig. 5 shows the relationship between isolated infected cases and hospitalised infected cases. It is clear that as time goes on both populations will tend to zero. Finally Fig. 6 shows the dynamics of various infection cases. It can be seen that isolated and hospitalised populations are very small compared to the asymptomatic and symptomatic cases. Hence there is need for more effort in tracing out the infected individuals.

Fig. 1

Dynamics of different populations in the model.

Fig. 2

Asymptomatic versus symptomatic.

Fig. 3

Recovered versus Dead.

Fig. 4

Total Infected cases versus total death.

Fig. 5

Isolated infective cases versus hospitalised infective cases.

Fig. 6

Various infection cases.

Many informations on the dynamics of COVID -19 in Oman were obtained using this model. Also many informations on the qualitative behaviour of the model were obtained.

Conclusions

In conclusion, this paper consists of a system of eight non-linear fractional order differential equations in Caputo sense. The existence and uniqueness of solution of the proposed model using fixed point theorems is discussed. Stability analysis in the frame of Ulam-Hyers and generalized Ulam-Hyers criteria is established. Numerical simulations were carried out using real data from Federal ministry of health Oman. It was numerically shown that although the disease is not fatal in Oman, but there will be many death cases. Hence there is need for relevant authorities to take every available measure to curtail the spread of the disease.

CRediT authorship contribution statement

Evren Hincal: Visualization, Investigation, Supervision, Software, Validation, Writing - review & editing. Sultan Hamed Alsaadi: Conceptualization, Methodology, Software, Data curation, Writing - original draft.

Declaration of Competing Interest

None.