Application of piecewise fractional differential equation to COVID-19 infection dynamics.

We proposed a new mathematical model to study the COVID-19 infection in piecewise fractional differential equations. The model was initially designed using the classical differential equations and later we extend it to the fractional case. We consider the infected cases generated at health care and formulate the model first in integer order. We extend the model into Caputo fractional differential equation and study its background mathematical results. We show that the fractional model is locally asymptotically stable when R 0 < 1 at the disease-free case. For R 0 ≤ 1 , we show the global asymptotical stability of the model. We consider the infected cases in Saudi Arabia and determine the parameters of the model. We show that for the real cases, the basic reproduction is R 0 ≈ 1 . 7372 . We further extend the Caputo model into piecewise stochastic fractional differential equations and discuss the procedure for its numerical simulation. Numerical simulations for the Caputo case and piecewise models are shown in detail.

Introduction

Mathematical models in epidemiology are considered important to study the dynamics and predictions of the disease’s long-time behaviors. With the emergence of the new infectious diseases, the mathematical models were considered the primary tools to study the predictions and eliminations of the disease outbreaks. The emergence of the coronavirus infection that given a lot of deaths and infected cases throughout the world, and at the same time many countries faced financial crises. In Saudi Arabia, a lot of infected cases with deaths are recorded. To date, the total cases reported in the kingdom are 751 518, with 9054 deaths. One of the important reasons to minimize the infected cases is the lockdown, minimizing social distance, and other related essential measures recommended by the world health organization (WHO). Due to prevention and the government instructions of the KSA government and recommendations of the (WHO), the KSA government succeeded to minimize the infected cases of the COVID-19 in the country.

Mathematical models based on classical and fractional derivatives are considered useful in scientistic and engineering problems see [1], [2], [3], [4] and especially in epidemiology, see [5], [6], [7], [8]. Due to the heredity, crossover behavior, and memory, the factional models become superior to the integer order models. The application of fractional derivatives in various fields of scientific and engineering can be seen in literature, see for example the applications to integrodifferential equations [9], the new advancement and development in fractional operators [10], [11], application to epidemiology  [12], [13], [14], in wave dynamical equations,  [15], [16] etc. Some more related works on the applications of fractional derivatives in literature, we discussed in detail. The use of lockdown in the modeling of the COVID-19 infection has been discussed in [17]. In [18], the authors constructed a mathematical model using Continuous Markov-Chain to analyze the dynamics of coronavirus infection. A SARS-CoV-2 model using the variable fractional order has been investigated in [19]. A COVID-19 infection model using the control intervention is discussed in [20]. A mathematical study established the results for the coronavirus infection by considering the Ethiopian cases in [21]. The impact of stress due to coronavirus infection has been modeled mathematically and discussed their effects in [22]. The authors considered a modified model to analyze the disease dynamics of the coronavirus infection by taking the real cases from Saudi Arabia [23]. The authors considered the self-isolated cases to study the COVID-19 infection in [24]. A mathematical study of the infected cases in Saudi Arabia has been investigated in [25]. A COVID-19 infection model with the use of the face mask has been studied in [26]. A confection model to study cholera and COVID-19 and its dynamical analysis have been given in [27]. The authors studied the COVID-19 infection in Turkey and South Africa with the available real data [28]. The mathematical modeling approach has been used in [29] to study the COVID-19 infection. The mathematical modeling and related results are explored in [30]. A mathematical study to focus on HIV and COVID-19 and is alert to the countries of the world has been suggested in [31]. The considered diffusion reaction mathematical model and its applications to COVID-19 infection have been suggested in [32]. A delay COVID-19 disease model has been studied in [33].

This paper investigates the dynamics of the COVID-19 disease using the assumptions of the mortality rates in the disease-infected classes through the new approach of the piecewise fractional differential equations. Real data from March 2020 to March 2021 of Saudi Arabia is considered to obtain the numerical values of the realistic parameters by utilizing the procedure of the least-square curve fitting method. We formulate the model in classical differential equations initially and then update it to Caputo fractional differential equations. The model further extended to piecewise fractional differential equations. We study the details mathematical results of the Caputo model and the numerical simulation of the stochastic and Caputo model is studied with the important parameters that have a great impact on the disease elimination. A section-wise detail of the present work is given is as: We study in Section “Model formulation” the mathematical modeling of the problem. Section “Related definitions and formulation of the fractional order model” discusses the background fractional calculus results and their application to the model. The stability analysis of the fractional model is considered in Section “Stability analysis”. The piecewise differential equations and their numerical procedures is given in Section “A piecewise differential equation model”. Estimation of the parameters using the real cases is given in Section “Estimations of the parameters”. Section “Numerical simulation” studies the numerical simulation of the Caputo and the piecewise model. A recommendation on the obtained results is discussed in Section “Conclusion”.

Model formulation

We consider here to formulate a mathematical model for SARS-COV-2 infection based on the disease death in the exposed, asymptomatic, symptomatic, and hospitalized classes. We consider further the infection not only spread due to the asymptomatic, or asymptomatic but also due to the individuals in the exposed class. We denote the total population of the humans by and splitting into six different epidemiological compartments, such as, the healthy or susceptible individuals, (individuals that attract the disease but are not yet infected), individuals after interacting with infected people become exposed, (individuals to complete their incubations periods), the exposed individuals after completing their incubation period, a portion of the individuals that do not show disease symptoms shall join the class known as asymptomatic individuals , individuals that complete their incubation period with disease clinical symptoms join the class , individuals infected are hospitalized given by and those individuals that are recovered from the infection are become recovered are shown by . We consider in this model the disease death rate in the compartments, , , and , so the death class that represents the number of death for each time is designated by . We have . The above discussion leads to the following evolutionary differential equations: The initial conditions of the variables given in the system (1) are non-negative. The individuals in the susceptible population are recruited to the birth rate given by while its natural death rate is . The force of infection is given by where is the effective contact rate by which the healthy individuals are contacted by the exposed, and are the disease contact rates of healthy with asymptomatic, and with the symptomatic individuals respectively. The parameter measures the infected generated through the hospitalized individuals. The proportion of the exposed individuals who do not exhibit disease symptoms clearly joins the class with a rate while the rest of the individuals that show disease symptoms join the class . The disease death rate for respectively represent the death in the exposed, asymptomatic (individuals with no disease signs), symptomatic (with disease signs), and hospitalized individuals. The individuals in the asymptomatic and symptomatic classes are hospitalized at the rate given by and respectively. The parameter denotes the removal or recovery of the hospitalized individuals. The disease death rate of the infected compartments that join the death class can be formulated by the following equation:

Related definitions and formulation of the fractional order model

Based on [34], we have the following definitions:

A and , , the Caputo derivative is given by

The fractional integral of (3) is

The fixed point is known to be the solution of the fractional system in Caputo sense if it satisfy the equation below:

The following theorems are given based on the results presented in [35], [36].

Assume that is the solution of the system (5) and is the domain containing . Let is a continuous differentiable function such that and

for every and , and for are the positive definite functions that continuous on , then, of (5) is called the uniformly asymptotically stable equilibrium.

A fractional model in Caputo derivative

We extend the model (1) into fractional operator in Caputo sense given by, with the initial conditions

Solution existence and its positivity

To show the positivity of the model (8), we consider, We give the following results based on the theorem given in [37], given by.

Assume that and , then with , .

Consider that and , where , then if

We prove now the result in the following:

A unique and positive solution exists of the model (8) and remains in .

It follows from [38] that , the model (8) possess a non-negative solution, given by

It follows from (10) that the solution is positive and will remain in , and thus, we define the biologically feasible region for the model (8), given by

Equilibria

This section determines the possible endemic equilibria of the disease model. There exist two equilibria for the model (8), namely the disease free and the endemic equilibria which are obtained in the following: Next, we determine the expression for the basic reproduction number denoted by of the system (8), by using the approach suggested by the authors in [39]. After utilizing the technique, we get the following matrices, The spectral radius gives the required basic reproduction number for the model (8), where,

Stability analysis

This section discusses the stability analysis of the model (8) at the disease-free equilibrium . For the local asymptotical stability of the fractional model (8), we obtain the following Jacobian matrix at .

For any, such that. Letand suppose, then the model(8)is locally asymptotically stableif, for every roots ofof Eq. (12) of .

The characteristics equation of is given by, where The coefficients for given above of Eq. (13) can be shown here to be positive in order to fulfill one of the conditions of the Routh–Hurtwiz . It can be that the first coefficient for are positive when the sub-reproduction number is less than (which is obviously less than 1). The coefficient is positive when , so all . Next, one can easily prove the last condition, . Upon satisfying the last condition, that will make sure that the infection model at the disease-free equilibrium of the system (8) is locally asymptotically stable if . □

We show the global asymptotical stability of the fractional system (8) at by following Theorem 1.

If and , then the model (8) at is globally asymptotically stable.

We construct the following Lyapunove function Here are the positive constant and later we will determine their value. Applying the Caputo derivative on (15), we get It follows from (8), and with some arrangements, we get Let us choose the constants for , ,  , ,  and . After some calculations, we get finally Here, if then and if , we have . Thus, the model (8) at the disease free case is stable globally asymptotically on when . □

Endemic equilibria

The endemic equilibria of the system (8) can be obtained by denoting it by , given by Inserting Eq. (19) into the following, we have, where The coefficient in Eq. (22) is positive while is positive when and hence is negative. If , we can have a unique positive endemic equilibrium. For such existence of the unique positive endemic equilibrium, it ensures that there is no possibility of the endemic equilibria. In other words, the only is enough to decrease in order to reduce the infected cases.

A piecewise differential equation model

Piecewise fractional differential equation has been introduced recently in literature. The purpose of this piecewise approach is to study effectively the model with real data having many waves. We extend the model (8) by applying the theory given in [41]. In system (23) the interval has been used. The interval is used for the following system: and the interval is finally for , thus, we have, where for are the stochastic environment and for represent the standard Brownian motion.

Numerical procedure for piecewise model

The following numerical procedure can be used to solve the model (23)–(25) by the procedure given in [41].

The numerical scheme follows from [41], [42] and its recent application in disease modeling, see [43] and heat diffusion [44], has been given in the following:

By dividing into three parts given by, After some calculations, we then have the numerical scheme finally given by: where

Estimations of the parameters

We consider the updated cases in the Kingdom of Saudi Arabia started from March 02, 2020, till March 6, 2021. The cumulative cases reported in the Kingdom of Saudi Arabia till March 6, 2021, are 379 474, among these cases, the reported number of death is 6539 which is 2% of the cases while the recovered cases are 371 338 which is 98% of the total cases [45]. Using the mentioned period of the cases in KSA, we wish to parameterize our propped system (8) in order to better fit the model and predict the future trends of cases in KSA. We consider the outbreak year is 2020 with the total population in KSA by and taking the time unit is per day. We can estimate the parameter that defines the birth rate of the healthy population while accounts for the natural mortality in each class of the model are respectively given by per day and , where denotes the average life span in KSA. The initial conditions of the model fitting to the data and the rest of the numerical simulations are considered to be , , , , and . The other parameters of the model are fitted to the data and have been listed in Table 1. The least square procedure the desired fitting to the infected data has been shown graphically in Fig. 1. The basic reproduction number using the obtained numerical values of the parameters is .

Table 1

Details of the parameters.

Not	Details	Numeric value	Ref
Λ	Birth rate	d×N(0)	Estimated
d	Natural death rate	174.87×365	[40]
τ_1	Rate of contact among exposed and susceptible	0.7478	Fitted
τ_2	Rate of contact rate among asymptomatic and susceptible	0.0594	Fitted
τ_3	Rate of contact among asymptomatic and susceptible	0.0412	Fitted
τ_4	Rate of contact among hospitalized and susceptible	0.01	Fitted
δ	Incubation period	0.7106	Fitted
η	Incubation period	0.9634	Fitted
d_1	Disease death of exposed individuals	0.0310	Fitted
d_2	Disease death of asymptomatic individuals	0.0100	Fitted
d_3	Disease death of symptomatic individuals	0.0011	Fitted
d_4	Disease death of hospitalized individuals	0.0394	Fitted
ψ_1	The rate by which asymptomatic are hospitalized	0.7637	Fitted
ψ_2	The rate by which symptomatic are hospitalized	0.0011	Fitted
π	Recovery rate of hospitalized individuals	0.9997	Fitted

Fig. 1

Reported number of coronavirus cases in KSA versus model fitting.

Numerical simulation

We consider here the simulations of the model (8) in Caputo sense and the model (23)–(25) in piecewise differential equations. We use the parameter values obtained in Table 1 with the suggested initial conditions of the model variables given in Section “Estimations of the parameters”. The numerical results of the model (8) are shown graphically in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7. Fig. 2 shows the dynamics of the total infected population for the parameter . Decreasing the contact between the healthy and the individuals with disease symptoms may decrease the infected cases further. The parameter that defines the contact among the asymptomatically infected with the healthy individuals can decrease the infected population if the contact rate is decreased, which is shown graphically in Fig. 3 and demonstrate this fact. The parameter measures the contact rate among healthy and symptomatically infected populations. Decreasing the value of , a decrease in the infected population is observed see 4. The individuals that show disease symptoms should be isolated in order to reduce the infection further in society. The health care individuals that work in the hospitals also contracted the infection while dealing with the patients, that is why follows the kits and another necessary measure in order to protect themselves from the infection. We have shown in Fig. 5 that by decreasing the contact rate a decrease in the infected cases can be seen. The proportion of the infected cases generated through the parameter at the exposed period and their further distribution in asymptomatic and symptomatic classes are shown graphically in Fig. 6. A slight variation in value makes decreased the number of total infected cases. Similarly making a decrease in the value of , one can see a decrease in the number of total infected individuals.

Fig. 2

The variation in on the total infected population.

Fig. 3

The variation in on the total infected population.

Fig. 4

The variation in on the total infected population.

Fig. 5

The variation in on the total infected population.

Fig. 6

The variation in on the total infected population.

Fig. 7

The variation in on the total infected population.

We consider the stochastic model (23)–(25) by using the above numerical procedure with same initial conditions and parameters values shown in Table 1 and obtain the graphical results given in Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12. Fig. 8 shows the dynamics of the total infected population when using various values of for in the absence of . Fig. 9(a–d) represents the dynamics of the total infected population for different values of the and , , , , , . The sub-graph in Fig. 9(e) demonstrate the dynamics of the total infected population for different fractional order parameter . Fig. 10 is the phase portraits of SE, SI, EI plane and SEI. Fig. 11 is the phase portraits of SA, SI, AI plane and SAI, while Fig. 12 is the phase portraits of SH, SI, HI plane and SHI different planes.

Fig. 8

Simulation of the piecewise model when .

Fig. 9

Simulation of the piecewise model when , , , , , .

Fig. 10

Simulation of the piecewise model when , , , , , .

Fig. 11

Simulation of the piecewise model when , , , , , .

Fig. 12

Simulation of the piecewise model when , , , , , .

Conclusion

A new mathematical model investigating the COVID-19 infection in Caputo and the piecewise stochastic differential equation has been formulated. The detailed model formulation of the model for the COVID-19 infection is first formulated in integer case and then it is extended to Caputo fractional differential equations. The mathematical results for the Caputo fractional model are obtained and discussed. The local and global asymptotical stability of the model is obtained on the basis of the basic reproduction number . We further extended the model into a piecewise differential equation and present a novel approach for its numerical simulations. The real data of Saudi Arabia for the COVID-19 infection were considered to obtain the parameter’s numerical values. The basic reproduction number computed using the real data is . We solved the Caputo model numerically and presented the graphical results for the key parameters. It was found that the prevention or isolation of healthy individuals from individuals exposed, infected with symptoms, infected with no symptoms, and those hospitalized can best decrease the infection cases in the country. We solved the stochastic fractional model piecewise using the novel numerical procedure and obtained the results graphically for various values of the sensitive parameters in the presence of stochastic noises and without noises. We plotted some phase portraits for the different variables of the model in a stochastic environment.

CRediT authorship contribution statement

Xiao-Ping Li: Software, Data curation, Writing – review & editing, Validation. Haifaa F. Alrihieli: Software, Data curation, Writing – review & editing, Validation. Ebrahem A. Algehyne: Software, Data curation, Writing – review & editing, Validation. Muhammad Altaf Khan: Conceptualization, Methodology, Software, Data curation, Writing – original draft, Visualization, Investigation, Validation, Writing – review & editing. Mohammad Y. Alshahrani: Software, Data curation, Writing – reviewi & editing, Validation. Yasser Alraey: Software, Data curation, Writing – review & editing, Validation. Muhammad Bilal Riaz: Software, Data curation, Writing – review & editing, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.