Fractional epidemic model of coronavirus disease with vaccination and crowding effects

Most of the countries in the world are affected by the coronavirus epidemic that put people in danger, with many infected cases and deaths. The crowding factor plays a significant role in the transmission of coronavirus disease. On the other hand, the vaccines of the covid-19 played a decisive role in the control of coronavirus infection. In this paper, a fractional order epidemic model (SIVR) of coronavirus disease is proposed by considering the effects of crowding and vaccination because the transmission of this infection is highly influenced by these two factors. The nonlinear incidence rate with the inclusion of these effects is a better approach to understand and analyse the dynamics of the model. The positivity and boundedness of the fractional order model is ensured by applying some standard results of Mittag Leffler function and Laplace transformation. The equilibrium points are described analytically. The existence and uniqueness of the non-integer order model is also confirmed by using results of the fixed-point theory. Stability analysis is carried out for the system at both the steady states by using Jacobian matrix theory, Routh–Hurwitz criterion and Volterra-type Lyapunov functions. Basic reproductive number is calculated by using next generation matrix. It is verified that disease-free equilibrium is locally asymptotically stable if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{0}<1$$\end{document}R0<1 and endemic equilibrium is locally asymptotically stable if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{0}>1$$\end{document}R0>1. Moreover, the disease-free equilibrium is globally asymptotically stable if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{0}<1$$\end{document}R0<1 and endemic equilibrium is globally asymptotically stable if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${R}_{0}>1$$\end{document}R0>1. The non-standard finite difference (NSFD) scheme is developed to approximate the solutions of the system. The simulated graphs are presented to show the key features of the NSFD approach. It is proved that non-standard finite difference approach preserves the positivity and boundedness properties of model. The simulated graphs show that the implementation of control strategies reduced the infected population and increase the recovered population. The impact of fractional order parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α is described by the graphical templates. The future trends of the virus transmission are predicted under some control measures. The current work will be a value addition in the literature. The article is closed by some useful concluding remarks.

To fill these gaps and to address the challenges arise in the COVID-19 modeling, the present study focuses on analyzing an SIVR epidemic model using the Caputo fractional order operator.This choice is justified because the Caputo derivative has the ability to accommodate local initial conditions and it is compatible with biological and physical principles.
The primary objective of this research is to assess the impact of vaccination and crowding effect on COVID-19 dynamics in the perspective of fractional calculus.The insights gained from this study could assist in strategic planning by governments and public health authorities to bridge immunization gaps and prevent future outbreaks.Furthermore, this fractional order model will contribute to the ongoing research in COVID-19 mathematical modeling and will also assist the researchers to stimulate interest in fractional calculus modeling and mathematical epidemiology.
The design of our paper is as follows.In "Description of the model" section, we have discussed the mathematical model and performed its analysis.Then, in the sub-sections, positivity, boundedness, existence, and uniqueness are studied.In "Qualitative analysis of the proposed model" section, qualitative analysis of the proposed model (local and global stability of the model) is presented.In "Numerical simulations" section, numerical simulations are presented to analyze the dynamics of the virus, graphically.In "Numerical method" section, numerical method (Grunwald-Letnikov non-standard finite difference method) of the proposed model is presented.Then, in the sub-sections, positivity and boundedness of the numerical method are discussed.Finally, the concluding remarks are presented in the closing section.Now, we quote some basic definitions of fractional calculus.
Definition 1 (Caputo Fractional Derivative) The Caputo derivative of fractional order α of function f (t) is defined as.

Laplace transform of Caputo fractional derivative
The Laplace transformation of Caputo fractional differential operator of order α is given by: Definition 2 (The Mittag-Leffler Functions) Two parametric Mittag-Leffler function is represented by the series.

Laplace transformation of the Mittag-Leffler Functions
Laplace transformation of the Mittag-Leffler function is given by.

Description of the model
In this segment, we present the fractional epidemic system of coronavirus disease.
In the proposed model, the whole human population N(t) is classified into four subclasses as S(t) (Suscepti- ble class), I(t) (Infected class), V (t) (Vaccinated class), and R(t) (Recovered or immune class).The principle of mass action is taken into account for the infection dynamics in the society.The transmission map of the model is shown in Fig. 1.
Parameters of the model are described as follows N (the recruitment rate of the population),β α I (the force of infection of virus), 1 1+α α 1 I (the crowding effect of population on the virus), µ α (the rate of mortality due to virus or natural of each subpopulation), δ α 1 (the rate at which infected population got vaccination during the period of quarantine, or isolation, etc.), δ α 2 (the rate at which susceptible population got vaccination under the program launched by World Health Organization (WHO)), γ α (the rate at which infected population may recover due to its internal immunity and natural circumstances), and σ α (the rate of doses in the population who recovered or got immune after vaccination).Deterministic model is based on the following assumptions.Susceptible population becomes immune against the disease after vaccination.Recovered population do not become infected.Only direct contact of infected individuals and susceptible population is considered in this model, all other types of interactions are neglected.

Model equations
The system of equations obtained from the transmission map of the virus is as follows:

Invariant region
The total dynamics of the system (1) is obtained by adding the four equations as follows where N(t) = S(t) + I(t) + V (t) + R(t).
The feasible region for the system (1) is defined by

Properties
In this section, we shall study the basic properties of our proposed model (1).The model will be biological meaningful if all the variables are non-negative for t ≥ 0. It means that solution with non-negative initial conditions will remain non-negative for all time.We ensure this result by Theorem 1.
Theorem 1 (Positivity) For any initial data (S(0), Proof We define the norm as f ∞ = sup t∈D f f (t) .Firstly, consider the class S(t), (1) www.nature.com/scientificreports/ By taking Laplace transformation on both sides of the above expression, we get By taking inverse Laplace transformation on both sides and by using the following result , we reach at the expression given below Next, for the function I(t), we proceed as explained below By taking Laplace transformation on both sides, we derive the following result By taking inverse Laplace transformation on both sides and by using the following result Similarly, for the class V(t) we have Let M 3 = σ α + µ α .Then, we conclude that By solving above inequality, we get Finally, for the function R(t) By solving above inequality, we get As desired.Now, we present the second property of model (1) i.e. boundedness of the solution.
Theorem 2 (Boundedness) For any time t, the system (1) is bounded and lies in the feasible region .
Proof By letting the population function N(t) = S(t) + I(t) + V (t) + R(t) , we advance as.
By taking Laplace transform on both sides of the above expression www.nature.com/scientificreports/By taking inverse Laplace transformation on both sides and by using the following formula , we can write above inequality in the form Thus N(t) is bounded uniformly and hence the solution Therefore, the model ( 1) is well-posed, biologically and mathematically in the invariant set , as desired.

Existence and uniqueness
This subsection presents the existence and uniqueness of solution of the proposed model using the technique of fixed-point theory, for this we will apply the following Lemma.
Theorem 3 (Existence and Uniqueness) For any time t, the solution of system (1) will exist and the solution will be unique 47 .

Proof
To study the existence and uniqueness of system (1), let us consider the region × [t 0 , γ] , where.

Qualitative analysis of the proposed model
This section presents the equilibrium points, basic reproductive number and stability analysis, analytically.The order of the results for the stability is described as: • Asymptotically local stability at corona virus-free equilibrium is established by Theorem 4.
• Asymptotically local stability at corona existing equilibrium is ensured by Theorem 5.
• Asymptotically global stability at corona virus-free equilibrium is guaranteed by Theorem 6.
• Asymptotically global stability at corona existing equilibrium is confirmed by Theorem 7.

Equilibria
We determine the equilibria of the system (1) by assuming the state variables are constant and by putting the right side equal to zero.Equation (1) admits two types of equilibria as follows:

Reproduction number
We determine the reproduction number of the system (1) by using the well-known results like the next-generation matrix method after substituting the value of coronavirus free equilibrium.We get: Mathematically R 0 is expressed by the following parametric relation

Stability analysis
In this section, we test the local and global stability of the system (1), considering the two defined equilibria.
Theorem 4 (Local stability at C 0 ) The system (1) at C 0 = (S 0 , Proof The Jacobian matrix is obtained for the system (1) as follows.
The Jacobian matrix at C 0 is computed as follows Consider |J − I| = 0 , and hence Vol:.( 1234567890) Consequently, we draw the following result It is clear that, system (1) is locally asymptotically stable at C 0 when R 0 < 1, as desired.

Proof
The Jacobian matrix at C 1 of the system (1) is as follows.
Consider |J − I| = 0 , and hence After solving above determinant, where Since, A 1 , A 0 both are positive when R 0 > 1 , by the Routh-Hurwitz criterion for the second order polynomial, the system (1) is locally asymptotically stable at C 1 when R 0 > 1, as desired.
The following lemma is provided to improve the global stability analysis of the system.
We tackle now the global asymptotic stability of the system at the equilibrium points 46 .
Theorem 6 (Global stability at C 0 ) The system (1) at C 0 = (S 0 , Proof Firstly, we define the Lyapunov function as.Observe that 0 c D α t L < 0 when R 0 < 1.Hence, the system is globally asymptotically stable at the disease-free equilibrium C 0 .Theorem 7 (Global stability at C 1 ) The system (1) at C 1 = S * , I * , V * , R * is globally asymptotically stable if R 0 > 1.
Proof The proof of this theorem is similar to the previous theorem.To prove this theorem, we construct the Lyapunov functional at C 1 as: By using Lemma 1: After some simplifications, we get: Vol:.( 1234567890 www.nature.com/scientificreports/ Therefore, the system is globally asymptotically stable at the endemic equilibrium C 1 , as desired.

Numerical simulations
In this section, parametric values for the simulations are described.This section is devoted for investigating the key properties of the simulated graphs against the set of parametric values as mentioned in Table 1.Further, these graphs are plotted when the disease is prevailing in the human population and attains the endemic steady state with in due course of time.To analyze the dynamics of the susceptible individuals at endemic equilibrium, four appropriate values of α are selected.We have practi- cally noticed in the outburst of COVID-19 that it propagated with different rates in the different countries of the world.These rates are biologically meaning full as every individual or group of individuals have different physical environments, immunity levels, health conditions and leaving standards etc.In this regard, fractional parameter α handles the rate of spread for the corona virus.Figure 1 shows the flow map of coronavirus model.All the graphs in Fig. 2 depicts the dynamical behavior of the susceptible population.Every trajectory of the curved graph reflects that how the susceptible compartment attains the endemic steady state for different values of α , with a particular set of parametric values including the crowding effect.It can be observed that each graph indicates the convergence towards the exact fixed point.Also, each graph possesses a specific trajectory and rate of convergence according to the fractional order parameter α.
Similarly, the curved graph in Fig. 3 shows the convergence of infected compartment, towards the exact fixed point.Infected graphs capture the dynamical evolution in the presence of viral load in the infected populace for the highlighted values of parameter.It is worth mentioning that each parameter has its own biological meaning and importance as described in the description of the model in section "Description of the model".All the graphs show the different phases of the disease dynamics according to the value of α .If value of α is less then the rate of convergence is slow and vice versa.Moreover, the graphical behavior is in line with the real behavior of the continues system.

Symbols
Value/per day Source The set of the graphs in Fig. 4 demonstrate the vital behavior of vaccinated the populace in Covid-19 model.These graphs describe the evolutionary behavior of the vaccinated individuals during the propagation of Covid-19.Every graph has the specific trajectory and a different rate of convergence to attain the required steady state.Thus, fractional order parameter α plays an important role in describing the disease dynamics of every compartment.
Finally, the graphs in Fig. 5 shows the dynamical behavior of the recovered population, when the Covid-19 disease persists in the community.The graphs bring an important fact into the lime light that the fast recovery rate can be captured by a large value of α .Similarly, the slower rate of recovery may be represented by a smaller value of α.

Effect of vaccination
Vaccine of any disease plays a vital role in controlling the disease.Likewise, vaccination is an important strategy to control the Covid-19.The graphs in Fig. 6 depict the role of vaccinating for stopping the spread of the virus.All the graphs are drawn by choosing the α = 0.8 and values of vaccination parameter δ 1 are shown in the values of vaccination.When the value of δ 1 is less i.e. 0.1 , the number of infected individuals are greater as compared to the number of individuals against the δ 1 = 0.2, 0.3 and 0.4.
Consequently, the number of infected individuals I(t) are inversely proportional to the vaccination parameters δ 1 .Hence, the parameter δ 1 imparts a significant role to slow down the disease dynamics and the transmission of the virus can be controlled effectively by vaccinating a large part of population.
The main advantage of the theory, applied in this work is that it may be applied to examine the positivity, boundedness, stability analysis and uniquely existence of the solution for the fractional order epidemic models.On the other hand, if we look on the drawbacks or disadvantages of this study, one can notice that any epidemic model cannot be captured all the factors related to disease dynamics.So, there is a chance of error and slight deviation from the actual behavior of the disease.

Grunwald-Letnikov non-standard finite difference method
Let us now apply the Grunwald-Letnikov definition to the following fractional differential equation using the Caputo operator: And assume there exists a unique solution y = y(T ) in the interval [0, T ] and let y k denote the approxima- tion of the true solution y(T k ) .Then the explicit or implicit Grunwald-Letnikov method for an equidistant grid is given by:   www.nature.com/scientificreports/

Discussion
In our current world, interdisciplinary research stands as a vital tool for ensuring the well-being and standard of living for all humanity.Over time, mathematical prediction modelling has become indispensable for understanding the behavior of epidemics, aiding policymakers in making crucial decisions and preparing for future challenges.Our study aimed to develop a mathematical model to predict the dynamics of COVID-19, employing the SIVR modelling framework.Scientists globally are striving to identify effective control measures to curb the spread of the COVID-19 virus.Strategies such as social distancing, quarantine for exposed individuals, isolation of infected persons, and mask-wearing are widely adopted.The primary objective in containing the virus is to minimize contact between susceptible and infected individuals, thereby reducing the transmission rate from the susceptible (S) to the infected (I) class.Graphs plotted at the endemic equilibrium point illustrate the dynamical behavior of the susceptible population for varying fractional order parameter α, as depicted in Fig. 2. Each curve's trajectory demonstrates the susceptible compartment's convergence to the endemic steady state for different α values and specific parameter sets.Notably, the graphs display distinctive trajectories and convergence rates corresponding to α values, influencing the disease dynamics.Similarly, Fig. 3 exhibits curved graphs demonstrating the convergence of the infected compartment towards the fixed point, showcasing different phases of disease dynamics based on α values.Lower α values indicate slower convergence rates and vice versa, aligning with real-world system behaviors.
Figure 4 presents a set of graphs illustrating the critical role of vaccination in the COVID-19 model.These graphs depict the evolutionary behavior of vaccinated individuals during the disease propagation, with each graph exhibiting unique trajectories and convergence rates.The fractional order parameter α significantly influences the disease dynamics of each compartment.Furthermore, Fig. 5 displays the dynamic behavior of the recovered population in persistent COVID-19 scenarios.The graphs highlight that a higher α value captures faster recovery rates, while lower α values represent slower recovery rates.Vaccination plays a pivotal role in disease control, particularly in managing COVID-19.Figure 5 illustrates the impact of vaccination on halting virus spread, with graphs drawn at α = 0.8 and varying vaccination parameter values δ 1 .Lower δ 1 values result in a greater number of infected individuals compared to higher values of δ 1 (e.g., δ 1 = 0.2, 0.3, and 0.4).Thus, the number of infected individuals (I(t)) is inversely proportional to vaccination parameters δ 1 , underscoring its significant role in mitigating disease dynamics and effectively controlling virus transmission through widespread vaccination efforts.

Concluding remarks
The fractional order COVID-19 model is developed to study the virus propagation in the population.To this end, analytical and numerical results are established.These results ascertained the unique positive solution of the underlying model.For numerical solutions, the GL-NSFD scheme is formulated to study the numerical aspects of the proposed scheme.This scheme has provided the positive and bounded numerical solutions, which are the salient features of the compartmental models.It is also identified that the disease model has two equilibrium state i.e. virus free and virus existing state.The simulated graphs and rendered that the endemic equilibrium may be obtained physically.Moreover, the disease-free state may also be attained in real scenario.Stability analysis is arranged and results for local and global stability are established for different values of R 0 i.e.R 0 < 1 or R 0 > 1 .The results have shown that the fractional COVID-19 system preserves the stability at both the steady states depending upon the values of R 0 .The crowding effect of the corona virus on the disease dynamics is presented.The role of vaccination for controlling the virus spread is also studied graphically.It is observed that the virus may be controlled, significantly if maximum portion of the population is vaccinated.Moreover, the SOP's planned by the world health organization can reduce the rate of virus propagation in the society.Consequently, our proposed fractional epidemic model for the COVID-19 may be considered for studying the disease dynamics in the community.In addition, crowding effect and vaccination strategy can play a vital role devising the health policies and pre cautionary measures.Moreover, the fractional order epidemic model will be helpful for studying the dynamics of various disease in future.
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Figure 1 .
Figure 1.Flow map of coronavirus model.

Figure 2 .
Figure 2. Graphical behavior of susceptible population (disease present in the population) for various values of fractional order α.

Figure 3 .
Figure 3. Graphical behavior of infected population (disease present in the population) for various values of fractional order α.

Figure 4 .
Figure 4. Graphical behavior of vaccinated population (disease present in the population) for various values of fractional order α.

Figure 5 .
Figure 5. Graphical behavior of recovered population (disease present in the population) for various values of fractional order α.

Figure 6 .
Figure 6.Effect of vaccination on active cases when fractional order value α = 0.8.