key: cord-0857162-3oquc3lo authors: Mahata, Animesh; Paul, Subrata; Mukherjee, Supriya; Roy, Banamali title: Stability analysis and Hopf bifurcation in fractional order SEIRV epidemic model with a time delay in infected individuals date: 2022-02-02 journal: Partial Differential Equations in Applied Mathematics DOI: 10.1016/j.padiff.2022.100282 sha: f6475db6c050e875760b50a60fe57e54339e6911 doc_id: 857162 cord_uid: 3oquc3lo Infectious diseases have been a constant cause of disaster in human population. Simultaneously, it provides motivation for math and biology professionals to research and analyze the systems that drive such illnesses in order to predict their long-term spread and management. During the spread of such diseases several kinds of delay come into play, owing to changes in their dynamics. Here, we have studied a fractional order dynamical system of susceptible, exposed, infected, recovered and vaccinated population with a single delay incorporated in the infectious population accounting for the time period required by the said population to recover. We have employed Adam-Bashforth-Moulton technique for deriving numerical solutions to the model system. The stability of all equilibrium points has been analyzed with respect to the delay parameter. Utilizing actual data from India COVID-19 instances, the parameters of the fractional order SEIRV model were calculated. Graphical demonstration and numerical simulations have been done with the help of MATLAB (2018a). Threshold values of the time delay parameter have been found beyond which the system exhibits Hopf bifurcation and the solutions are no longer periodic. Vaccination is one of the most effective measures in the prevention and control of highly contagious diseases like chicken pox, small pox, HIV, SARS, Swine flu, polio etc. It has been proved that vaccination may be considered as a key component in the anti-spread drive of such diseases. Among other measures, complete lockdown, semi lockdown, rationing, improvement of health services etc. may be mentioned. Considering the formidable challenge posed by the social, cultural, economic, demographic and geographical impact of such diseases on human population, it becomes necessary to discover methods of their prevention. From the inception of viral invasion into human community, scientists have constantly made efforts in the study of causes of newly infected cases of susceptible and exposed population, and the effect of vaccination on recovered population. Mathematical modelling of these epidemic diseases is very common in the related research where in the total population is primarily compartmentalized into the susceptible individuals ( ), the exposed individuals ( ), the infected individuals ( ) and the The construction of the model, as well as the establishing of non-negativity and boundedness of the solution and the calculation of the basic reproduction number, are all covered in Section 2. Section 3 comprises of the stability analysis of equilibrium points. Section 4 consists of the numerical solution of the model using Adam-Bashforth-Moulton method. Numerical Simulation using MATLAB is presented in Section 5. Section 6 consists of the conclusion. The total population ( ) is compartmentalized into five classes, namely, the susceptible individuals ( ), the exposed individuals ( ), the infected individuals ( ), the recovered individuals ( ) and the vaccinated individuals ( )at any time ≥ 0. Thus ( ) = ( ) + ( ) + ( ) + ( ) + ( ). where − 1 < < . Let ℎ ∈ [ , ], < . The fractional derivative in Caputo sense or order 0 < ≤ 1 is defined as where the normalization function is denoted by ( ) with (0) = (1) = 1. The Laplace transform for the fractional operator of order 0 < ≤ 1 is defined as (2.5) One-parametric and two-parametric Mittag-Leffler functions are described as where 1 , 2 ∈ ℝ + . Definition 6 For 1 , 2 ∈ ℝ + and ∈ ℂ × where ℂ denotes complex plane, then ( 1 , 2 ,…, ) calculated at the equilibrium points satisfies |arg( )| > 2 . Lemma 2 Let ℎ( ) ∈ ℝ + be a differentiable function. Then, for any > 0, [ℎ( ) − ℎ * − ℎ * ℎ( ) ℎ * ] ≤ (1 − ℎ * ℎ( ) ) (ℎ( )), ℎ * ∈ ℝ + , ∀ ∈ (0,1). The integral order model 46, 47 with vaccination as a dynamical variable is as follows: In this presentation, we analyze the model with time delay using Caputo operator of order 0 < ≤ 1. The time dimension of the system (2.8) is confirmed to be valid, even though both sides have dimension ( ) − . Let 0 = 0 and ignore the super script and the system becomes: (2.10) where ψ =( 1 , 2 , 3 , 4 , 5 ) ∈ , such that ( ) ≥ 0, = 1,2,3,4,5. Where B denotes the Banach space of continuous mapping from the interval [− 1 , 0] to ℝ 5 . We presume, by biological meaning, that ( ) > 0 for = 1,2,3,4,5. Proof We have ( + + + + )( ) = − 0 ( + + + )(t). (2.11) Appling Laplace transforms, we get Taking inverse Laplace transform, we have According to Mittag-Leffler function, (2.14) And hence the model (2.9) is bounded above by 0 . Thus , , , , and are all non-negative, and the model (2.9) is non-negative invariant. The basic reproduction number ℜ 0 provides the number of secondary cases induced by single susceptible individual. Using next generation matrix method 48, 49 , ℜ 0 can be determined from the maximum eigen value of ℱ −1 where, Therefore, the reproduction number ℜ 0 = 1 ( 0 + )( 0 + 1 )( 0 + 2 ) . (2.15) The disease-free equilibrium points 0 and the epidemic equilibrium point 1 of are obtained from ) and 1 = ( * , * , * , * , * ), . Now we consider the community matrix of the model (2.9) at 0 is given by Therefore the point 0 is locally asymptotically stable or unstable according as ℜ 0 < 1or ℜ 0 > 1. Putting 2 = in equation (3.4), then we have 5 + 4 4 + 3 3 + 2 2 + 1 + 0 = 0. Hence if ℜ 0 < 1, then Ƒ(t) < 0. As a result of LaSalle's extension to Lyapunov's principle 50, 51 , 0 is globally asymptotically stable and unstable if ℜ 0 > 1. Using . . ≥ . ., we get: Thus ( ) ≤ 0 for ℜ 0 > 1. Therefore 1 is globally asymptotically stable, according to LaSalle's Invariance Principle 51 . For fractional order initial value situations, the Adams-Bashforth-Moulton approach is the most commonly used numerical technique. Let ( ) = ( , ( ), ( − 1 )), ∈ [− 1 , 0], (0) = 0 , (4.1) = 0,1,2, … , ⌈ ⌉, ∈ ℕ where 0 ∈ ℝ, > 0 and is same as Volterra integral equation in the Caputo sense. Let ℎ =̂, = ℎ, = 0,1,2, … ,̂. Corrector formulae: Predictor formulae: We have studied and analyzed the dynamical behavior of the solutions of (2.9) using an extensive numerical simulation. In this section, we use MATLAB to analyze the solutions generated by Adam's-Bashforth-Moulton scheme. The results of model simulations and the associated findings have been classified as follows: In this case, we analyze the dynamical characteristics of all population for various fractional order with 1 = 0. In this case, we analyze the dynamical characteristics of all population for various fractional order with 1 = 0, 1 = 0.5 and 1 = 2. The values of parameters in Table 2 Figs. 4 (a) to 4 (e) shows the behavior of all individuals with time corresponding to 1 = 0 for different fractional order . Fig. 4 (a) depicts that the number of susceptible individuals increase when changes to 0.9 to 0.7. An increase value of leads to decrease in the exposed rate in the exposed population in Fig. 4 (b) . We see in Fig. 4 (c) that number of infected individuals increase when changes to 0.9 to 0.7. Fig. 4 (d) depicts that the number of recovered individuals increase with time when increases. The existence of the Hopf bifurcation of the model system (2.9) with fractional order = 1 is discussed in this case. The following set of parametric values is chosen: Fig. 12 . Diagram of a single parameter bifurcation with respect to 1 . We have studied the model (2.9) considering a single time delay parameter 1 . The stability analysis of the system depicts that point 0 of the system (2.9) is locally asymptotically stable when ℜ 0 < 1, and unstable when ℜ 0 > 1 in the absence of time delay. The endemic equilibrium 1 = ( * , * , * , * , * ) is locally asymptotically stable if ℜ 0 > 1 , when 1 = 0. However, in the presence of time delay parameter 1 , both the points 0 and 1 are asymptotically stable in the interval [0, 1 * ] where 1 * is given by 1 * = 1 10, −1 ( + 2 + 2 ). Numerical computations reveal that if 1 > 41.56 then the system (2.9) exhibits Hopf bifurcation. Thus, it becomes apparent that beyond the value of 1 * = 41.56 the dynamics of the system becomes unstable. It may be recalled that the time delay parameter was incorporated in (2.9) to justify the argument that the infected population will take some time to recover. When the time delay owing to the time period required by the infected individuals to recover from the disease surpasses a threshold value, the model described here produces a Hopf bifurcation around the endemic equilibrium point. An SVEIR model for assessing potential impact of an imperfect anti-SARS vaccine Global stability of an SEIR epidemic model with vaccination Multigroup SIR epidemic model with stochastic perturbation Asymptotic stability of a two-group stochastic SEIR model with infinite delays Search for adequate closed form wave solutions to space-time fractional nonlinear equations Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations Global stability of two-group SIR model with random perturbation The mathematical study of climate change model under nonlocal fractional derivative Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions SEIR order parameters and eigenvectors of the three stages of completed COVID-19 epidemics: with an illustration for Thailand Transmission of 2019-nCoV infection from an asymptomatic contact in Germany Outbreak of pneumonia of unknown etiology in Wuhan, China: the mystery and the miracle Forecasting the outcome and estimating the epidemic model parameters from the fatality time series in COVID-19 outbreaks The stability analysis and numerical simulation based on Sinc Legendre collocation method for solving a fractional epidemiological model of the Ebola virus Effectiveness of airport screening at detecting travellers infected with novel coronavirus (2019-ncov) Phase-adjusted estimation of the number of Coronavirus Disease Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing Dynamics of SIQR epidemic model with fractional order derivative Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak Now casting and forecasting the potential domestic and international spread of the 2019-ncov outbreak originating in wuhan, china: a modelling study SEIR epidemic model and scenario analysis of COVID-19 pandemic Delay differential equations with applications in population dynamics Mathematical models in population biology and epidemiology Influence of multiple time delays on bifurcation of fractional-order neural networks Fractional-order bidirectional associate memory (BAM) neural networks with multiple delays: The case of Hopf bifurcation Further exploration on bifurcation of fractional-order sixneuron bi-directional associative memory neural networks with multi-delays Bifurcation Properties for Fractional Order Delayed BAM Neural Networks Further investigation on bifurcation and their control of fractional-order bidirectional associative memory neural networks involving four neurons and multiple delays Impact of leakage delay on bifurcation in fractional-order complex-valued neural networks Delay-induced periodic oscillation for fractional-order neural networks with mixed delays Bifurcation of a delayed SEIS epidemic model with a changing delitescence and nonlinear incidence rate Hopf bifurcation of a delayed SIQR epidemic model with constant input and nonlinear incidence rate Stability and bifurcation analysis in a viral infection model with delays Mathematical analysis of an influenza a epidemic model with discrete delay The threshold of a stochastic delayed SIR epidemic model with temporary immunity Traveling waves in a delayed SIR epidemic model with nonlinear incidence The analysis of an epidemic model with time delay on scalefree networks Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model Stability analysis and estimation of domain of attraction for the endemic equilibrium of an SEIQ epidemic model Theory and Applications of Fractional Differential Equations A new definition of fractional derivative without singular kernel Laplace transform of fractional order differential equations. Electron Laplace transform and fractional differential equations Fractional-Order Nonlinear Systems: Modeling Aanlysis and Simulation Modeling the virus dynamics in computer network with SVEIR model and nonlinear incident rate Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delay Estimating the size of COVID-19 epidemic outbreak Dynamics analysis and optimal control strategy for a SIRS epidemic model with two discrete time delays Differential Equations and Dynamical Systems The authors wish to thank the anonymous referees for providing insightful remarks and suggestions that helped to improve the performance of this paper.