Stability behavior of a nonlinear mathematical epidemic transmission model with time delay

Goel, Kanica; Nilam, .

doi:10.1007/s11071-019-05276-z

Cited by 31 publications

(20 citation statements)

References 35 publications

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“…It can be employed for both ordinary differential equations and delay differential equations. Many examples of its use can be found in economics [10][11][12][13][14] and in other sciences [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

2020

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In this paper, we consider a model of economic growth with a distributed time-delay investment function, where the time-delay parameter is a mean time delay of the gamma distribution. Using the linear chain trick technique, we transform the delay differential equation system into an equivalent one of ordinary differential equations (ODEs). Since we are dealing with weak and strong kernels, our system will be reduced to a three-and four-dimensional ODE system, respectively. The occurrence of Hopf bifurcation is investigated with respect to the following two parameters: time-delay parameter and rate of growth parameter. Sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation are presented in case of time-delay parameter. Numerical studies with the Dana and Malgrange investment

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Section: Introductionmentioning

confidence: 99%

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

2020

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“…The compartmental mathematical modeling of infectious diseases is a powerful tool that focuses on predicting, assessing, and controlling potential outbreaks. Kermack and McKendrick [9] proposed a compartmental SIR (susceptible-infected-recovered) epidemic model, and following up the work of Kermack and McKendrick, many researchers studied epidemic models with different modifications (for instance, SIS [10], SIR [11][12][13][14][15][16], SIRS [17], SVEIR [18], SEIR [19], SVIRS [20], SAAIR [21], SFIR [22], etc. ).…”

Section: Introductionmentioning

confidence: 99%

“…The classical Kermack and McKendrick epidemic model [9] addresses the mass-action incidence rate in the disease transmission cycle, suggesting that the probability of infection of a susceptible is analogous to the number of contacts with infected individuals, which is not realistic for a large population. Therefore, to model the realistic scenario, several authors [16,19,[22][23][24][25] focused on considering nonlinear incidence rates such as Holling type II, Monod-Haldane, Beddington-DeAngelis, etc. Crowley and Martin [26] introduced an incidence rate of the form β S I (1+αS)(1+γ I ) , which includes the forms of other incidences also, for instance, bilinear incidence rate if α = 0, γ = 0; saturated incidence rate with susceptibles if γ = 0; Holling-II incidence rate if α = 0.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a logistic growth epidemic model with two explicit time-delays, the nonlinear incidence and treatment rates

Goel

Kumar

Nilam

2021

J. Appl. Math. Comput.

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In the present study, a time-delayed SIR epidemic model with a logistic growth of susceptibles, Crowley-Martin type incidence, and Holling type III treatment rates is proposed and analyzed mathematically. We consider two explicit time-delays: one in the incidence rate of new infection to measuring the impact of the latent period, and another in the treatment rate of infectives to analyzing the effect of late treatment availability. The stability behavior of the model is analyzed for two equilibria: the disease-free equilibrium (DFE) and the endemic equilibrium (EE). We derive the threshold quantity, the basic reproduction number R 0 , which determines the eradication or persistence of infectious diseases in the host population. Using the basic reproduction number, we show that the DFE is locally asymptotically stable when R 0 < 1, linearly neutrally stable when R 0 = 1, and unstable when R 0 > 1 for the time-delayed system. We analyze the system without a latent period, revealing the forward bifurcation at R 0 = 1, which implies that keeping R 0 below unity can diminish the disease. Further, the stability behavior for the EE is investigated, demonstrating the occurrence of oscillatory and periodic solutions through Hopf bifurcation concerning every possible grouping of two time-delays as the bifurcation parameter. To conclude, the numerical simulations in support of the theoretical findings are carried out.

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“…In 1979, Cooke introduced a "time delay" to represent the disease incubation period in studying the spread of an infectious disease transmitted by a vector in [1]. Since then, many authors have incorporated time delays in epidemic models in different scenarios, such as vaccination period [2], asymptomatic carriage period [3], immune period [4] and incubation period or latent period [3][4][5][6][7]. More precisely, in [3], a disease transmission model with two delays in incubation and asymptomatic carriage periods is investigated.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], a latent period and relapse are considered in a general mathematical model for disease transmission. In [7], the authors studied a time-delayed SIR model with nonlinear incidence rate and Holling functional type II treatment rate for epidemic transmission. Also, many authors studied time-delayed epidemic models with vaccination [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate

Al-Darabsah

2020

Nonlinear Dyn

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In this paper, we study the dynamics of an infectious disease in the presence of a continuous-imperfect vaccine and latent period. We consider a general incidence rate function with a non-monotonicity property to interpret the psychological effect in the susceptible population. After we propose the model, we provide the well-posedness property and calculate the effective reproduction number . Then, we obtain the threshold dynamics of the system with respect to by studying the global stability of the disease-free equilibrium when and the system persistence when . For the endemic equilibrium, we use the semi-discretization method to analyze its linear stability. Then, we discuss the critical vaccination coverage rate that is required to eliminate the disease. Numerical simulations are provided to implement a case study regarding data of influenza patients, study the local and global sensitivity of , construct approximate stability charts for the endemic equilibrium over different parameter spaces and explore the sensitivity of the proposed model solutions.

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Stability behavior of a nonlinear mathematical epidemic transmission model with time delay

Cited by 31 publications

References 35 publications

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

Bifurcations in an economic growth model with a distributed time delay transformed to ODE

Stability analysis of a logistic growth epidemic model with two explicit time-delays, the nonlinear incidence and treatment rates

Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate

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