On a Seir Epidemic Model with Delay (original) (raw)
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SEIR epidemic model with delay
The ANZIAM Journal, 2006
A disease transmission model of SEIR type with exponential demographic structure is formulated, with a natural death rate constant and an excess death rate constant for infective individuals. The latent period is assumed to be constant, and the force of the infection is assumed to be of the standard form, namely, proportional to I .t/=N .t/ where N .t/ is the total (variable) population size and I .t/ is the size of the infective population. The infected individuals are assumed not to be able to give birth and when an individual is removed from the I -class, it recovers, acquiring permanent immunity with probability f .0 ≤ f ≤ 1/ and dies from the disease with probability 1 − f . The global attractiveness of the disease-free equilibrium, existence of the endemic equilibrium as well as the permanence criteria are investigated. Further, it is shown that for the special case of the model with zero latent period, R 0 > 1 leads to the global stability of the endemic equilibrium, which completely answers the conjecture proposed by Diekmann and Heesterbeek.
SEIR epidemiological model with varying infectivity and infinite delay
Math. Biosci. Eng, 2008
A new SEIR model with distributed infinite delay is derived when the infectivity depends on the age of infection. The basic reproduction number R 0 , which is a threshold quantity for the stability of equilibria, is calculated. If R 0 < 1, then the disease-free equilibrium is globally asymptotically stable and this is the only equilibrium. On the contrary, if R 0 > 1, then an endemic equilibrium appears which is locally asymptotically stable. Applying a permanence theorem for infinite dimensional systems, we obtain that the disease is always present when R 0 > 1.
Stochastic Epidemic SEIRS Models with a Constant Latency Period
Mediterranean Journal of Mathematics, 2017
In this paper we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of r consecutive days, where r is a fixed positive integer, in the "exposed" individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, that under suitable assumptions the free disease equilibrium is still stable.
Delay epidemic models determined by latency, infection, and immunity duration
Mathematical Biosciences, 2024
We propose new single and two-strain epidemic models represented by systems of delay differential equations and based on the number of newly exposed individuals. Transitions between exposed, infectious, recovered, and back to susceptible compartments are determined by the corresponding time delays. Existence and positiveness of solutions are proved. Reduction of delay differential equations to integral equations allows the analysis of stationary solutions and their stability. In the case of two strains, they compete with each other, and the strain with a larger individual basic reproduction number dominates the other one. However, if the basic reproduction number exceeds some critical values, stationary solution loses its stability resulting in periodic time oscillations. In this case, both strains are present and their dynamics is not completely determined by the basic reproduction numbers but also by other parameters. The results of the work are illustrated by comparison with data on seasonal influenza.
An epidemic model with time delays determined by the infectivity and disease durations
Mathematical Biosciences and Engineering, 2023
We propose an epidemiological model with distributed recovery and death rates. It represents an integrodifferential system of equations for susceptible, exposed, infectious, recovered and dead compartments. This model can be reduced to the conventional ODE model under the assumption that recovery and death rates are uniformly distributed in time during disease duration. Another limiting case, where recovery and death rates are given by the delta-function, leads to a new point-wise delay model with two time delays corresponding to the infectivity period and disease duration. Existence and positiveness of solutions for the distributed delay model and point-wise delay model are proved. The basic reproduction number and the final size of the epidemic are determined. Both, the ODE model and the delay models are used to describe COVID-19 epidemic progression. The delay model gives a better approximation of the Omicron data than the conventional ODE model from the point of view of parameter estimation.
Global stability of a diffusive SEIR epidemic model with distributed delay
Elsevier eBooks, 2022
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
Advances in Difference Equations, 2010
This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-byimmunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.
Sveir epidemiological model with varying infectivity and distributed delays
Mathematical Biosciences and Engineering, 2011
In this paper, based on an SEIR epidemiological model with distributed delays to account for varying infectivity, we introduce a vaccination compartment, leading to an SVEIR model. By employing direct Lyapunov method and LaSalle's invariance principle, we construct appropriate functionals that integrate over past states to establish global asymptotic stability conditions, which are completely determined by the basic reproduction number R V 0 . More precisely, it is shown that, if R V 0 ≤ 1, then the disease free equilibrium is globally asymptotically stable; if R V 0 > 1, then there exists a unique endemic equilibrium which is globally asymptotically stable. Mathematical results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number. However, there is a necessary condition for successful elimination of disease. If the time for the vaccinees to obtain immunity or the possibility for them to be infected before acquiring immunity can be neglected, this condition would be satisfied and the disease can always be eradicated by some suitable vaccination strategies. This may lead to overevaluating the effect of vaccination.
The stability of an SIR epidemic model with time delays
Mathematical Biosciences and Engineering, 2005
In this paper, an SIR epidemic model for the spread of an infectious disease transmitted by direct contact among humans and vectors (mosquitoes ) which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point (if it exists) is globally stable with respect to a "weak delay". Some known results are generalized.
A SIR Epidemic Model with Primary Immunodeficiency and Time Delay
Journal of Informatics and Mathematical Sciences, 2017
In this paper, we have proposed a SIR (Susceptible-Infected-Recovered) epidemic model incorporating Primary Immunodeficiency and distributed delays. We discretize the model using a variation of Backward Euler method. We divide the susceptible population into two groups based on their immunity levels and apply the transmission rate for these two populations. We derive a threshold value known as the basic reproduction number denoted by \(R_0\). We have two equilibria namely the disease free and endemic equilibrium. We analyze the global stability of the disease free and endemic equilibrium using Lyapunov functional techniques. Finally, We prove our theoretical results using numerical simulations through MATLAB.