Exact Solution to a Dynamic SIR Model (original) (raw)
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Discrete Type SIR Epidemic Model with Nonlinear Incidence Rate in Presence of Immunity
WSEAS transactions on biology and biomedicine, 2020
Mathematical modeling is playing an incredible role for providing quantitative insight into multiple fields. It has already contributed to a better understanding of the mechanisms of various diseases. Mathematical modeling has gotten attention because modeling and simulation of any physical phenomena allows us for rapid assessment. So it is mainly used to describe the real phenomena which lead to design better prediction, management and control strategies.Infections and infectious diseases are massive burden on many societies, including the countries. Mathematical models narrating the population dynamics of infectious diseases have been playing an important role in better understanding epidemiological patterns and disease control for a long time. In many cases a simple mathematical model can reveal the nature of the infectious disease transmission, which plays an significant role in the control and prevention of the infectious diseases. Mainly Differential equations and difference equations are two allegory mathematical approaches to modeling epidemic dynamical systems. Fortunately, mathematical models are uniquely positioned to provide a tool amenable for rigorous analysis, hypothesis generation, and connecting results from isolated in vitro experiments with results from in vivo and whole organism studies. In particular, one of the most successful combinations of expertise is that of experimental and biological sciences with the mathematical and computational sciences. The use of mathematical models was proven to be fundamental towards advancing physics in the 20th century, and many are projecting mathematics to play a similar role in advancing biological discovery in the 21st century. Mathematical and computational models have already begun to play an increasingly large role in the advancement of biological and biochemical research because they make it possible to quantitatively bridge the gap between data
2017
We consider a SIR model with birth and death terms and time-varying infectivity parameter β (t). In the particular case of a sinusoidal parameter, we show that the average Basic Reproduction Number ¯ R o , introduced in [Bacaer & Guernaoui, 2006], is not the only relevant parameter and we emphasize the role played by the initial phase, the amplitude and the period. For a (general) periodic infectivity parameter β (t) a periodic orbit exists, as already proved in [Katriel, 2014]. In the case of a slowly varying β (t) an approximation of such a solution is given, which is shown to be asymptotically stable under an extra assumption on the slowness of β (t). For a non necessarily periodic β (t) , all the trajectories of the system are proved to be attracted into a tubular region around a suitable curve, which is then an approximation of the underlying attractor. Numerical simulations are given.
Exact and approximate analytic solutions in the SIR epidemic model
arXiv (Cornell University), 2020
In this work, some new exact and approximate analytical solutions are obtained for the SIR epidemic model, which is formulated in terms of dimensionless variables and parameters, reducing the number of independent parameters from 4 (I0, S0, , ) to 2 (i0 = I0/S0 and R0 = S0/). The susceptibles population is in this way explicitly related to the infectives population using the Lambert W function (both the principal and the secondary branches). A simple and accurate relation for the fraction of the population that does not catch the disease is also obtained. The explicit time dependences of the susceptibles, infectives and removed populations, as well as that of the epidemic curve are also modelled with good accuracy for any value of R0 using simple functions that are modified solutions of the R0 → limiting case (logistic curve). It is also shown that for small i0 (i0 < 10-2) the effect of a change in this parameter on the population evolution curves amounts to a time shift, their shape and relative position being unaffected.
Mathematical modeling of optimized SIRS epidemic model and some dynamical behavior of the solution
2017
In this paper, a generalized mathematical model of spread of infectious disease as SIRS epidemic model is considered as a nonlinear system of differential equation. We prove that for positive initial conditions the resulting equivalence system has positive solution and under some hypothesis, this system with initial positive condition, has a positive TTT-periodic solution which is globally asymptotically stable. For numerical simulations the fourth order Runge-Kutta method is applied to the nonlinear system of differential equations.
SIR model with general distribution function in the infectious period
Physica A: Statistical Mechanics and its Applications, 2009
The Susceptible-Infected-Removed or SIR model, as it was formulated by Kermack and McKendrick, is the key model for epidemic dynamics. Most applications of such a basic scheme use a constant rate for the removal term. However, that assumption corresponds to the rather unrealistic exponential distribution of infectious times. On the other hand, recent approaches, like numerical simulations, frequently assume a fixed and uniform duration for the infectious state-which is unrealistic too. The extreme assumptions in those different schemes are a hurdle that can frustrate any intention of drawing comparison between results from them. In the present contribution we study the delay equations for the SIR model, comparing the solutions for many typical cases with the simulation counterpart and with the standard SIR model. Using delay equations, where each infected individual is removed at a specific time after being infected, the dynamics for the infected and susceptible agree almost exactly with the numerical implementation. Even in the general case of distributed infective periods, the agreement is excellent.
In this paper, the exact analytical solution of the Susceptible-Infected-Recovered (SIR) epidemic model is obtained in a parametric form. By using the exact solution we investigate some explicit models corresponding to fixed values of the parameters, and show that the numerical solution reproduces exactly the analytical solution. We also show that the generalization of the SIR model, including births and deaths, described by a nonlinear system of differential equations, can be reduced to an Abel type equation. The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The general solution of the Abel equation is obtained by using a perturbative approach, in a power series form, and it is shown that the general solution of the SIR model with vital dynamics can be represented in an exact parametric form.
Global Behaviors of a Class of Discrete SIRS Epidemic Models with Nonlinear Incidence Rate
Abstract and Applied Analysis, 2014
We study a class of discrete SIRS epidemic models with nonlinear incidence rate ( ) ( ) and disease-induced mortality. By using analytic techniques and constructing discrete Lyapunov functions, the global stability of disease-free equilibrium and endemic equilibrium is obtained. That is, if basic reproduction number R 0 < 1, then the disease-free equilibrium is globally asymptotically stable, and if R 0 > 1, then the model has a unique endemic equilibrium and when some additional conditions hold the endemic equilibrium also is globally asymptotically stable. By using the theory of persistence in dynamical systems, we further obtain that only when R 0 > 1, the disease in the model is permanent. Some special cases of ( ) ( ) are discussed. Particularly, when ( ) ( ) = /(1 + ), it is obtained that the endemic equilibrium is globally asymptotically stable if and only if R 0 > 1. Furthermore, the numerical simulations show that for general incidence rate ( ) ( ) the endemic equilibrium may be globally asymptotically stable only as R 0 > 1.
Analysis of an Epidemic Spreading Model with Exponential Decay Law
Mathematical Sciences and Applications E-Notes, 2020
Mathematical modeling of infectious diseases has shown that combinations of isolation, quarantine, vaccine, and treatment are often necessary in order to eliminate most infectious diseases. Continuous mathematical models have been used to study the dynamics of infectious diseases within a human host and in the population. We have used in this study a SIR model that categorizes individuals in a population as susceptible (S), infected (I) and recovered (R). It also simulates the transmission dynamics of diseases where individuals acquire permanent immunity. We have considered the SIR model using the Caputo-Fabrizio and we have obtained special solutions and numerical simulations using an iterative scheme with Laplace transform. Moreover, we have studied the uniqueness and existence of the solutions.
A Sir Epidemic Model Structured by Immunological Variables
Journal of Biological Systems, 2013
Standard mathematical models for analyzing the spread of a disease are usually either epidemiological or immunological. The former are mostly ordinary differential equation (ODE)-based models that use classes like susceptibles, recovered, infectives, latently infected, and others to describe the evolution of an epidemic in a population. Some of them also use structure variables, such as size or age. The latter describe the evolution of the immune system/pathogen in the infected host — evolution that usually results in death, recovery or chronic infection. There is valuable insight to be gained from combining these two types of models, as that may lead to a better understanding of the severity of an epidemic. In this article, we propose a new type of model that combines the two by using variables of immunological nature as structure variables for epidemiological models. We prove the well-posedness of the proposed model under some restrictions and conclude with a look at a practical a...