Sir model differential equations solutions
Sir model differential equations solutions. Jul 18, 2022 · We will use the SIR model to address two fundamental questions: (1) Under what condition does an epidemic occur? (2) If an epidemic occurs, what fraction of a well-mixed population gets sick? Let \(\left(\hat{S}_{*}, \hat{I}_{*}, \hat{R}_{*}\right)\) be the fixed points of (4. Results for the simulated means were obtained using the BioSimulator package in Julia and averaging results over r = 100 runs. The proposed discrete model is also equivalent to a discrete susceptible–infected–recovered (SIR) model [16–25] for the spread of biological virus infections with a constant infectious period because quarantining all infected persons after a certain period can be equivalent to a temporary infection. Semi-time case. Poisson's equation is: dφ/dx 2 + dφ/dy 2 = f(x, y). The convergence of the series solution is given SIR Model Project. Keywords: SIR model, systems of ordinary differential equations, qualitative investigation, Lyapunov function, Lambert W-function, COVID-19 pandemics. Sep 20, 2022 · Barlow NS, Weinstein SJ. A qualitative analysis is carried out of the stationary Jun 28, 2018 · Stack Exchange Network. − bI, (2) dR. 1) The model has a susceptible group gr designated by S, an infected group I, and a recov overed group R with permanent immunity, rc is the intrinsic growth rate ra of susceptible, k is the carrying capacity of the susceptible in the absence of infective 4 YUTA TANAKA AND KEN-ICHI MARUNO which is the system of linear ordinary differential equations. Solving a system of differential equations means finding the values of the variables (here \(S\), \(I\) and \(R\)) at a number of points in time. / 10 # A grid of time points (in days) t = np. In this paper Abel differential equations play an important role in establishing the exact solution of SEIR differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an Abel Apr 20, 2021 · The SIRW model is a simple ordinary differential equation model that extends the classic SIR framework by adding a compartment (W) that tracks the pathogen concentration in water. The latter part of this section focuses on Abel differential equations play an important role in establishing the exact solution of the SEIRD differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an initial value problem for an Abel differential equation. Apr 1, 2014 · This research implemented two analytical techniques, namely the variational iteration method (VIM) and the homotopy perturbation method (HPM) for solving the SIR model of dengue fever, and found that the HPM and the RK4 were in excellent conformance. (4) This form allows you to solve the differential equations of the SIR model of the spread of disease. Feb 11, 2021 · Differential equations, initial conditions, exact and approximate solutions for the classical SIR model are presented. Since most processes involve something changing, derivatives come into play resulting in a differential equation. 3, γ = 0. After normalizing the dependent variables, the model is a system of two non-linear differential equations for the Sep 28, 2020 · 2. Use your helper application's differential equations solver, with the sample values of b = 1/2 and k = 1/3 in your worksheet, to generate graphical solutions of the SIR equations, starting from s(0) = 1, i(0) = 1. 2020; 408:132540. Jun 28, 2020 · Our SIR model is given by the same, simple system of three ordinary differential equations (ODEs) with the classic SIR model that can be easily implemented and used to gain a better understanding of how the COVID-19 virus spreads within communities of variable populations in time, including the possibility of surges in the susceptible populations. Additionally, we treat the generalization of the SIR model including births and natural Jun 6, 2020 · data or it is a good fit because this type of functions are the solution of the differential equations that govern the dynamics of the epidemic evolution. Part 5. This Abel equation Jul 17, 2020 · This is a tutorial for the mathematical model of the spread of epidemic diseases. Mar 1, 2023 · The SIR model of epidemic spreading can be reduced to a nonlinear differential equation with an exponential nonlinearity. The desired approximate solution can be obtained discretely or analytically based on individual preference. ) Jun 30, 2023 · The SIR model is a simple system of nonlinear differential equations that has a rich dynamic. History Epidemiological models based on first-order, linear and multidimensional differential equations can be traced back to Daniel Bernoulli []. −aS I N , (1) dI. You can modify the default SEIR model to an SIR model by turning off the incubation period. Delay differential equations (DDEs) such as the above have been a classic subject of Sep 17, 2020 · Developing algorithms for solving high-dimensional uncertain differential equations has been an exceedingly difficult task. The famous susceptible–infected–removed (SIR) and SIRD (plus dead) systems were developed by Sir Ronald Ross nearly a hundred years ago; see []. This manuscript presents an alternative numerical tool for the SIR and SEIR models. com/course/cs222. Beginning with the basic mathematics, we introduce the susceptible-infected-recovered (SIR) model. In section II, we begin by revisiting the transcendental equation for removals and discussing the essential properties of the SIR model. The independent variable is time t, measured in days. I'm wanting to find where dI/dt = 0, for the time wh. We show that the minimum wave speed of traveling waves for the three-dimensional non-monotonic system can S0 = N-I0-R0 # Contact rate, beta, and mean recovery rate, gamma, (in 1/days). 5025. Theorem2 (Yao and Chen (2013a)) Let Xt and Xα t be the solution and α-path of the uncertain differential equation dXt = f(t,Xt)dt +g(t,Xt)dCt, respectively. In this paper, we study the deviation of the spatial stochastic model from the Mar 16, 2024 · In the context of the SIR model’s nonlinear differential equations representing the changing rates of S, I, and R populations, ADM effectively breaks down these equations into a series of functions. May 1, 2019 · We investigate an epidemic model based on Bailey’s continuous differential system. i384100. We consider a general incidence rate function and the recovery rate as functions of the number of hospital beds. This is a suite for numerically solving differential equations written in Julia and available for use in Julia, Python, and R. Build a model similar to the SIR model in M-Box 26. For the SIR model with births and deaths we have shown that the non-linear system of differential equations governing it can be reduced to the Abel equation (57). 5). We construct a solution of the Cauchy problem for a system of two ordinary differential equations describing in integral form the concentration dynamics of infected and recovered individuals in an immune population. Parameter values are β = 0. 4×10 4, I 0 = 1 and parameter values η = 0. The technique of moment closure can be applicable to find out the approximate solutions of moment equations [26–28]. SEICR models have been used to control the viral infections. Ordinary Differential Equations. Apr 24, 2021 · The SIR model is a three-compartment model of the time development of an epidemic. We can solve the resulting set of linear ODEs, whereas we cannot, in general, solve a set of nonlinear differential equations. Simple formulas for probability of meeting an infected person and Jun 30, 2023 · Compartment models are implemented to understand the dynamic of a system. Nov 10, 2020 · In this example, we are free to choose any solution we wish; for example, \(y=x^2−3\) is a member of the family of solutions to this differential equation. Join me on Coursera: https://imp. udacity. The three classes, susceptible S, infected I and removed R, are all involved in the traveling wave solutions. We will investigate examples of how differential equations can model such processes. Mar 30, 2023 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Whenever there is a process to be investigated, a mathematical model becomes a possibility. Then the total number of vaccinated The SIR Model for Spread of Disease. linspace (0, 160, 160) # The SIR model differential equations. Nov 24, 2020 · The presented solution of the all-time SIR model can thus be used to replace the numerical solution of the SIR model. This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). This study investigates the application of differen tial transformation method and variational iteration method in finding the approxi mate solution of Epidemiology (SIR) model and revealed that both methods are in complete agreement, accurate and efficient for solving systems of ODE. To analyze the models, a numerical tool is required. The same idea could be applied to other compartment models. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities ear differential equations. This breakdown allows for the step-by-step determination of individual terms within the solution series. The model taken for the investigation is a compartmental model called SIR model, proposed by Kermack and McKendrick in 1927 ([10],[11]). 54. It is proven that there is a unique solution to the system. 6 are shown for different values of R0: The system of equations can be solved for several values of Jan 4, 2021 · In this paper, we study and analyze the susceptible-infectious-removed (SIR) dynamics considering the effect of health system. +aS I N. Jun 1, 2014 · The main properties of the exact solution were investigated numerically, and it was shown that it reproduces exactly the numerical solution of the model equations. The equations from the obtained sequence are treated by the Simple Equations Method (SEsM). An individual can be categorized as susceptible (S(t)), infected (I(t)), or removed (R(t), dead or cured), denoted by S, I and R respectively, along an independent variable, time . doi: 10. 15, and either N = 100 (top graphs) or N = 750 (bottom graphs). Hot Network Questions Sep 17, 2020 · An uncertain SIR model based on high-dimensional uncertain differential equations is built in Sect. The consistency of the two models is given by a law of large numbers. Analytical solution of the SIR-model for the temporal evolution of epidemics: part b. Jun 17, 2022 · The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive an explicit solution. Scott Dean, Kari Kuntz, T’Era Hartfield, and Bonnie Roberson Stability Analysis of an SIR Epidemic Model The SIR Model for Spread of Disease. This differential equation can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. This project will investigate the SIR model and use numeric methods to find solutions to the system of coupled, non-linear differential equations. Oct 1, 2019 · The study was further extended by considering non-integer differential equations, and the underlying model was studied both from fractional and fractal dimensions. Thus we can easily find the general solution of the system of linear differential equations (2. Mar 11, 2017 · The dashed curve is the ODE solution of I(t) in the SIR model and the other curves are four sample paths of the SIR CTMC model. Simple formulas for probability of meeting an infected person and calculations of the effective and basic reproduction numbers are given. Jun 28, 2024 · The solutions of the diffusion equation for the total population N t = δN xx is again known and R = N − S − I. S can be written as, [3]. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. May 31, 2024 · The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). The analytical solution is emphasized. 11) as follows: Stochastic Differential Equation for f SIR Model we consider the SIR model giv iven by : (2. Given a fixed population, let S(t) be the fraction that is susceptible to an infectious, but not deadly, disease at time t; let I(t) be the The SIR Model for Spread of Disease. SIR model. Our paper isfour-fold. The exact solution of the SIR epidemic model is presented in Section 2. This combination tries to give a physical explanation of infectious pathways. net/mathematics-f Apr 29, 2020 · The current need to model and predict viral epidemics motivates us to extend the application of asymptotic approximants to the commonly used Susceptible–Infected–Recovered (SIR) model. The results demonstrate that the SIR model with recovery time delay exhibits distinct patterns for the period of recovery. We will: Interpret the terms in a system of di erential equations. beta, gamma = 0. The present manuscript surveys new analytical results about the SIR model. Jun 13, 2022 · Comparison between the calculated and simulated means of SIR model outcomes in the stochastic SIR model simulated under the initial conditions S 0 = 3. 132540. Oct 18, 2021 · Both new models and the conventional SIR model were then used to simulate an infectious disease with a basic reproduction number (r0) of 3. This model is an example of compartment models mainly studied in epidemiology [1–3]. physd. The Dirichlet series satisfying the differential equation leads to an alternative numerical method to obtain the model’s solutions. Find equilibrium solutions and interpret them in the setting of the model. This model is formulated as a system of nonlinear ordinary differential equations. The solution of such type of differential equation is difficult and time consuming. This study analyzes the effect of recovery time delay on the SIR model using Fifth - Order Runge-Kutta (RK5) method, solving a system of delay differential equations (DDEs). Starting from the differential equation for the removed subjects presented by them in the original article, we rewrite it in a slightly different form to derive a formal solution, unless one integration. In the continuous time domain, we extend the classical model to time-dependent coefficients and present an alternative solution method to Gleissner’s approach. All new people in the population are susceptible. 2 How to Linearize a Model We shall illustrate the linearization process using the SIR model with births and deaths in a Sep 1, 2005 · Poisson's equation. Part 3: Euler's Method for Systems In Part 2, we displayed solutions of an SIR model without any hint of solution formulas. Most epidemics have an initial exponential curve and then gradually flatten out. 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 This project will investigate the SIR model and use numeric methods to find solutions to the system of coupled, non-linear differential equations. Nov 6, 2015 · We study the existence and nonexistence of traveling waves of a general diffusive Kermack–McKendrick SIR model with standard incidence where the total population is not constant. The SIR model measures the number of susceptible, infected, and recovered individuals in a host population. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The derived Dirichlet solu- The first two equations can be solved for I and S as in [3] The variation of I versus S can be seen from the figure provided Figure 2. SIR Model Variant Suppose in our SIR model that our population is growing at a rate proportional to the total population with rate r. [PMC free article] [Google Scholar] Schlickeiser R, Kröger M. If the coefficients are constant, both solution methods yield the same result. Through differentiation over t, Eqs. The Contact Number. Apr 1, 2020 · The derivation of the three differential equations for the SIR model (Susceptible, Infective, Removed) of an epidemic disease. Introduction We consider a classical epidemic model and we used to describe a chickenpox infection. Enter the following data, then click on Show Solution below. J Phys A Math Theor. Jul 28, 2009 · Introduction. The SIR (Susceptible-Infected-Recovered) model for the spread of infectious diseases is a very simple model of three linear differential equations. Subsequently, we present the numerical and exact analytical solutions of the SIR model. 9), (2. Then M{Xt ≤ Xα t, ∀t > 0}=α, (2) M{Xt Ordinary Differential Equation Solution. Introduction The SIR model of infectious disease propagation was proposed some 100 years ago by the Scot scientists William Ogilvy Kermack and Anderson Gray McKendrick [1]. DifferentialEquations. Conformable Fractional Derivative and Some Properties Apr 12, 2021 · The earlier analytical analysis (part A) of the susceptible–infectious–recovered (SIR) epidemics model for a constant ratio k of infection to recovery rates is extended here to the semi-time case which is particularly appropriate for modeling the temporal evolution of later (than the first) pandemic waves when a greater population fraction from the first wave has been infected. Having an analytical solution should be also advantageous to understand the effect of interventions reflected by a(t) much easier, and in a more transparent fashion. jl: Efficient Differential Equation Solving in Julia. The result starts with transforming the SIR model to an equivalent differential equation. Support Material; Examples. Phys D. 2. Coupled system of first-order nonlinear ODEs. This paper presents an $$\\alpha $$ α -path-based approach that can handle the proposed high-dimensional uncertain SIR model. 1 for this scenario. Apr 3, 2023 · Numerical solution to the set of differential equations shows that quarantining pre-symptomatic and asymptomatic patients is effective in controlling the pandemic. The SIR model is a representation that divides a population with respect to a disease’s impact on an individual over time. Initial conditions: Duration of solution: (Maximum duration is 1000. 1) The model has a susceptible group gr designated by S, an infected group I, and a recov overed group R with permanent immunity, rc is the intrinsic growth rate ra of susceptible, k is the carrying capacity of the susceptible in the absence of infective Building a Basic Model COVID-19 Parameters Implementations Extensions Two-Group Model General Recursive Model Differential SIR Model Definition (Differential SIR Model) The differential SIR model uses a system of differential equations to model disease. In this paper, we introduce a new solution of the SIR model by using the differential fractional transformation method. The model can be changed while retaining three compartments to give a steady-state endemic solution by adding some input to the S compartment. A particular solution can often be uniquely identified if we are given additional information about the problem. The EMOD generic simulation uses an SEIR-like disease model by default. Exact Solutions Using Lie Symmetries X 3 and X 4 If solutions of Equation 3 are bounded and the equilibrium X of Equation 4 is globally asymptotically stable, then any solution X(t) of Equation 3 satisfies X(t) !X as t !1: Therefore, we may now consider only the limit system. N = S + I + R. As our main goal, we establish an implicit time-discrete SIR Jan 5, 2021 · I've used ode45 to solve a simple SIR model, I've got the graph to work as I wish but I'm strugling to output any numerical values to discuss. As the first step in the modeling process, we identify the independent and dependent variables. Including births and deaths in the Apr 30, 2022 · – constitutes a more realistic representation of spreading epidemic dynamics than the standard SIR model. We can numerically solve differential equations in R thanks to the ode() function of the deSolve package. These values will depend on the parameters’ values. The SIR Model for Spread of Disease. May 15, 2020 · The proposed SIR model has three differential equations. It is also true for epidemic processes which does not form the closed system. 667 with and Jul 22, 2020 · This video explains the numerical technique pf solving a system of three nonlinear coupled ordinary differential equations. In SIR models, individuals in the recovered state gain total immunity to the pathogen; in SIRS models, that immunity wanes over time and individuals can become reinfected. May 1, 2019 · Exact solutions of the SEIR epidemic model are derived, and various properties of solutions are obtained directly from the exact solution. Scientific Reports - New Apr 28, 2022 · closed-form solutions to the full set of Kermack and McKendrick’s integro-differential equations; 2) An approximation to the full equations, known as the SIR (Susceptible, Infected, Recovered) model and its variants, are accepted as reasonable representatives of the full equations; and 3) the final number of uninfected individuals, ∞ Jun 2, 2024 · A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The differential equations of mean as well as higher-order moments could be easily obtained. FRACTIONAL DERIVATIVES AND INTEGRALS Fractional Calculus is a branch of mathematics that deals with the study of integrals and derivatives of non-integer orders, plays an outstanding role and have found several applications in large areas of Jun 4, 2023 · Build a model similar to the SIR model in M-Box 26. The SIR model has no analytical solution, but we can conduct numerical simulation for the approximate solution. The model consists of three compartments: “S” for Jul 1, 2023 · compartment models. This is modeled by SIR model of epidemic spread. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 2020. 6. First of all, it is attractive to test the SIR-model for the Painlevé property to estimate related analytical solution perspectives. Mar 15, 2021 · We present an analytical solution for the Susceptible–Infective–Removed (SIR) model introduced initially by Kermack–McKendrick in 1927. The SIR model is often represented with the following flow diagram that shows the three states (S, I, and R) and arrows depicting the direction of flow between the states. = dt. net Feb 1, 2021 · Introduction. 2, 1. Assume that the time-dependent function u = u (t) represents the vaccination rate. It examines how an infected population spreads a disease to a susceptible population, which transforms into a recovered population. In this research, the susceptible–infected–recovered (SIR) model of dengue fever is considered. 27 x 10-6, and r(0) = 0. The sum. The whole shape of the differential rate j depends on k only. The Painlevé test and the first integrals of the SIR model. The To construct and interpret models using systems of ordinary di erential equations in various settings. Oct 7, 2020 · Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. In 1766 Daniel Bernoulli published an article where he described the effects of smallpox variolation (a precursor of vaccination) on life expectancy using mathematical life table analysis (Dietz and Heesterbeek 2000). 0 and a herd immunity threshold (HIT) of 0. Join me on Coursera: https://imp. Dec 5, 2023 · We discuss the spread of a piece of news in a population. 4 YUTA TANAKA AND KEN-ICHI MARUNO which is the system of linear ordinary differential equations. This is called a particular solution to the differential equation. We consider two related sets of dependent variables. We also found the analytical solutions using Homotopy Perturbation Method (HPM) to find out the solution of nonlinear ordinary differential equation systems. Mar 10, 2014 · The reduction of the complex SIR model with vital dynamics to an Abel type equation can greatly simplify the analysis of its properties. The stability of fractional ordered model was studied by using the Hyers–Ulam (HU) approach. Sep 1, 2018 · The continuous model is described with a Riccati equation. We will work through the derivation of the model and some assumptions. 53. The SIR model and its adaptive versions have been used broadly to investigate the dynamic of a considered system. SIR epidemiology model. The nonlinear system of differential equations governing the SIR model with deaths is reduced to an Abel type equation, and the general solution of the model equations is obtained in an exact parametric form in Section 3. Expand Jun 21, 2019 · In this study, we have analyzed Hepatitis B virus to find out the analytical solutions for reducing the HBV infection. Dec 14, 2012 · This video is part of an online course, Differential Equations in Action. The model is defined by the following differential equations: dS. Continuous SIR model Weinvestigateasusceptible–infected–removed(SIR)modelproposedbyNormanBaileyin[1]oftheform Exact solution to a dynamic SIR model Aug 11, 2021 · The method presented in this work can easily be used to perform the non-trivial task of simultaneously fitting differential equation solutions to different epidemiological data sets, regardless of The numerical solution of the SIR model differential equations, and an explanation of the meaning of an epidemic. Sep 20, 2022 · The modified SIR model for the spread of an infectious disease and vaccination (SIRV) The standard notations in the SIR model are: S(t) denotes the number of susceptible, I(t) – infectives, and R(t) – removed individuals. This suggests the use of a numerical solution method, such as Euler's Method, which was discussed in Part 4 of An Introduction to Differential Equations. SIR model (3 differential equations) 1. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. def deriv (y, t, N, beta, gamma): S, I, R = y dSdt =-beta * S * I / N dIdt = beta * S * I / N-gamma * I dRdt = gamma * I return dSdt Apr 5, 2023 · This is a theoretical study of the SIR model — a popular mathematical model of the propagation of infectious diseases. differential equation ( ) ( ) ( ) (1) ( ) ( ) i t dt dr i t n i c s t dt di n i t c s t dt ds γ β γ β = = − =− This model is appropriate to viral diseases such as measles, mumps and rubella. The solutions of I vs. 7194, δ = . (3) Here S is the number of susceptible people, I is the number of infected (and therefore infectious) people, and R is the number of recovered (and therefore immune) people. We investigate all possible steady-state solutions of the model and their stability. 10), (2. We apply the $$\\alpha $$ α -path-based approach to calculating the uncertainty distributions and related expected values of the solutions The following theorem shows that the solution of an uncertain differential equation is related to a class of ordinary differential equations. 3. 5 estimates parameters and designs a 99-method to solve the proposed uncertain differential system. The Dirichlet series satisfying the Feb 16, 2023 · We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. +bI. Check out the course here: https://www. We study the difference between the integral Riemann-Liouville, differential Apr 24, 2018 · KeywordsFractional order, SIR model , Differential equations, Stability, Generalized Euler method. 1016/j. The idea that transmission and spread of infectious diseases follows laws that can be formulated in mathematical language is old. 3. This completes the exact solution for all the variables of the SIR model. Accurate closed-form solution of the sir epidemic model. Disadvantages: can be more difficult to explain & understand cannot be explicitly solved 27 Solving differential equations in R. In Part 4 we took it for granted that the parameters b and k could be estimated somehow, and therefore it would be possible to generate numerical solutions of the differential equations. (2) I(S) = S + 1 R0 lnS +1: The graphs of this equation 2. We have implemented two analytical techniques The SIR Model for Spread of Disease. Therefore we have used Euler’s method for solving these three differential equations. Infected individuals shed the pathogen into water compartments, and new infections arise both through exposure to contaminated water as well as by the classic SIR At its most basic level, the SIR model is a set of equations that describes the number (or proportion) of people in each compartment at every point in time. If equations that govern the behavior of the system by linear differential equations. Section 4 introduces α-path and proves the theorem for numerical solution, and Sect. A calibration on uncertain SIR model is discussed in Sect. Introduction to numerical analysis: modelling a badminton shuttlecock; Epidemic model SIR; Gravity; Damping harmonic oscillator; Lotka-Volterra equations; Molecular Dynamics; Zombie invasion model; Exercices; Image Processing; Optimization; Machine Learning Jun 1, 2014 · The present paper is organized as follows. (1) and appear to be reducible to the second-order nonlinear differential equation: Dec 4, 2021 · 2. Part 2: The Differential Equation Model . This allows us to obtain exact solutions to some of The only steady state solution to the classic SIR model as defined by the differential equations above is I=0, S and R can then take any values. This study investigates the application of differen tial transformation method and variational iteration Jan 13, 2021 · In this case, the conformable fractional-order system of the SIR model will be transformed to one conformable fractional equation and are solved using the variational iteration method and the conformable differential transformation method for numerical comparison. 1. We prove the existence, uniqueness, and boundedness of the model. gkwfjwe zhefk zti zle fzh kdcvu rknq iug mcrfp agut