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Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates
Tiberiu Harko, Francisco S. N. Lobo, M. K. Mak
TL;DR
The paper addresses the lack of exact analytical solutions for SIR epidemic dynamics and their birth–death extension. It derives parametric solutions for the standard model, reduces the vital-dynamics system to an Abel equation, and reports exact agreement with numerical integration for the standard case. The Abel formulation also yields a parametric solution for the model with vital dynamics.
Problem
The paper considers how to obtain exact analytical solutions for the standard SIR model and its nonlinear extension with births and deaths.
Method
The standard SIR system is solved in parametric form, while the equal-birth-and-death system is reduced to an Abel equation solved by an iterative approach.
Results
The standard model’s analytical solution exactly reproduces numerical integration, and the vital-dynamics model has a general exact parametric representation once the Abel equation is solved.
Takeaways & Limitations
The Abel reduction simplifies analysis of the SIR model with vital dynamics, while exact parametric forms describe both model versions.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper introduces the fixed-population SIR model and its vital-dynamics extension, motivating an exact analytical treatment alongside prior numerical approaches. It frames the extension with equal birth and death rates as a nonlinear system that can be reduced to an Abel equation.
- Model formulation: The SIR model divides a fixed population into susceptible, infected, and recovered compartments.The compartment variables are x(t), y(t), and z(t), with x(t)+y(t)+z(t)=N.
- Model formulation: The infection and recovery transitions are governed by the contact rate β and recovery rate γ.The paper interprets γ as the inverse infection duration, γ = 1/D.
- Vital dynamics: New susceptible arrivals and deaths motivate extending the model to equal birth and death rates.This extension is presented for diseases such as measles, where susceptible individuals enter the population.
- Prior work: Earlier studies analyzed the epidemic equations numerically using decomposition, iteration, homotopy perturbation, and differential transformation methods.The introduction also cites stochastic and network-based epidemic-model studies.
- Paper objective: The paper derives an exact parametric SIR solution and reduces the vital-dynamics system to an Abel equation solved iteratively.The stated aim is to obtain exact parametric forms for both model versions.
II. THE EXACT SOLUTION OF THE SIR EPIDEMIC MODEL
The derivation begins by combining the SIR equations to obtain a population-conservation relation. Positivity of the compartments then identifies the total population as a positive integration constant.
- Population conservation: Adding the three SIR equations yields a differential equation for the total population.The resulting relation is immediately integrated in the paper.
- Population conservation: The integrated relation shows that x(t), y(t), and z(t) remain positive for t ≥ 0 under the stated conditions.The total population N=N1+N2+N3 is therefore an arbitrary positive integration constant.
- Population conservation: The constant-population result is consistent with the standard SIR model’s three-compartment fixed-population assumption.The paper explicitly connects the integration constant to the model interpretation.
A. The general evolution equation for the SIR model
The standard SIR system is transformed into a single second-order differential equation by differentiating one equation and eliminating the other compartment variables. The resulting scalar equation is equivalent to the original three-dimensional system.
- Scalar reduction: Differentiating the first SIR equation with respect to time produces a second-order differential equation.The prime denotes differentiation with respect to time.
- Scalar reduction: Substituting the original equations into the differentiated relation transforms it into an equation involving the remaining model quantities.This substitution is the next algebraic step in the reduction.
- Scalar reduction: Eliminating y from the first and third equations yields an integrable relation for the susceptible variable.The paper integrates this relation and introduces a positive integration constant x0.
- Scalar reduction: The derived scalar equation describes the spread of a non-fatal disease in a given population.The paper obtains it by inserting the preceding relations into the reduced equation.
- Equivalence: The resulting equation is equivalent to the original SIR system of three differential equations.Thus, the scalar formulation retains the dynamics of the full model.
B. The general solution of the evolution equation of the SIR model
The paper solves the reduced SIR evolution equation through successive transformations and expresses the complete solution parametrically. For selected initial values and rates, the analytical solution exactly reproduces numerical integration.
- Parametric solution: A new function u(t) is introduced to solve the nonlinear reduced differential equation.This substitution begins the parametric solution construction.
- Parametric solution: The transformed equation becomes a Bernoulli equation with a general solution containing an arbitrary integration constant.The time variable is then represented through an integral involving the transformed solution.
- Parametric solution: The complete exact SIR solution is obtained in parametric form with u as the parameter.The arbitrary time origin may be chosen as t0=0 without loss of generality.
- Parametric solution: C1 is a negative integration constant determined alongside u0 by the model parameters and initial conditions.The solution remains valid for arbitrary β, γ, and admissible initial conditions.
- Numerical comparison: For N1=20, N2=15, N3=10, β=0.01, and γ=0.02, the analytical solution perfectly reproduces numerical integration.The paper also reports agreement with earlier numerical results.
- Vital-dynamics extension: The vital-dynamics model is likewise analyzed through a single first-order Abel equation and an iterative solution approach.This extends the reduction strategy beyond the standard SIR model.
III. THE SIR MODEL WITH EQUAL DEATH AND BIRTH RATES
This section extends the simple SIR model by incorporating equal birth and death rates and presents numerical trajectories for that vital-dynamics system.
- The simple SIR model is extended by including equal rates of births and deaths.
- The total-population equation introduces an arbitrary integration constant N0, which is fixed to N0 = 0 to keep the total number of individuals constant.
- For the simple SIR model with β = 0.01, γ = 0.02, and initial conditions (20, 15, 10), numerical and analytical solutions completely overlap.
- The analytical solution also gives z(t) variations for four different initial-condition combinations at β = 0.01 and γ = 0.02.
- The section represents the vital-dynamics trajectories in Fig. 3 using numerical integration of the governing differential equations.
- For β = 0.01, γ = 0.02, and initial conditions (20, 15, 10), the vital-dynamics model’s x(t), y(t), and z(t) variations are plotted.
A. Qualitative properties of the SIR model with vital dynamics
The SIR systems are two-dimensional on the invariant plane x + y + z = N, with simple dynamics and no chaotic behavior. For vital dynamics, the equilibrium is shown to be globally asymptotically stable using a Lyapunov function.
- Both the simple SIR model and the vital-dynamics model are two-dimensional systems in the invariant plane x + y + z = N.
- Neither model exhibits chaotic behavior in the invariant plane, and the vital-dynamics system has no chaotic attractor outside it because trajectories approach the plane exponentially.
- The systems include a bifurcation at βN = γ + µ.
- For µ ≠ 0 and β > 0, the equilibrium is described as a global attractor that is a stable node or focus.
- For positive initial x(0) and y(0), the equilibrium (x*, y*) is rigorously shown to be globally asymptotically stable using the Lyapunov direct method.
- After scaling by population size, the paper defines L(x, y) = x − x*ln x + y − y*ln y on x, y ∈ (0, 1).
- The existence of L as a Lyapunov function establishes global asymptotic stability of the equilibrium point.
B. The evolution equation of the SIR models with vital dynamics
The vital-dynamics SIR equations are transformed into a second-order evolution equation and then reduced to a first-order Abel equation. The resulting Abel formulation is equivalent to the original system and recovers the no-death case when µ = 0.
- The vital-dynamics SIR system is differentiated and manipulated to derive a higher-order differential equation for its evolution.
- The derived evolution equation is integrated with initial conditions z(0) = N3, z′(0) = γN2 − µN3, and z′′(0) = βγN1N2 − γ(γ + 2µ)N2 + µ2N3.
- A set of transformations is introduced to simplify the evolution equation before reducing it to a lower-order form.
- The transformed equation takes the form of the standard Abel type first-order differential equation of the first kind.
- The Abel equation uses the initial condition w(γN2) = 1/γ [βN1 − (γ + µ)]N2.
- Introducing v reduces the Abel equation to an equation equivalent to the nonlinear SIR system, with v(γN2) = 1/[βN1 − (γ + µ)].
- When µ = 0, the Abel equation becomes separable, and the resulting equation is equivalent to the SIR model without deaths.
D. The iterative solution of the Abel equation
The Abel equation is solved iteratively through successive linear approximations, then used to represent the SIR model with vital dynamics parametrically. At twenty iterations, the resulting solution approximately overlaps the numerical solution.
- Iterative approximation: The n-th Abel-equation approximation is constructed by solving a linear differential equation after replacing nonlinear terms with lower-order approximations.The first-order approximation substitutes Ψ0 for nonlinear terms containing Ψ; the same iterative structure extends to order n.
- Iterative approximation: The Abel equation’s general solution is obtained in a power-series-style iterative form.The construction begins with a zeroth-order solution and proceeds through higher-order corrections.
- Parametric SIR solution: The SIR variables are recovered from the Abel solution through v(ψ) = e^Ψ(ψ) and w(ψ) = e^Ψ(ψ)/ψ, followed by time evolution in the parameter ψ.Initial conditions determine integration constants for the reconstructed variables.
- Numerical comparison: After twenty iterative steps, the Abel-based approximation and the numerical solution approximately overlap.Figure 4 compares the numerical y(t) curve with approximations at n = 1, 2, 3, 5, 10, and 20 for y(0) = 15.
IV. CONCLUSIONS
The paper derives exact parametric solutions for the SIR model and reduces the vital-dynamics version to an Abel equation. Numerical investigation confirms the exact solution for the basic model, while the Abel reduction simplifies analysis of the extended model.
- Conclusions: The SIR model without births and deaths has an exact analytical solution in parametric form that reproduces the model’s numerical solution.The paper also investigates numerical properties of this exact solution for fixed parameter values.
- Conclusions: The nonlinear SIR system with births and deaths is reduced to an Abel equation.The Abel equation can be studied with semianalytical or numerical methods, simplifying analysis of the model.
- Conclusions: The exact solution is presented as useful for experiments using natural initial conditions to study epidemic spread and control.This stated application connects the analytical solution to experimental investigation of infectious-disease dynamics.