# Approximate and Parametric Solutions to SIR Epidemic Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Solutions to System (1)–(3)

#### 2.1. Upper and Lower Bounds of Solutions

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

#### 2.2. Solutions in Parametric Forms

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

## 3. Analytic Approximate Solutions via Adomian Decomposition Method

#### 3.1. Convergence Analysis

#### 3.1.1. Uniqueness Theorem

**Theorem**

**3.**

#### 3.1.2. Convergence Theorem

**Theorem**

**4.**

## 4. Numerical Explorations and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. List of the Adomian Polynomials A_{i} for N(u) = e^{u} [11,12,13]

## References

- Kermack, W.O.; McKendrick, A.G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A
**1927**, 115, 700–721. [Google Scholar] - Bacaër, N. A Short History of Mathematical Population Dynamics; Springer: London, UK, 2011. [Google Scholar]
- Yin, Z.; Yu, Y.; Lu, Z. Stability analysis of an age-structured SEIRS model with time delay. Mathematics
**2020**, 8, 455. [Google Scholar] [CrossRef] - Janssen, M.A.; Ostrom, E. Empirically based, agent-based models. Ecol. Soc.
**2006**, 11, 37. [Google Scholar] [CrossRef] - Mollalo, A.; Vahedi, B.; Rivera, K.M. GIS-based spatial modeling of COVID-19 incidence rate in the continental United States. Sci. Total Environ.
**2020**, 728, 138884. [Google Scholar] [CrossRef] - Sporns, O. Contributions and challenges for network models in cognitive neuroscience. Nat. Neurosci.
**2014**, 17, 652–660. [Google Scholar] [CrossRef] [PubMed] - Sun, J.; Chen, X.; Zhang, Z.; Lai, S.; Zhao, B.; Liu, H.; Wang, S.; Huan, W.; Zhao, R.; Ng, M.T.A.; et al. Forecasting the long-term trend of COVID-19 epidemic using a dynamic model. Sci. Rep.
**2020**, 10, 21122. [Google Scholar] [CrossRef] - Raval, M.; Sivashanmugam, P.; Pham, V.; Gohel, H.; Kaushik, A.; Wan, Y. Automated predictive analytics tool for rainfall forecasting. Sci. Rep.
**2021**, 11, 17704. [Google Scholar] [CrossRef] [PubMed] - Khan, H.; Mohapatra, R.N.; Vajravelu, K.; Liao, S.J. The explicit series solution of SIR and SIS epidemic models. Appl. Math. Comput.
**2009**, 215, 653. [Google Scholar] [CrossRef] - Barlow, N.S.; Weinstein, S.J. Accurate closed-form solution of the SIR epidemic model. Physica D
**2020**, 408, 132540, Erratum in Physica D**2021**, 416, 132807. [Google Scholar] [CrossRef] [PubMed] - Adomian, G. Solving Frontier Problems of Physics: The Decomposition Method; Kluwer Academic: Dordrecht, The Netherlands, 1994. [Google Scholar]
- Adomian, G. Modification of decomposition approach to the heat equation. J. Math. Anal. Appl.
**1987**, 124, 290–291. [Google Scholar] [CrossRef] - Adomian, G.; Rach, R.C. Inhomogeneous nonlinear partial differential equations with variable coefficients. Appl. Math. Lett.
**1992**, 5, 11–12. [Google Scholar] [CrossRef] - Adomian, G.; Rach, R.C. Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition. J. Math. Anal. Appl.
**1993**, 174, 118–137. [Google Scholar] [CrossRef] - Duan, J.-S.; Rach, R. A new modification of the Adomian decomposition method for solving boundary value problems for higher order differential equations. Appl. Math. Comput.
**2011**, 218, 4090–4118. [Google Scholar] [CrossRef] - Duan, J.-S.; Rach, R.; Wazwaz, A.-M. Solution of the model of beam-type micro- and nano-scale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems. Int. J. Non-Linear Mech.
**2013**, 49, 159–169. [Google Scholar] [CrossRef] - Duan, J.-S.; Rach, R.; Wazwaz, A.-M.; Chaolu, T.; Wang, Z. A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions. Appl. Math. Modell.
**2013**, 37, 8687–8708. [Google Scholar] [CrossRef] - Duan, J.-S.; Rach, R.; Wazwaz, A.-M. A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method. Open Eng.
**2014**, 5, 59–74. [Google Scholar] [CrossRef] - Wazwaz, A.-M. Partial Differential Equations and Solitary Waves Theory; Higher Education Press: Beijing, China; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Bougoffa, L.; Rach, R.C.; El-Manouni, S. A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Int. J. Comput. Math.
**2013**, 90, 360–375. [Google Scholar] [CrossRef] - Bougoffa, L.; Rach, R.C.; Wazwaz, A.M.; Duan, J.S. On the Adomian decomposition method for solving the Stefan problem. Int. J. Numer. Methods Heat Fluid Flow
**2015**, 25, 912–928. [Google Scholar] [CrossRef] - Bougoffa, L.; Bougouffa, S. Adomian method for solving some coupled systems of two equations. Appl. Math. Comput.
**2006**, 177, 553–560. [Google Scholar] [CrossRef] - Bougoffa, L.; Bougouffa, S. Solutions of the two-wave interactions in quadratic nonlinear media. Mathematics
**2020**, 8, 1867. [Google Scholar] [CrossRef] - Polyanin, A.D.; Zaitsev, V.F. Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed.; Chapman and Hall/CRC: Boca Raton, FL, USA, 2003. [Google Scholar]
- Cherruault, Y. Convergence of Adomian’s method. Math. Comput. Model.
**1990**, 14, 83–86. [Google Scholar] [CrossRef] - El-Kalla, I.L. Convergence of the Adomian method applied to a class of nonlinear integral equations. Appl. Math. Lett.
**2008**, 21, 372–376. [Google Scholar] [CrossRef] - Enright, W.H.; Jackson, K.R.; Norsett, S.P.; Thomsen, P.G. Interpolants for Runge-Kutta Formulas. ACM Trans. Math. Softw.
**1986**, 12, 193–218. [Google Scholar] [CrossRef] - Fehlberg, E. Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Waermeleitungsprobleme. Computing
**1970**, 6, 61–71. [Google Scholar] [CrossRef] - Forsythe, G.E.; Malcolm, M.A.; Moler, C.B. Computer Methods for Mathematical Computations; Prentice Hall: Hoboken, NJ, USA, 1977. [Google Scholar]
- Shampine, F.L.; Corless, M.R. Initial Value Problems for ODEs in Problem-Solving Environments. J. Comp. Appl. Math.
**2000**, 125, 31–40. [Google Scholar] [CrossRef] - John Hopkins University. CSSE Novel Coronavirus (COVID-19) Cases. Available online: https://github.com/CSSEGISandData/COVID-19 (accessed on 10 March 2023).

**Figure 1.**The numerical solutions to the SIR epidemic model (1) and (3) where the susceptible (S), infected (I), and recovered (R) populations are plotted versus time, with the initial conditions (2). (

**Left panel**) $\alpha =2.,r=1/5,I\left(0\right)=25,S\left(0\right)=75,R\left(0\right)=0$ from [9]. (

**Mid panel**) $\alpha =2.73,r=0.0178,I\left(0\right)=7,S\left(0\right)=254,R\left(0\right)=0$ from [9]. (

**Right panel**) $\alpha =0.0164,r=2.9236\times {10}^{-5},I\left(0\right)=2,S\left(0\right)=4206,R\left(0\right)=0$ from [10]. The representation is as follows: the solid black line corresponds to $S\left(t\right)$, the dashed blue line to $I\left(t\right)$, and the dash-dotted red line to $R\left(t\right)$. The solutions are in units of people and t is in months.

**Figure 3.**ADM and numerical solution to the SIR epidemic model (1) and (3) with the initial conditions (2). For the case $\alpha =2,r=1/5,I\left(0\right)=25,S\left(0\right)=75,R\left(0\right)=0$ from [9]. The representation is as follows: the solid red line corresponds to the numerical solution, and the dashed black line is the fourth-order solution obtained using the ADM technique.

**Figure 4.**ADM and numerical solutions to the SIR epidemic model (1) and (3) with the initial conditions (2). For the case $\alpha =2.73,r=0.0178,I\left(0\right)=7,S\left(0\right)=254,R\left(0\right)=0$ from [9]. (

**Left panel**) The representation is as follows: the solid lines correspond to the numerical solution, and the dashed lines are the fifth-order solution obtained using the ADM technique. (

**Right panel**) The dotted lines represent the sixth-order solution obtained using the ADM. Black line: S, blue lines: I, and red lines: R.

**Figure 5.**ADM and numerical solutions to the SIR epidemic model (1) and (3) with the initial conditions (2). For the case $\alpha =0.0164,r=2.9236\times {10}^{-5},I\left(0\right)=2,S\left(0\right)=4206,R\left(0\right)=0$ from Japan’s COVID-19 outbreak data [10,31]. (

**Left panel**) The representation is as follows: the solid lines correspond to the numerical solution, and the dashed lines are the seventh-order solution obtained using the ADM technique. (

**Right panel**) The eighth-order solutions obtained using the ADM and the numerical solutions. Black line: S, blue lines: I, and red lines: R. Here, $t=0$ is 22 January 2020 [10,31].

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**MDPI and ACS Style**

Bougoffa, L.; Bougouffa, S.; Khanfer, A.
Approximate and Parametric Solutions to SIR Epidemic Model. *Axioms* **2024**, *13*, 201.
https://doi.org/10.3390/axioms13030201

**AMA Style**

Bougoffa L, Bougouffa S, Khanfer A.
Approximate and Parametric Solutions to SIR Epidemic Model. *Axioms*. 2024; 13(3):201.
https://doi.org/10.3390/axioms13030201

**Chicago/Turabian Style**

Bougoffa, Lazhar, Smail Bougouffa, and Ammar Khanfer.
2024. "Approximate and Parametric Solutions to SIR Epidemic Model" *Axioms* 13, no. 3: 201.
https://doi.org/10.3390/axioms13030201