# A deterministic time-delayed SIR epidemic model: mathematical modeling and analysis

- 34 Downloads

## Abstract

In this paper, a deterministic model for transmission of an epidemic has been proposed by dividing the total population into three subclasses, namely susceptible, infectious and recovered. The incidence rate of infection is taken as a nonlinear functional along with time delay, and treatment rate of infected is considered as Holling type III functional. We have structured a deterministic transmission model of the epidemic taking into account the factors that affect the epidemic transmission such as social and natural factors, inhibitory effects and numerous control measures. The delayed model has been analyzed mathematically for two equilibria, namely disease-free equilibrium (DFE) and endemic equilibrium. It is found that DFE is locally and globally asymptotically stable when the basic reproduction number \( (R_{0} ) \) is less than unity. It has also been shown that the delayed system for DFE at \( R_{0} = 1 \) is linearly neutrally stable. The existence of an endemic equilibrium has been shown and found that under some conditions, endemic equilibrium is locally asymptotically stable, and is globally asymptotically stable when \( R_{0} > 1 \). Further, the endemic equilibrium exhibits Hopf bifurcation under some conditions. Finally, an undelayed system has been analyzed, and it is shown that at \( R_{0} = 1 \), DFE exhibits a forward bifurcation.

## Keywords

Epidemic Delay SIR model Holling type III treatment rate Nonlinear incidence rate Stability Bifurcation## Notes

### Acknowledgements

The authors acknowledged Delhi Technological University for providing the monetary help for this research. The authors thank the handling editor and anonymous reviewers for their careful reading of our manuscript and their insightful comments and suggestions.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- Buonomo B, Cerasuolo M (2015) The effect of time delay in plant-pathogen interactions with host demography. Math Biosci Eng 12(3):473–490CrossRefGoogle Scholar
- Capasso V, Serio G (1978) A generalization of the Kermack–McKendrick deterministic epidemic model. Math Biosci 42:43–61CrossRefGoogle Scholar
- Castillo-Chavez C, Song B (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1:361–404CrossRefGoogle Scholar
- Cui Q, Qiu Z, Liu W, Hu Z (2017) Complex dynamics of an SIR epidemic model with nonlinear saturated incidence rate and recovery rate. Entropy. https://doi.org/10.3390/e19070305 Google Scholar
- Dubey B, Patara A, Srivastava PK, Dubey US (2013) Modelling and analysis of a SEIR model with different types of nonlinear treatment rates. J Biol Syst 21(3):1350023CrossRefGoogle Scholar
- Dubey B, Dubey P, Dubey Uma S (2015) Dynamics of a SIR model with nonlinear incidence rate and treatment rate. Appl Appl Math 2(2):718–737Google Scholar
- Dubey P, Dubey B, Dubey US (2016) An SIR model with nonlinear incidence rate and Holling type III treatment rate. Appl Anal Biol Phys Sci Springer Proc Math Stat 186:63–81CrossRefGoogle Scholar
- Goel K, Nilam (2019) A mathematical and numerical study of a SIR epidemic model with time delay. Nonlinear Incid Treat Rates Theory Biosci. https://doi.org/10.1007/s12064-019-00275-5 Google Scholar
- Gumel AB, Connell Mccluskey C, Watmough J (2006) An SVEIR model for assessing the potential impact of an imperfect anti-SARS vaccine. Math. Biosci. Eng. 3:485–494CrossRefGoogle Scholar
- Hale J, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations. Springer, New YorkCrossRefGoogle Scholar
- Hattaf K, Yousfi N (2009) Mathematical model of influenza A (H1N1) infection. Adv Stud Biol 1(8):383–390Google Scholar
- Hattaf K, Lashari AA, Louartassi Y, Yousfi N (2013) A delayed SIR epidemic model with general incidence rate. Electr J Qual Theory Differ Equ 3:1–9Google Scholar
- Huang G, Takeuchi Y, Ma W, Wei D (2010) Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72:1192–1207CrossRefGoogle Scholar
- Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of Epidemics. Proc R Soc Lond A 115(772):700–721CrossRefGoogle Scholar
- Korobeinikov A, Maini PK (2005) Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. 22:113–128CrossRefGoogle Scholar
- Kuang Y (1993) Delay differential equations with applications in population dynamics. Academic Press, San DiegoGoogle Scholar
- Kumar A, Nilam (2018) Stability of a time delayed SIR epidemic model along with nonlinear incidence rate and Holling type II treatment rate. Int J Comput Methods 15(6):1850055CrossRefGoogle Scholar
- Kumar A, Nilam (2019a) Dynamical model of epidemic along with time delay; Holling type II incidence rate and Monod–Haldane treatment rate. Differ Equ Dyn Syst 27(1–3):299–312CrossRefGoogle Scholar
- Kumar A, Nilam (2019b) Mathematical analysis of a delayed epidemic model with nonlinear incidence and treatment rates. J Eng Math 115(1):1–20CrossRefGoogle Scholar
- Kumar A, Nilam, Kishor R (2019) A short study of an SIR model with inclusion of an alert class, two explicit nonlinear incidence rates and saturated treatment rate. SeMA J 10:10. https://doi.org/10.1007/s40324-019-00189-8 Google Scholar
- Li M, Liu X (2014) An SIR epidemic model with time delay and general nonlinear incidence rate. Abstr Appl Anal 2014, Article ID 131257Google Scholar
- Li Michael Y, Graef JR, Wang L, Karsai J (1999) Global dynamics of a SEIR model with varying total population size. Math Biosci 160:191–213CrossRefGoogle Scholar
- Li GH, Zhang YH (2017) Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence rate and recovery rates. PLoS ONE 12(4):e0175789CrossRefGoogle Scholar
- Liu WM, Levin SA, Iwasa Y (1986) Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J Math Biol 23(2):187–204CrossRefGoogle Scholar
- Naresh R, Tripathi A, Tchuenche JM, Sharma D (2009) Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Comput Math Appl 58:348–359CrossRefGoogle Scholar
- Ruan S, Wei J (2003) On the zeros of transcendental functions with applications to stability of delay differential equations with two. Dyn Contin Discrete Impuls Syst Ser A Math Anal 10:863–874Google Scholar
- Sastry S (1999) Analysis, stability and control. Springer, New YorkGoogle Scholar
- van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartment models of disease transmission. Math Biosci 180:29–48CrossRefGoogle Scholar
- Wang WD (2002) Global behavior of an SEIRS epidemic model with time delays. Appl Math Lett 15:423–428CrossRefGoogle Scholar
- Wang W, Ruan S (2004) Bifurcation in an epidemic model with constant removal rates of the infectives. J Math Anal Appl 21:775–793CrossRefGoogle Scholar
- Xiao D, Ruan S (2007) Global analysis of an epidemic model with nonmonotone incidence rate. Math Biosci 208(2):419–429CrossRefGoogle Scholar
- Xu R, Ma Z (2009) Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solut Fractals 41:2319–2325CrossRefGoogle Scholar
- Zhang Z, Suo S (2010) Qualitative analysis of an SIR epidemic model with saturated treatment rate. J Appl Math Comput 34:177–194CrossRefGoogle Scholar
- Zhou L, Fan M (2012) Dynamics of a SIR epidemic model with limited medical resources revisited. Nonlinear Anal RWA 13:312–324CrossRefGoogle Scholar