A new perspective on infection forces with demonstration by a DDE infectious disease model

Tianyu Cheng; Xingfu Zou; Tianyu Cheng; Xingfu Zou

doi:10.3934/mbe.2022227

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 5: 4856-4880. doi: 10.3934/mbe.2022227

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A new perspective on infection forces with demonstration by a DDE infectious disease model

Tianyu Cheng ,
Xingfu Zou ^,

Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

Academic Editor: Rongsong Liu

Received: 30 November 2021 Revised: 21 February 2022 Accepted: 28 February 2022 Published: 14 March 2022

In this paper, we revisit the notion of infection force from a new angle which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain many existing infection force functions in the literature, it can also motivate new forms of infection force functions, particularly infection forces depending on disease surveillance of the past. As a demonstration, we propose an SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, particularly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of the past epidemics in the infection force. We confirm Hopf bifurcations both theoretically and numerically. The results show that, depending on how recent the disease surveillance data are, their assigned weight may have a different impact on disease control measures.
- infection forces,
- practically susceptible,
- SIRS model,
- delay differential equation,
- Hopf bifurcation
Citation: Tianyu Cheng, Xingfu Zou. A new perspective on infection forces with demonstration by a DDE infectious disease model[J]. Mathematical Biosciences and Engineering, 2022, 19(5): 4856-4880. doi: 10.3934/mbe.2022227

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Abstract

In this paper, we revisit the notion of infection force from a new angle which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain many existing infection force functions in the literature, it can also motivate new forms of infection force functions, particularly infection forces depending on disease surveillance of the past. As a demonstration, we propose an SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, particularly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of the past epidemics in the infection force. We confirm Hopf bifurcations both theoretically and numerically. The results show that, depending on how recent the disease surveillance data are, their assigned weight may have a different impact on disease control measures.

References

[1]	R. M. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, 1992.
[2]	M. Alexander, S. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75–96. https://doi.org/10.1016/j.mbs.2004.01.003 doi: 10.1016/j.mbs.2004.01.003
[3]	M. Alexander, S. Moghadas, Bifurcation analysis of an SIRS epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794–1816. https://doi.org/10.1137/040604947 doi: 10.1137/040604947
[4]	A. Korobeinikov, P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57–60. https://doi.org/10.3934/mbe.2004.1.57 doi: 10.3934/mbe.2004.1.57
[5]	A. Korobeinikov, P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113–128. https://doi.org/10.1093/imammb/dqi001 doi: 10.1093/imammb/dqi001
[6]	W. Liu, S. A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187–204. https://doi.org/10.1007/BF00276956 doi: 10.1007/BF00276956
[7]	W. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380. https://doi.org/10.1007/BF00277162 doi: 10.1007/BF00277162
[8]	M. Lu, J. Huang, S. Ruan, P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Diff. Eqns., 267 (2019), 1859–1898. https://doi.org/10.1016/j.jde.2019.03.005 doi: 10.1016/j.jde.2019.03.005
[9]	M. Lu, J. Huang, S. Ruan, P. Yu, Global dynamics of a susceptible-infectious-recovered epidemic model with a generalized non-monotone incidence rate, J. Dyn. Diff. Eqns., 33 (2021), 1625–1661. https://doi.org/10.1007/s10884-020-09862-3 doi: 10.1007/s10884-020-09862-3
[10]	S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Diff. Eqns., 188 (2003), 135–163. https://doi.org/10.1016/S0022-0396(02)00089-X doi: 10.1016/S0022-0396(02)00089-X
[11]	W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267–279. https://doi.org/10.3934/mbe.2006.3.267 doi: 10.3934/mbe.2006.3.267
[12]	D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 20 (2007), 419–429. https://doi.org/10.1016/j.mbs.2006.09.025 doi: 10.1016/j.mbs.2006.09.025
[13]	H. McCallum, N. Barlow, J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol., 6 (2001), 295–300. https://doi.org/10.1016/s0169-5347(01)02144-9 doi: 10.1016/s0169-5347(01)02144-9
[14]	R. Liu, J. Wu, H. Zhu, Media/psychological impact on multiple outbreaks of emerging infections diseases, Comp. Math. Meth, Medic., 8 (2007), 153–164. https://doi.org/10.1080/17486700701425870 doi: 10.1080/17486700701425870
[15]	J. Cui, Y. Song, H. Zhu, The impact of media on the control of infectious diseases, J. Dyn. Diff. Eqns., 20 2008, 31–53. https://doi.org/10.1007/s10884-007-9075-0 doi: 10.1007/s10884-007-9075-0
[16]	P. Song, Y. Xiao, Global hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, J. Math. Biol., 76 (2018), 1249–1267. https://doi.org/10.1007/s00285-017-1173-y doi: 10.1007/s00285-017-1173-y
[17]	P. Song, Y. Xiao, Analysis of an epidemic system with two response delays in media impact function, Bull. Math. Biol., 81 (2019), 1582–1612. https://doi.org/10.1007/s11538-019-00586-0 doi: 10.1007/s11538-019-00586-0
[18]	K. L. Cooke, Stability analysis of a vector disease model, Rocky Mountain J. Math., 5 (1979), 31–42. https://doi.org/10.1216/RMJ-1979-9-1-31 doi: 10.1216/RMJ-1979-9-1-31
[19]	G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207. https://doi.org/10.1007/s11538-009-9487-6 doi: 10.1007/s11538-009-9487-6
[20]	G. Huang, Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2010), 125–139. https://doi.org/10.1007/s00285-010-0368-2 doi: 10.1007/s00285-010-0368-2
[21]	C. C. McCluskey, Complete global stability for an SIR epidemic model with delay–distributed or discrete, Nonlin. Anal. RWA., 11 (2010), 55–59. https://doi.org/10.1016/j.nonrwa.2008.10.014 doi: 10.1016/j.nonrwa.2008.10.014
[22]	C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlin. Anal. RWA., 11 (2010), 3106–3109. https://doi.org/10.1016/j.nonrwa.2009.11.005 doi: 10.1016/j.nonrwa.2009.11.005
[23]	A. M. Rahman, X. Zou, Global dynamics of a two-strain disease model with latency and saturating incidence rate, Can. Appl. Math. Quat., 20 (2012), 51–73.
[24]	R. Xu, Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlin. Anal. RWA., 10 (2009), 3175–3189. https://doi.org/https://doi.org/10.1016/j.nonrwa.2008.10.013 doi: 10.1016/j.nonrwa.2008.10.013
[25]	R. Xu, Z. Ma, Global dynamics of a vector disease model with saturation incidence and time delay, IMA J. Appl. Math., 76 (2011), 919–937. https://doi.org/10.1093/imamat/hxr013 doi: 10.1093/imamat/hxr013
[26]	J. K. Hale, S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, 1993.
[27]	C. Castillo-Chavez, H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. Arino, et al.), Wuerz Publishing Ltd., (1995), 33–50.
[28]	S. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, Nat. Sci. J. Hainan Teach. Coll., 2 (1989), 91–98.
[29]	S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863–874.
[30]	R. M. Corless, G. H. Gonnet, D. E. Hare, D. Jeffrey, D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 5 (1996), 329–359.
[31]	K. Engelborghs, T. Luzyanina, G. Samaey, A Matlab package for bifurcation analysis of delay differential equations, Tech. rep. Leuven, 2 (2001).
[32]	K. Engelborghs, T. Luzyanina, D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 28 (2002), 1–21. https://doi.org/10.1145/513001.513002 doi: 10.1145/513001.513002
[33]	J. A. Collera, Numerical continuation and bifurcation analysis in a Harvested Predator-prey model with time delay using DDE-Biftool, in Dynamical Systems, Bifurcation Analysis and Applications (eds. M. Mohd, et al.), Springer, (2018), 225–241. https://doi.org/10.1007/978-981-32-9832-3_12
[34]	Y. Xiao, S. Tang, J. Wu, Media impact switching surface during an infectious disease outbreak, Sci. Rep., 7838 (2015), 1–9. https://doi.org/10.1038/srep07838 doi: 10.1038/srep07838
[35]	A. Li, Y. Wang, P. Cong, X. Zou, Re-examination of the impact of some non-pharmaceutical interventions and media coverage on the COVID-19 outbreak in Wuhan, Infect. Dis. Model., 6 (2021), 975–987. https://doi.org/10.1016/j.idm.2021.07.001 doi: 10.1016/j.idm.2021.07.001

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