Abstract
In this paper, we proposed a susceptible-infected-susceptible (SIS) reaction–diffusion model with spatial and behavioral heterogeneity. We established the basic reproduction number \(R_0\) based on the next generation infection operators and then derived the threshold dynamics in terms of \(R_0\). We obtained the monotonicity of \(R_0\) and its asymptotic properties as diffusion rates of the infected individuals approach zero or infinity. In particular, the basic reproduction number \(R_0\) decreases with increasing diffusion rates of the infected individuals given the balance of transition from adopting normal to altered behaviors and from altered to normal behaviors, which agrees well with the existing conclusion. Interestingly, \(R_0\) may increase or exhibit non-monotonicity with increasing diffusion rate for non-balanced transition of different behaviors. Further, we numerically examined the effect of behavior changes on \(R_0\) and main results reveal that increasing transition rate from normal to altered behaviors may either decrease or increase \(R_0\) or non-monotonically affect on \(R_0\), depending on the choice of spatial transition rate from altered to normal behaviors. Our results reveal the importance of behavior changes on threshold level, and non-monotonic variation patterns in \(R_0\) are induced by the spatial transitions between groups with different behaviors.
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References
Allen LJ, Bolker BM, Lou Y et al (2008) Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model. Discrete Contin Dyn Syst 21(1):1–20. https://doi.org/10.3934/dcds.2008.21.1
Ball JM (1982) Geometric theory of semilinear parabolic equations (Lecture Notes in Mathematics, 840), vol 14
Bauch CT (2005) Imitation dynamics predict vaccinating behaviour. Proc R Soc B Biol Sci 272(1573):1669–1675. https://doi.org/10.1098/rspb.2005.3153
Bauch CT, Earn DJ (2004) Vaccination and the theory of games. Proc Natl Acad Sci 101(36):13391–13394. https://doi.org/10.1073/pnas.040382310
Bavel JJV, Baicker K, Boggio PS et al (2020) Using social and behavioural science to support COVID-19 pandemic response. Nat Hum Behav 4(5):460–471. https://doi.org/10.1038/s41562-020-0884-z
Betsch C (2020) How behavioural science data helps mitigate the COVID-19 crisis. Nat Hum Behav 4(5):438–438. https://doi.org/10.1038/s41562-020-0866-1
Buonomo B, d’Onofrio A, Lacitignola D (2008) Global stability of an SIR epidemic model with information dependent vaccination. Math Biosci 216(1):9–16. https://doi.org/10.1016/j.mbs.2008.07.011
Cui R, Lou Y (2016) A spatial SIS model in advective heterogeneous environments. J Differ Eq 261(6):3305–3343. https://doi.org/10.1016/j.jde.2016.05.025
Deng K (2019) Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete Contin Dyn Syst Ser B. https://doi.org/10.3934/dcdsb.2019114
Diekmann O, Heesterbeek JAP, Metz JA (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382. https://doi.org/10.1007/BF00178324
Dung L (1997) Dissipativity and global attractors for a class of quasilinear parabolic systems. Commun Partial Differ Eq 22(3–4):413–433. https://doi.org/10.1080/03605309708821269
Epstein JM (2009) Modelling to contain pandemics. Nature 460(7256):687–687. https://doi.org/10.1038/460687a
Fair KR, Karatayev VA, Anand M et al (2022) Estimating COVID-19 cases and deaths prevented by non-pharmaceutical interventions, and the impact of individual actions: A retrospective model-based analysis. Epidemics 39:100557. https://doi.org/10.1016/j.epidem.2022.100557
Ferguson N (2007) Capturing human behaviour. Nature 446(7137):733–733. https://doi.org/10.1038/446733a
Funk S, Gilad E, Watkins C et al (2009) The spread of awareness and its impact on epidemic outbreaks. Proc Natl Acad Sci 106(16):6872–6877. https://doi.org/10.1073/pnas.0810762106
Funk S, Gilad E, Jansen VA (2010) Endemic disease, awareness, and local behavioural response. J Theor Biol 264(2):501–509. https://doi.org/10.1016/j.jtbi.2010.02.032
Garnett GP, Anderson RM (1996) Sexually transmitted diseases and sexual behavior: insights from mathematical models. J Infect Dis 174(Supplement_2):S150–S161. https://doi.org/10.1093/infdis/174.Supplement_2.S150
Ge J, Kim KI, Lin Z et al (2015) A SIS reaction-diffusion-advection model in a low-risk and high-risk domain. J Differ Eq 259(10):5486–5509. https://doi.org/10.1016/j.jde.2015.06.035
Henry D (2006) Geometric theory of semilinear parabolic equations, vol 840. Springer, New York
Hess P (1991) Periodic-parabolic boundary value problems and positivity. Longman, Harlow
Huang W, Han M, Liu K (2010) Dynamics of an SIS reaction–diffusion epidemic model for disease transmission. Math Biosci Eng 7(1):51–66. https://doi.org/10.3934/mbe.2010.7.51
Jentsch PC, Anand M, Bauch CT (2021) Prioritising COVID-19 vaccination in changing social and epidemiological landscapes: a mathematical modelling study. Lancet Infect Dis 21(8):1097–1106. https://doi.org/10.1016/S1473-3099(21)00057-8
Koch-Medina P, Daners D (1992) Abstract evolution equations, periodic problems and applications. Chapman and Hall/CRC, London
Krein MG, Rutman MA (1962) Linear operators leaving invariant a cone in a banach space. Am Math Soc Transl 10:199–325
Kucharski AJ, Klepac P, Conlan AJ et al (2020) Effectiveness of isolation, testing, contact tracing, and physical distancing on reducing transmission of SARS-CoV-2 in different settings: a mathematical modelling study. Lancet Infect Dis 20(10):1151–1160. https://doi.org/10.1016/S1473-3099(20)30457-6
Lam KY, Lou Y (2016) Asymptotic behavior of the principal eigenvalue for cooperative elliptic systems and applications. J Dyn Differ Equ 28:29–48. https://doi.org/10.1007/s10884-015-9504-4
Liu S, Lou Y (2022) Classifying the level set of principal eigenvalue for time-periodic parabolic operators and applications. J Funct Anal 282(4):109338. https://doi.org/10.1016/j.jfa.2021.109338
Magal P, Webb GF, Wu Y (2019) On the basic reproduction number of reaction–diffusion epidemic models. SIAM J Appl Math 79(1):284–304. https://doi.org/10.1137/18M1182243
Moyles IR, Heffernan JM, Kong JD (2021) Cost and social distancing dynamics in a mathematical model of COVID-19 with application to ontario, canada. R Soc Open Sci 8(2):201770. https://doi.org/10.1098/rsos.201770
Peng R (2009) Asymptotic profiles of the positive steady state for an SIS epidemic reaction–diffusion model. Part i. J Differ Equ 247(4):1096–1119. https://doi.org/10.1016/j.jde.2009.05.002
Peng R, Liu S (2009) Global stability of the steady states of an SIS epidemic reaction–diffusion model. Nonlinear Anal Theory Methods Appl 71(1–2):239–247. https://doi.org/10.1016/j.na.2008.10.043
Peng R, Yi F (2013) Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement. Phys D Nonlinear Phenom 259:8–25. https://doi.org/10.1016/j.physd.2013.05.006
Peng R, Zhao XQ (2012) A reaction–diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25(5):1451. https://doi.org/10.1088/0951-7715/25/5/1451
Poletti P, Caprile B, Ajelli M et al (2009) Spontaneous behavioural changes in response to epidemics. J Theor Biol 260(1):31–40. https://doi.org/10.1016/j.jtbi.2009.04.029
Poletti P, Ajelli M, Merler S (2012) Risk perception and effectiveness of uncoordinated behavioral responses in an emerging epidemic. Math Biosci 238(2):80–89. https://doi.org/10.1016/j.mbs.2012.04.003
Protter MH, Weinberger HF (2012) Maximum principles in differential equations. Springer Science & Business Media, New York
Reluga TC, Galvani AP (2011) A general approach for population games with application to vaccination. Math Biosci 230(2):67–78. https://doi.org/10.1016/j.mbs.2011.01.003
Reluga TC, Bauch CT, Galvani AP (2006) Evolving public perceptions and stability in vaccine uptake. Math Biosci 204(2):185–198. https://doi.org/10.1016/j.mbs.2006.08.015
Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems: an introduction to the theory of competitive and cooperative systems. 41, American Mathematical Soc
Song P, Lou Y, Xiao Y (2019) A spatial SEIRS reaction–diffusion model in heterogeneous environment. J Differ Equ 267(9):5084–5114. https://doi.org/10.1016/j.jde.2019.05.022
Tang B, Zhou W, Wang X et al (2022) Controlling multiple COVID-19 epidemic waves: an insight from a multi-scale model linking the behaviour change dynamics to the disease transmission dynamics. Bull Math Biol 84(10):106. https://doi.org/10.1007/s11538-022-01061-z
Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1–2):29–48. https://doi.org/10.1016/S0025-5564(02)00108-6
Wang W, Zhao XQ (2012) Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J Appl Dyn Syst 11(4):1652–1673. https://doi.org/10.1137/120872942
Wu Y, Zou X (2016) Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J Differ Equ 261(8):4424–4447. https://doi.org/10.1016/j.jde.2016.06.028
Xiao Y, Xiang C, Cheke RA et al (2020) Coupling the macroscale to the microscale in a spatiotemporal context to examine effects of spatial diffusion on disease transmission. Bull Math Biol 82(5):58. https://doi.org/10.1007/s11538-020-00736-9
Zhao XQ (1995) Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Can Appl Math Quart 3(4):473–495
Zhao XQ (2003) Dynamical systems in population biology, vol 16. Springer, New York
Acknowledgements
This work is supported by Major International (Regional) Joint Research Project of National Natural Science Foundation of China (Nos. 12220101001, 12031010).
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Li, L., Xiao, Y. An SIS reaction–diffusion model with spatial/behavioral heterogeneity. Comp. Appl. Math. 43, 164 (2024). https://doi.org/10.1007/s40314-024-02690-x
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DOI: https://doi.org/10.1007/s40314-024-02690-x
Keywords
- SIS epidemic model
- Spatial heterogeneity
- Basic reproduction number
- Behavioral changes
- Persistence/extinction
- Asymptotic profile