April  2018, 15(2): 461-483. doi: 10.3934/mbe.2018021

Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions

1. 

Botswana International University of Science and Technology, Department of Mathematics and Statistical Sciences, Private Bag 16, Palapye, Botswana

2. 

University of Waterloo, Department of Applied Mathematics, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada

* Corresponding author: N. Ringa

Received  October 20, 2016 Accepted  May 05, 2017 Published  June 2017

Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number $R_0$ and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

Citation: Notice Ringa, Chris T. Bauch. Spatially-implicit modelling of disease-behaviour interactions in the context of non-pharmaceutical interventions. Mathematical Biosciences & Engineering, 2018, 15 (2) : 461-483. doi: 10.3934/mbe.2018021
References:
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K. A. Alexander and J. W. McNutt, Human behavior influences infectious disease emergence at the human-animal interface, Frontiers in Ecology and the Environment, 8 (2010), 522-526. 

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C. T. Bauch, A versatile ODE approximation to a network model for the spread of sexually transmitted diseases, Journal of Mathematical Biology, 45 (2002), 375-395.  doi: 10.1007/s002850200153.

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C. T. Bauch, The spread of infectious diseases in spatially structured populations: An invasory pair approximation, Mathematical Biosciences, 198 (2005), 217-237.  doi: 10.1016/j.mbs.2005.06.005.

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C. T. BauchA. d'Onofrio and P. Manfredi, Behavioral epidemiology of infectious diseases: An overview, Modeling the interplay between human behavior and the spread of infectious diseases, Springer-Verlag, (2013), 1-19.  doi: 10.1007/978-1-4614-5474-8_1.

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C. T. Bauch and A. P. Galvani, Using network models to approximate spatial point-process models, Mathematical Biosciences, 184 (2003), 101-114.  doi: 10.1016/S0025-5564(03)00042-7.

[8]

C. T. Bauch and D. A. Rand, A moment closure model for sexually transmitted disease transmission through a concurrent partnership network, The Royal Society, 267 (2000), 2019-2027.  doi: 10.1098/rspb.2000.1244.

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J. BenoitA. Nunes and M. Telo da Gama, Pair approximation models for disease spread, The European Physical Journal B, 50 (2006), 177-181.  doi: 10.1140/epjb/e2006-00096-x.

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E. P. FenichelC. Castillo-ChavezM. G. GeddiaG. ChowellP. A. Gonzalez ParraG. J. HicklingG. HollowayR. HoranB. MorinC. PerringsM. SpringbornL. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, Proceedings of the National Academy of Sciences, 108 (2011), 6306-6311.  doi: 10.1073/pnas.1011250108.

[13]

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[14]

M. J. FerrariS. BansalL. A. Meyers and O. N. Bjϕrnstad, Network frailty and the geometry of head immunity, Proceedings of the Royal Society B, 273 (2006), 2743-2748. 

[15]

S. FunkE. GiladC. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences, 106 (2009), 6872-6877.  doi: 10.1073/pnas.0810762106.

[16]

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[23]

L. Mao and Y. Yang, Coupling infectious diseases, human preventive behavior, and networks--a conceptual framework for epidemic modeling, Social Science Medicine, 74 (2012), 167-175.  doi: 10.1016/j.socscimed.2011.10.012.

[24]

J. P. McGowanS. S. ShahC. E. GaneaS. BlumJ. A. ErnstK. L. IrwinN. Olivo and P. J. Weidle, Risk behavior for transmission of Human Immunodeficiency Virus (HIV) among HIV-seropositive individuals in an urban setting, Clinical Infectious Diseases, 38 (2004), 122-127.  doi: 10.1086/380128.

[25]

T. Modie-Moroka, Intimate partner violence and sexually risky behavior in Botswana: Implications for HIV prevention, Health Care for Women International, 30 (2009), 230-231.  doi: 10.1080/07399330802662036.

[26]

S. S. Morse, Factors in the emergence of infectious diseases, Factors in the Emergence of Infectious Diseases, (2001), 8-26.  doi: 10.1057/9780230524248_2.

[27]

S. MushayabasaC. P. Bhunu and M. Dhlamini, Impact of vaccination and culling on controlling foot and mouth disease: A mathematical modeling approach, World of Journal Vaccines, 1 (2011), 156-161. 

[28]

S. O. OyeyemiE. Gabarron and R. Wynn, Ebola, Twitter, and misinformation: A dangerous combination?, British Medical Journal, 349 (2014), g6178.  doi: 10.1136/bmj.g6178.

[29]

P. E. Parham and N. M. Ferguson, Space and contact networks: Capturing of the locality of disease transmission, Journal of Royal Society, 3 (2005), 483-493.  doi: 10.1098/rsif.2005.0105.

[30]

P. E. ParhamB. K. Singh and N. M. Ferguson, Analytical approximation of spatial epidemic models of foot and mouth disease, Theoretical Population Biology, 72 (2008), 349-368. 

[31]

F. M. PillemerR. J. BlendonA. M. Zaslavsky and B. Y. Lee, Predicting support for non-pharmaceutical interventions during infectious outbreaks: A four region analysis, Disasters, 39 (2014), 125-145. 

[32]

D. A. Rand, Correlation equations and pair approximations for spatial ecologies, CWI Quarterly, 12 (1999), 329-368. 

[33]

C. T. Reluga, Game theory of social distancing in response to an epidemic Plos Computational Biology 6 (2010), e1000793, 9pp. doi: 10.1371/journal.pcbi.1000793.

[34]

N. Ringa and C. T. Bauch, Dynamics and control of foot and mouth disease in endemic countries: A pair approximation model, Journal of Theoretical Biology, 357 (2014), 150-159.  doi: 10.1016/j.jtbi.2014.05.010.

[35]

A. RizzoM. Frasca and M. Porfiri, Effect of individual behavior on epidemic spreading in activity-driven networks, Physical Review E, 90 (2014), 042801.  doi: 10.1103/PhysRevE.90.042801.

[36]

M. Salathe and S. Bonhoeffer, The effect of opinion clustering on disease outbreaks, Journal of The Royal Society Interface, 5 (2008), 1505-1508.  doi: 10.1098/rsif.2008.0271.

[37]

L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Physical Review E, 77 (2008), 066101, 10pp.  doi: 10.1103/PhysRevE.77.066101.

[38]

R. L. Stoneburner and D. Low-Beer, Population-level HIV decline and behavioral risk avoidance in Uganda, Science, 304 (2004), 714-718.  doi: 10.1126/science.1093166.

[39]

R. R. SwensonW. S. HadleyC. D. HouckS. K. Dance and L. K. Brown, Who accepts a rapid HIV antibody test? The role of race/ethnicity and HIV risk behavior among community adolescents, Journal of Adolescent Health, 48 (2011), 527-529.  doi: 10.1016/j.jadohealth.2010.08.013.

[40]

J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission ISRN Biomathematics2012 (2012), Article ID 581274, 10pp. doi: 10.5402/2012/581274.

show all references

References:
[1]

K. A. Alexander and J. W. McNutt, Human behavior influences infectious disease emergence at the human-animal interface, Frontiers in Ecology and the Environment, 8 (2010), 522-526. 

[2]

M. C. Auld, Estimating behavioral response to the AIDS epidemic, Contributions to Economic Analysis and Policy 5 (2006), Art. 12.

[3]

N. Bacaer, Approximation of the basic reproduction number for vector-borne disease with periodic vector population, Bulleting of Mathematical Biology, 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9.

[4]

C. T. Bauch, A versatile ODE approximation to a network model for the spread of sexually transmitted diseases, Journal of Mathematical Biology, 45 (2002), 375-395.  doi: 10.1007/s002850200153.

[5]

C. T. Bauch, The spread of infectious diseases in spatially structured populations: An invasory pair approximation, Mathematical Biosciences, 198 (2005), 217-237.  doi: 10.1016/j.mbs.2005.06.005.

[6]

C. T. BauchA. d'Onofrio and P. Manfredi, Behavioral epidemiology of infectious diseases: An overview, Modeling the interplay between human behavior and the spread of infectious diseases, Springer-Verlag, (2013), 1-19.  doi: 10.1007/978-1-4614-5474-8_1.

[7]

C. T. Bauch and A. P. Galvani, Using network models to approximate spatial point-process models, Mathematical Biosciences, 184 (2003), 101-114.  doi: 10.1016/S0025-5564(03)00042-7.

[8]

C. T. Bauch and D. A. Rand, A moment closure model for sexually transmitted disease transmission through a concurrent partnership network, The Royal Society, 267 (2000), 2019-2027.  doi: 10.1098/rspb.2000.1244.

[9]

J. BenoitA. Nunes and M. Telo da Gama, Pair approximation models for disease spread, The European Physical Journal B, 50 (2006), 177-181.  doi: 10.1140/epjb/e2006-00096-x.

[10]

K. Dietz, The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research, 2 (1993), 23-41.  doi: 10.1177/096228029300200103.

[11]

S. P. Ellner, Pair approximation for lattice models with multiple interaction scales, Journal of Theoretical Biology, 210 (2001), 435-447.  doi: 10.1006/jtbi.2001.2322.

[12]

E. P. FenichelC. Castillo-ChavezM. G. GeddiaG. ChowellP. A. Gonzalez ParraG. J. HicklingG. HollowayR. HoranB. MorinC. PerringsM. SpringbornL. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, Proceedings of the National Academy of Sciences, 108 (2011), 6306-6311.  doi: 10.1073/pnas.1011250108.

[13]

N. M. FergusonC. A. Donnelly and R. M. Anderson, The foot and mouth epidemic in Great Britain: pattern of spread and impact of interventions, Science, 292 (2001), 1155-1160.  doi: 10.1126/science.1061020.

[14]

M. J. FerrariS. BansalL. A. Meyers and O. N. Bjϕrnstad, Network frailty and the geometry of head immunity, Proceedings of the Royal Society B, 273 (2006), 2743-2748. 

[15]

S. FunkE. GiladC. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences, 106 (2009), 6872-6877.  doi: 10.1073/pnas.0810762106.

[16]

R. J. GlassL. M. GlassW. E. Beyeler and H. J. Min, Targeted social distancing designs for pandemic influenza, Emerging Infectious Diseases, 12 (2016), 1671-1681. 

[17]

D. Hiebeler, Moment equations and dynamics of a household SIS epidemiological model, Bulletin of Mathematical Biology, 68 (2006), 1315-1333.  doi: 10.1007/s11538-006-9080-1.

[18]

M. J. Keeling, The effects of local spatial structure on epidemiological invasions, Proceedings of The Royal Society of London B, 266 (1999), 859-867.  doi: 10.1098/rspb.1999.0716.

[19]

M.J. KeelingD. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proceedings of The Royal Society of London B, 264 (1997), 1149-1156.  doi: 10.1098/rspb.1997.0159.

[20]

J. LiD. Blakeley and R. J. Smith, The failure of $\mathbb{R}_0$, Computational and Mathematical Methods in Medicine, 12 (2011), 1-17. 

[21]

C. N. L. Macpherson, Human behavior and the epidemiology of parasitic zoonoses, International Journal for Parasitology, 35 (2005), 1319-1331. 

[22]

S. Maharaj and A. Kleczkowski, Controlling epidemic spread by social distancing: Do it well or not at all, BMC Public Health, 12 (2012), p679.  doi: 10.1186/1471-2458-12-679.

[23]

L. Mao and Y. Yang, Coupling infectious diseases, human preventive behavior, and networks--a conceptual framework for epidemic modeling, Social Science Medicine, 74 (2012), 167-175.  doi: 10.1016/j.socscimed.2011.10.012.

[24]

J. P. McGowanS. S. ShahC. E. GaneaS. BlumJ. A. ErnstK. L. IrwinN. Olivo and P. J. Weidle, Risk behavior for transmission of Human Immunodeficiency Virus (HIV) among HIV-seropositive individuals in an urban setting, Clinical Infectious Diseases, 38 (2004), 122-127.  doi: 10.1086/380128.

[25]

T. Modie-Moroka, Intimate partner violence and sexually risky behavior in Botswana: Implications for HIV prevention, Health Care for Women International, 30 (2009), 230-231.  doi: 10.1080/07399330802662036.

[26]

S. S. Morse, Factors in the emergence of infectious diseases, Factors in the Emergence of Infectious Diseases, (2001), 8-26.  doi: 10.1057/9780230524248_2.

[27]

S. MushayabasaC. P. Bhunu and M. Dhlamini, Impact of vaccination and culling on controlling foot and mouth disease: A mathematical modeling approach, World of Journal Vaccines, 1 (2011), 156-161. 

[28]

S. O. OyeyemiE. Gabarron and R. Wynn, Ebola, Twitter, and misinformation: A dangerous combination?, British Medical Journal, 349 (2014), g6178.  doi: 10.1136/bmj.g6178.

[29]

P. E. Parham and N. M. Ferguson, Space and contact networks: Capturing of the locality of disease transmission, Journal of Royal Society, 3 (2005), 483-493.  doi: 10.1098/rsif.2005.0105.

[30]

P. E. ParhamB. K. Singh and N. M. Ferguson, Analytical approximation of spatial epidemic models of foot and mouth disease, Theoretical Population Biology, 72 (2008), 349-368. 

[31]

F. M. PillemerR. J. BlendonA. M. Zaslavsky and B. Y. Lee, Predicting support for non-pharmaceutical interventions during infectious outbreaks: A four region analysis, Disasters, 39 (2014), 125-145. 

[32]

D. A. Rand, Correlation equations and pair approximations for spatial ecologies, CWI Quarterly, 12 (1999), 329-368. 

[33]

C. T. Reluga, Game theory of social distancing in response to an epidemic Plos Computational Biology 6 (2010), e1000793, 9pp. doi: 10.1371/journal.pcbi.1000793.

[34]

N. Ringa and C. T. Bauch, Dynamics and control of foot and mouth disease in endemic countries: A pair approximation model, Journal of Theoretical Biology, 357 (2014), 150-159.  doi: 10.1016/j.jtbi.2014.05.010.

[35]

A. RizzoM. Frasca and M. Porfiri, Effect of individual behavior on epidemic spreading in activity-driven networks, Physical Review E, 90 (2014), 042801.  doi: 10.1103/PhysRevE.90.042801.

[36]

M. Salathe and S. Bonhoeffer, The effect of opinion clustering on disease outbreaks, Journal of The Royal Society Interface, 5 (2008), 1505-1508.  doi: 10.1098/rsif.2008.0271.

[37]

L. B. Shaw and I. B. Schwartz, Fluctuating epidemics on adaptive networks, Physical Review E, 77 (2008), 066101, 10pp.  doi: 10.1103/PhysRevE.77.066101.

[38]

R. L. Stoneburner and D. Low-Beer, Population-level HIV decline and behavioral risk avoidance in Uganda, Science, 304 (2004), 714-718.  doi: 10.1126/science.1093166.

[39]

R. R. SwensonW. S. HadleyC. D. HouckS. K. Dance and L. K. Brown, Who accepts a rapid HIV antibody test? The role of race/ethnicity and HIV risk behavior among community adolescents, Journal of Adolescent Health, 48 (2011), 527-529.  doi: 10.1016/j.jadohealth.2010.08.013.

[40]

J. M. Tchuenche and C. T. Bauch, Dynamics of an infectious disease where media coverage influences transmission ISRN Biomathematics2012 (2012), Article ID 581274, 10pp. doi: 10.5402/2012/581274.

Figure 1.  Typical network distributions of susceptible contacts, $S$, neighbors who practice social distancing techniques, $S_p$ (as well as the respective calculations of the basic reproduction number) around the initial infection source, where all other members of the host population are fully susceptible (i.e. state $S$). The population size is $N=40000$, each individual has $n=4$ neighbors and model parameters are $\tau=0.75$ $day^{-1}$, $\tau_p=0.1$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$, $\xi=\rho=0.5$ $day^{-1}$ and $\kappa=0.01$ $day^{-1}$
Figure 2.  The basic reproduction number as a function of social learning from protective contacts at a rate $\xi$, and from infectious contacts at a rate $\rho$, where the transmission rate to protective individuals is $\tau_p=0.1$ $day^{-1}$ (a) and $\tau_p=0.5$ $day^{-1}$ (b). In all these plots $N = 40000,n=4$, $\tau=0.75$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$, $C_{S_pS_p}=0,C_{S_pS}=3/4$, $\kappa= 0$ $day^{-1}$ and $s_p=1/N$
Figure 3.  Infection peak versus initial distribution of single infected individuals with 4 state $S_p$ neighbors (a, d, g, j), time series for susceptible individuals who protect (b, e, h, k) and time series for infectious individuals (c, f, i, l), varying the number of 1 infected node plus 4 $S_p$ neighbors at the beginning of the outbreak (the rest of the population is fully susceptible). In (a to f) $\xi=0.25$ $day^{-1}$, $\rho=0$ $day^{-1}$; in (g to l) $\xi=0$ $day^{-1}$, $\rho=0.25$ $day^{-1}$; in (a, b, c and g, h, i) $\tau_p=0.6$ $day^{-1}$; in (d, e, f and j, k, l) $\tau_p=0.1$ $day^{-1}$. Model parameters common to all graphs are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$
Figure 4.  Infection peak versus rate of disease transmission to protective individuals, $\tau_p$, and the initial distribution of single infected individuals with 4 state $S_p$ neighbors (and all other members of the host population are fully susceptible, $S$), where $\xi=0.25$ $day^{-1}$, $\rho=0$ $day^{-1}$ (a) and $\xi=0$ $day^{-1}$, $\rho=0.25$ $day^{-1}$ (b). Other model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$
Figure 5.  Cumulative infections as a function of social learning from both infectious and state $S_p$ neighbors at rates $\rho$ and $\xi$, respectively, where the initial conditions are 1 infected node and 1 state $S_p$ neighbor while the rest of the population is fully susceptible (i.e. state $S$), and $\tau_p=0.1$ $day^{-1}$ (a), $\tau_p=0.2$ $day^{-1}$ (b), $\tau_p=0.3$ $day^{-1}$ (c). Other model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$.
Figure 6.  Cumulative infections as a function of social learning from both infectious and state $S_p$ neighbors at rates $\rho$ and $\xi$, respectively, where the initial conditions are 1 infected node and 1 state $S_p$ neighbor while the rest of the population is fully susceptible (i.e. state $S$). Model parameters are $\tau=0.8$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$
Figure 7.  Infection peak versus the rate at which protective susceptible individuals forget, $\kappa$, varying regimes for social contagion parameters $\xi$ and $\rho$. Initial conditions are 1 infected node and 2 state $S_p$ neighbors while the rest of the population is fully susceptible (i.e state $S$). Other model parameters are $\tau=0.8$ $day^{-1}$, $\tau_p=0.3$ $day^{-1}$ and $\sigma=0.25$ $day^{-1}$
Figure 8.  Cumulative infections as a function of the initial number of state $S_p$ individuals and the time at which the infection is introduced, varying $\xi$ and $\rho$, for the scenario of exposure learning only (dark grey surface) and social learning only (light grey surface). Other model parameters are $\tau=0.8$ $day^{-1}$, $\tau_p=0.001$ $day^{-1}$, $\sigma=0.25$ $day^{-1}$ and $\kappa=0$ $day^{-1}$
Table 1.  Summary of expressions of the basic reproduction number $R_0$ developed in this paper
(a) General expression of $R_0$ Equation (10)
(b) Expression of $R_0$ used in simulation results in this manuscript: obtained by assuming that initially the proportion of susceptible individuals who practice NPIs is very small $s_p\approx O(1/N)$ Equation (17)
(c) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through social learning only (i.e. $\xi>0$, $\rho=0$); once adopted NPIs are practised consistently (i.e. $\kappa = 0$); at initial stage there is 1 state $I$ with 1 state $S_p$ contact who has 1 state $S_p$, and the rest of the population is of state $S$ Equation (18)
(d) Simplification of $R_0$ in Part (c) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) Equation(19)
(e) Simplification of $R_0$ in Part (d) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (20)
(f) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through exposure learning only (i.e. $\xi=0$, $\rho>0$); other conditions are as in Part (c) above Equation(21)
(g) Simplification of $R_0$ in Part (f) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) acquired through exposure learning only Equation (22)
(h) Simplification of $R_0$ in Part (g) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (23)
(a) General expression of $R_0$ Equation (10)
(b) Expression of $R_0$ used in simulation results in this manuscript: obtained by assuming that initially the proportion of susceptible individuals who practice NPIs is very small $s_p\approx O(1/N)$ Equation (17)
(c) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through social learning only (i.e. $\xi>0$, $\rho=0$); once adopted NPIs are practised consistently (i.e. $\kappa = 0$); at initial stage there is 1 state $I$ with 1 state $S_p$ contact who has 1 state $S_p$, and the rest of the population is of state $S$ Equation (18)
(d) Simplification of $R_0$ in Part (c) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) Equation(19)
(e) Simplification of $R_0$ in Part (d) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (20)
(f) Simplification of $R_0$ in Part (b) above by further assumptions: adoption of NPIs is through exposure learning only (i.e. $\xi=0$, $\rho>0$); other conditions are as in Part (c) above Equation(21)
(g) Simplification of $R_0$ in Part (f) above by a further assumption: high efficacy NPIs (i.e. $\tau_p\approx 0$) acquired through exposure learning only Equation (22)
(h) Simplification of $R_0$ in Part (g) above by cancelling out insignificant terms dependent on the parameter regine: $N=40000$; initially $s_p=2/N$; $n=4$; $\tau = 1$; $\tau_p=0.0025$; $\sigma=0.25$; $\xi= 0.25$ Equation (23)
[1]

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