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February  2018, 15(1): 299-321. doi: 10.3934/mbe.2018013

Effect of seasonality on the dynamics of an imitation-based vaccination model with public health intervention

1. 

Department of Mathematics and Applications, University of Naples Federico Ⅱ, via Cintia, I-80126 Naples, Italy

2. 

International Prevention Research Institute, 95 cours Lafayette, 69006 Lyon, France

* Corresponding author

Received  November 18, 2016 Revised  April 02, 2017 Published  May 2017

We extend here the game-theoretic investigation made by d'Onofrio et al (2012) on the interplay between private vaccination choices and actions of the public health system (PHS) to favor vaccine propensity in SIR-type diseases. We focus here on three important features. First, we consider a SEIR-type disease. Second, we focus on the role of seasonal fluctuations of the transmission rate. Third, by a simple population-biology approach we derive -with a didactic aim -the game theoretic equation ruling the dynamics of vaccine propensity, without employing 'economy-related' concepts such as the payoff. By means of analytical and analytical-approximate methods, we investigate the global stability of the of disease-free equilibria. We show that in the general case the stability critically depends on the 'shape' of the periodically varying transmission rate. In other words, the knowledge of the average transmission rate (ATR) is not enough to make inferences on the stability of the elimination equilibria, due to the presence of the class of latent subjects. In particular, we obtain that the amplitude of the oscillations favors the possible elimination of the disease by the action of the PHS, through a threshold condition. Indeed, for a given average value of the transmission rate, in absence of oscillations as well as for moderate oscillations, there is no disease elimination. On the contrary, if the amplitude exceeds a threshold value, the elimination of the disease is induced. We heuristically explain this apparently paradoxical phenomenon as a beneficial effect of the phase when the transmission rate is under its average value: the reduction of transmission rate (for example during holidays) under its annual average over-compensates its increase during periods of intense contacts. We also investigate the conditions for the persistence of the disease. Numerical simulations support the theoretical predictions. Finally, we briefly investigate the qualitative behavior of the non-autonomous system for SIR-type disease, by showing that the stability of the elimination equilibria are, in such a case, determined by the ATR.

Citation: Bruno Buonomo, Giuseppe Carbone, Alberto d'Onofrio. Effect of seasonality on the dynamics of an imitation-based vaccination model with public health intervention. Mathematical Biosciences & Engineering, 2018, 15 (1) : 299-321. doi: 10.3934/mbe.2018013
References:
[1]

R. M. Anderson, The impact of vaccination on the epidemiology of infectious diseases, in The Vaccine Book, B. R. Bloom and P. -H. Lambert (eds. ) The Vaccine Book (Second Edition). Elsevier B. V. , Amsterdam, (2006), 3-31. doi: 10.1016/B978-0-12-802174-3.00001-1.  Google Scholar

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N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9.  Google Scholar

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C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc. R. Soc. B, 272 (2005), 1669-1675.  doi: 10.1098/rspb.2005.3153.  Google Scholar

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B. R. Bloom and P. -H. Lambert (eds. ), The Vaccine Book (Second Edition), Elsevier B. V. , Amsterdam, 2006. Google Scholar

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F. Brauer, P. van den Driessche and J. Wu (eds. ), Mathematical Epidemiology, Lecture Notes in Mathematics. Mathematical biosciences subseries, vol. 1945, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.  Google Scholar

[7]

B. BuonomoA. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404 (2013), 385-398.  doi: 10.1016/j.jmaa.2013.02.063.  Google Scholar

[8]

B. Buonomo, A. d'Onofrio and P. Manfredi, Public Health Intervention to shape voluntary vaccination: Continuous and piecewise optimal control, Submitted. Google Scholar

[9] V. Capasso, Mathematical Structure of Epidemic Models, 2nd edition, Springer, 2008.   Google Scholar
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C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer, New York, 2006.  Google Scholar

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M. ChoisyJ.-F. Guegan and P. Rohani, Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D, 223 (2006), 26-35.  doi: 10.1016/j.physd.2006.08.006.  Google Scholar

[13]

W. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.  Google Scholar

[14]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, 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 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[15]

A. d'Onofrio, Stability properties of pulses vaccination strategy in SEIR epidemic model, Math. Biosci., 179 (2002), 57-72.  doi: 10.1016/S0025-5564(02)00095-0.  Google Scholar

[16]

A. d'Onofrio and P. Manfredi, Impact of Human Behaviour on the Spread of Infectious Diseases: a review of some evidences and models, Submitted. Google Scholar

[17]

A. d'OnofrioP. Manfredi and P. Poletti, The impact of vaccine side effects on the natural history of immunization programmes: An imitation-game approach, J. Theor. Biol., 273 (2011), 63-71.  doi: 10.1016/j.jtbi.2010.12.029.  Google Scholar

[18]

A. d'OnofrioP. Manfredi and P. Poletti, The interplay of public intervention and private choices in determining the outcome of vaccination programmes, PLoS ONE, 7 (2012), e45653.  doi: 10.1371/journal.pone.0045653.  Google Scholar

[19]

A. d'OnofrioP. Manfredi and E. Salinelli, Vaccinating behaviour, information and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301-317.  doi: 10.1016/j.tpb.2007.01.001.  Google Scholar

[20]

A. d'OnofrioP. Manfredi and E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Math. Med. Biol., 25 (2008), 337-357.  doi: 10.1093/imammb/dqn019.  Google Scholar

[21]

D. Elliman and H. Bedford, MMR: Where are we now?, Archives of Disease in Childhood, 92 (2007), 1055-1057.  doi: 10.1136/adc.2006.103531.  Google Scholar

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B. F. Finkenstadt and B. T. Grenfell, Time series modelling of childhood diseases: A dynamical systems approach, Appl. Stat., 49 (2000), 187-205.  doi: 10.1111/1467-9876.00187.  Google Scholar

[23]

N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. Royal Soc. B, 273 (2006), 2541-2550.  doi: 10.1098/rspb.2006.3604.  Google Scholar

[24]

M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, vol. Ⅱ, Elsevier B. V. , Amsterdam, 2 (2005), 239-357.  Google Scholar

[25]

IMI. Innovative medicine initiative. First Innovative Medicines Initiative Ebola projects get underway, Available from: http://www.imi.europa.eu/content/ebola-project-launch (accessed September 2016). Google Scholar

[26]

A. Kata, A postmodern Pandora's box: Anti-vaccination misinformation on the Internet, Vaccine, 28 (2010), 1709-1716.  doi: 10.1016/j.vaccine.2009.12.022.  Google Scholar

[27]

A. Kata, Anti-vaccine activists, Web 2.0, and the postmodern paradigm-An overview of tactics and tropes used online by the anti-vaccination movement, Vaccine, 30 (2012), 3778-3789.  doi: 10.1016/j.vaccine.2011.11.112.  Google Scholar

[28] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008.   Google Scholar
[29]

W. P. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates, Am. J. Epidemiol., 98 (1973), 453-468.   Google Scholar

[30]

P. Manfredi and A. d'Onofrio (eds), Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer, New York, 2013. doi: 10.1007/978-1-4614-5474-8.  Google Scholar

[31]

M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[32]

E. Miller and N. J. Gay, Epidemiological determinants of pertussis, Dev. Biol. Stand., 89 (1997), 15-23.   Google Scholar

[33]

J. D. Murray, Mathematical Biology. I. An Introduction. Third Edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[34]

Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 325 (2010), 230-237.  doi: 10.1016/j.jmaa.2009.08.027.  Google Scholar

[35]

G. NapierD. LeeC. RobertsonA. Lawson and K. G. Pollock, A model to estimate the impact of changes in MMR vaccine uptake on inequalities in measles susceptibility in Scotland, Stat. Methods Med. Res., 25 (2016), 1185-1200.  doi: 10.1177/0962280216660420.  Google Scholar

[36]

L. F. Olsen and W. M. Schaffer, Chaos versus noisy periodicity: Alternative hypotheses for childhood epidemics, Science, 249 (1990), 499-504.  doi: 10.1126/science.2382131.  Google Scholar

[37]

T. Philipson, Private Vaccination and Public Health an empirical examination for U.S. Measles, J. Hum. Res., 31 (1996), 611-630.  doi: 10.2307/146268.  Google Scholar

[38]

I. B. Schwartz and H. L. Smith, Infinite subharmonic bifurcation in an SEIR epidemic model, J. Math. Biol., 18 (1983), 233-253.  doi: 10.1007/BF00276090.  Google Scholar

[39]

H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73.  doi: 10.2307/2341437.  Google Scholar

[40]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[41]

A. van LierJ. van de KassteeleP. de HooghI. Drijfhout and H. de Melker, Vaccine uptake determinants in The Netherlands, Eur. J. Public Health, 24 (2013), 304-309.   Google Scholar

[42]

G. S. Wallace, Why Measles Matters, 2014, Available from: http://stacks.cdc.gov/view/cdc/27488/cdc_27488_DS1.pdf (accessed November 2016). Google Scholar

[43]

Z. WangC. T. BauchS. BhattacharyyaA. d'OnofrioP. ManfrediM. PercN. PerraM. Salathé and D. Zhao, Statistical physics of vaccination, Physics Reports, 664 (2016), 1-113.  doi: 10.1016/j.physrep.2016.10.006.  Google Scholar

[44]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Eq., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[45]

WHO, Vaccine safety basics, Available from: http://vaccine-safety-training.org/adverse-events-classification.html (accessed November 2016). Google Scholar

[46]

R. M. Wolfe and L. K. Sharp, Anti-vaccinationists past and present, Brit. Med. J., 325 (2002), p430.  doi: 10.1136/bmj.325.7361.430.  Google Scholar

[47]

F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[48]

X. Q. Zhao, Dynamical Systems in Population Biology, CMS Books Math. , vol. 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

R. M. Anderson, The impact of vaccination on the epidemiology of infectious diseases, in The Vaccine Book, B. R. Bloom and P. -H. Lambert (eds. ) The Vaccine Book (Second Edition). Elsevier B. V. , Amsterdam, (2006), 3-31. doi: 10.1016/B978-0-12-802174-3.00001-1.  Google Scholar

[2] R. M. Anderson and R. M. May, Infectious Diseases of Humans. Dynamics and Control, Oxford University Press, Oxford, 1991.   Google Scholar
[3]

N. Bacaër, Approximation of the basic reproduction number $R_0$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.  doi: 10.1007/s11538-006-9166-9.  Google Scholar

[4]

C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc. R. Soc. B, 272 (2005), 1669-1675.  doi: 10.1098/rspb.2005.3153.  Google Scholar

[5]

B. R. Bloom and P. -H. Lambert (eds. ), The Vaccine Book (Second Edition), Elsevier B. V. , Amsterdam, 2006. Google Scholar

[6]

F. Brauer, P. van den Driessche and J. Wu (eds. ), Mathematical Epidemiology, Lecture Notes in Mathematics. Mathematical biosciences subseries, vol. 1945, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78911-6.  Google Scholar

[7]

B. BuonomoA. d'Onofrio and D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404 (2013), 385-398.  doi: 10.1016/j.jmaa.2013.02.063.  Google Scholar

[8]

B. Buonomo, A. d'Onofrio and P. Manfredi, Public Health Intervention to shape voluntary vaccination: Continuous and piecewise optimal control, Submitted. Google Scholar

[9] V. Capasso, Mathematical Structure of Epidemic Models, 2nd edition, Springer, 2008.   Google Scholar
[10]

L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 16, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[11]

C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer, New York, 2006.  Google Scholar

[12]

M. ChoisyJ.-F. Guegan and P. Rohani, Dynamics of infectious diseases and pulse vaccination: Teasing apart the embedded resonance effects, Physica D, 223 (2006), 26-35.  doi: 10.1016/j.physd.2006.08.006.  Google Scholar

[13]

W. Coppel, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965.  Google Scholar

[14]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, 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 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[15]

A. d'Onofrio, Stability properties of pulses vaccination strategy in SEIR epidemic model, Math. Biosci., 179 (2002), 57-72.  doi: 10.1016/S0025-5564(02)00095-0.  Google Scholar

[16]

A. d'Onofrio and P. Manfredi, Impact of Human Behaviour on the Spread of Infectious Diseases: a review of some evidences and models, Submitted. Google Scholar

[17]

A. d'OnofrioP. Manfredi and P. Poletti, The impact of vaccine side effects on the natural history of immunization programmes: An imitation-game approach, J. Theor. Biol., 273 (2011), 63-71.  doi: 10.1016/j.jtbi.2010.12.029.  Google Scholar

[18]

A. d'OnofrioP. Manfredi and P. Poletti, The interplay of public intervention and private choices in determining the outcome of vaccination programmes, PLoS ONE, 7 (2012), e45653.  doi: 10.1371/journal.pone.0045653.  Google Scholar

[19]

A. d'OnofrioP. Manfredi and E. Salinelli, Vaccinating behaviour, information and the dynamics of SIR vaccine preventable diseases, Theor. Popul. Biol., 71 (2007), 301-317.  doi: 10.1016/j.tpb.2007.01.001.  Google Scholar

[20]

A. d'OnofrioP. Manfredi and E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Math. Med. Biol., 25 (2008), 337-357.  doi: 10.1093/imammb/dqn019.  Google Scholar

[21]

D. Elliman and H. Bedford, MMR: Where are we now?, Archives of Disease in Childhood, 92 (2007), 1055-1057.  doi: 10.1136/adc.2006.103531.  Google Scholar

[22]

B. F. Finkenstadt and B. T. Grenfell, Time series modelling of childhood diseases: A dynamical systems approach, Appl. Stat., 49 (2000), 187-205.  doi: 10.1111/1467-9876.00187.  Google Scholar

[23]

N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proc. Royal Soc. B, 273 (2006), 2541-2550.  doi: 10.1098/rspb.2006.3604.  Google Scholar

[24]

M. W. Hirsch and H. Smith, Monotone dynamical systems, in Handbook of Differential Equations: Ordinary Differential Equations, vol. Ⅱ, Elsevier B. V. , Amsterdam, 2 (2005), 239-357.  Google Scholar

[25]

IMI. Innovative medicine initiative. First Innovative Medicines Initiative Ebola projects get underway, Available from: http://www.imi.europa.eu/content/ebola-project-launch (accessed September 2016). Google Scholar

[26]

A. Kata, A postmodern Pandora's box: Anti-vaccination misinformation on the Internet, Vaccine, 28 (2010), 1709-1716.  doi: 10.1016/j.vaccine.2009.12.022.  Google Scholar

[27]

A. Kata, Anti-vaccine activists, Web 2.0, and the postmodern paradigm-An overview of tactics and tropes used online by the anti-vaccination movement, Vaccine, 30 (2012), 3778-3789.  doi: 10.1016/j.vaccine.2011.11.112.  Google Scholar

[28] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Princeton, NJ, 2008.   Google Scholar
[29]

W. P. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates, Am. J. Epidemiol., 98 (1973), 453-468.   Google Scholar

[30]

P. Manfredi and A. d'Onofrio (eds), Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer, New York, 2013. doi: 10.1007/978-1-4614-5474-8.  Google Scholar

[31]

M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

[32]

E. Miller and N. J. Gay, Epidemiological determinants of pertussis, Dev. Biol. Stand., 89 (1997), 15-23.   Google Scholar

[33]

J. D. Murray, Mathematical Biology. I. An Introduction. Third Edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.  Google Scholar

[34]

Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 325 (2010), 230-237.  doi: 10.1016/j.jmaa.2009.08.027.  Google Scholar

[35]

G. NapierD. LeeC. RobertsonA. Lawson and K. G. Pollock, A model to estimate the impact of changes in MMR vaccine uptake on inequalities in measles susceptibility in Scotland, Stat. Methods Med. Res., 25 (2016), 1185-1200.  doi: 10.1177/0962280216660420.  Google Scholar

[36]

L. F. Olsen and W. M. Schaffer, Chaos versus noisy periodicity: Alternative hypotheses for childhood epidemics, Science, 249 (1990), 499-504.  doi: 10.1126/science.2382131.  Google Scholar

[37]

T. Philipson, Private Vaccination and Public Health an empirical examination for U.S. Measles, J. Hum. Res., 31 (1996), 611-630.  doi: 10.2307/146268.  Google Scholar

[38]

I. B. Schwartz and H. L. Smith, Infinite subharmonic bifurcation in an SEIR epidemic model, J. Math. Biol., 18 (1983), 233-253.  doi: 10.1007/BF00276090.  Google Scholar

[39]

H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73.  doi: 10.2307/2341437.  Google Scholar

[40]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[41]

A. van LierJ. van de KassteeleP. de HooghI. Drijfhout and H. de Melker, Vaccine uptake determinants in The Netherlands, Eur. J. Public Health, 24 (2013), 304-309.   Google Scholar

[42]

G. S. Wallace, Why Measles Matters, 2014, Available from: http://stacks.cdc.gov/view/cdc/27488/cdc_27488_DS1.pdf (accessed November 2016). Google Scholar

[43]

Z. WangC. T. BauchS. BhattacharyyaA. d'OnofrioP. ManfrediM. PercN. PerraM. Salathé and D. Zhao, Statistical physics of vaccination, Physics Reports, 664 (2016), 1-113.  doi: 10.1016/j.physrep.2016.10.006.  Google Scholar

[44]

W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Eq., 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[45]

WHO, Vaccine safety basics, Available from: http://vaccine-safety-training.org/adverse-events-classification.html (accessed November 2016). Google Scholar

[46]

R. M. Wolfe and L. K. Sharp, Anti-vaccinationists past and present, Brit. Med. J., 325 (2002), p430.  doi: 10.1136/bmj.325.7361.430.  Google Scholar

[47]

F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[48]

X. Q. Zhao, Dynamical Systems in Population Biology, CMS Books Math. , vol. 16, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

Figure 1.  Sinusoidal fluctuations of the transmission rate. The effective BRN of system (7)-(21), $\mathcal{R}(p_2,\sigma)$, as function of $\bar{\gamma}$ and $\sigma$. Figure (a): $\mathcal{R}(p_2,\sigma)$ vs $\sigma$ with $\bar{\gamma} = 1.4 \times 10^{-4}$. Figure (b): $\mathcal{R}(p_2,\sigma)$ vs $\sigma$ with $\bar{\gamma} = 1.2 \times 10^{-4}$. Figure (c): $\mathcal{R}(p_2,\sigma)$ vs $\bar{\gamma}$ with $\sigma = 0.3$. The other parameter values are taken as described in Section 5.1.
Figure 2.  The stabilizing role of seasonality. Left panels: dynamics of model (7)-(21) for $\sigma = 0.3$. Right panels: dynamics of model (7)-(21) for $\sigma= 0.95$. The initial conditions are $S_0 = 1/R_0$, $E_0 = 9\times 10^{-6}$, $I_0 = 8\times 10^{-6}$, $p_0 = 0.95$. The other parameter values are taken as described in Section 5.1, and $\bar\gamma = 1.41 \times 10^{-4}$.
Figure 3.  Extinction (left panels) and uniform persistence (right panels) of disease. Left panels: dynamics of model (7)-(21) for $\bar\gamma = 1.5 \times 10^{-4}$. Right panels: the dynamics of model (7)-(21) for $\bar\gamma = 1.2 \times 10^{-4}$. The initial condition are $S_0 = 1/R_0$, $E_0 = 9\times 10^{-6}$, $I_0 = 8\times 10^{-6}$, $p_0 = 0.95$. The other parameter values are taken as described in Section 5.1, and $\sigma = 0.3$.
Figure 4.  The stabilizing role of seasonality in case of piecewise contact rate. Plot of the Spectral Radius of the Floquet Matrix $F$ versus the parameter $\eta = c_L/c_H$ measuring the reduction of contacts. Parameters $\gamma$ and $\alpha$ are such that $p_2 = 0.99 p_{*}$. Left Panel: $f=0.5$ (i.e. $c(t)=c_L$ for half year); right panel: $f=0.25$ (i.e. $c(t)=c_L$ for one quarter of year). The other parameter values are taken as described in Section 5.1.
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