• Previous Article
    Optimal time to intervene: The case of measles child immunization
  • MBE Home
  • This Issue
  • Next Article
    A frailty model for intervention effectiveness against disease transmission when implemented with unobservable heterogeneity
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.

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

[4]

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

[5]

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

[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.

[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.

[8]

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

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

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

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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. 

[42]

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

[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.

[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.

[45]

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

[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.

[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.

[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.

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.

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

[4]

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

[5]

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

[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.

[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.

[8]

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

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

[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.

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

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[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.

[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.

[39]

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

[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.

[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. 

[42]

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

[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.

[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.

[45]

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

[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.

[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.

[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.

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.
[1]

Lianju Sun, Ziyou Gao, Yiju Wang. A Stackelberg game management model of the urban public transport. Journal of Industrial and Management Optimization, 2012, 8 (2) : 507-520. doi: 10.3934/jimo.2012.8.507

[2]

Eunha Shim, Beth Kochin, Alison Galvani. Insights from epidemiological game theory into gender-specific vaccination against rubella. Mathematical Biosciences & Engineering, 2009, 6 (4) : 839-854. doi: 10.3934/mbe.2009.6.839

[3]

Dennis L. Chao, Dobromir T. Dimitrov. Seasonality and the effectiveness of mass vaccination. Mathematical Biosciences & Engineering, 2016, 13 (2) : 249-259. doi: 10.3934/mbe.2015001

[4]

Ross Cressman, Vlastimil Křivan. Using chemical reaction network theory to show stability of distributional dynamics in game theory. Journal of Dynamics and Games, 2021  doi: 10.3934/jdg.2021030

[5]

Marta Faias, Emma Moreno-García, Myrna Wooders. A strategic market game approach for the private provision of public goods. Journal of Dynamics and Games, 2014, 1 (2) : 283-298. doi: 10.3934/jdg.2014.1.283

[6]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics and Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013

[7]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic and Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187

[8]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981

[9]

Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009

[10]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036

[11]

Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103

[12]

Amina-Aicha Khennaoui, A. Othman Almatroud, Adel Ouannas, M. Mossa Al-sawalha, Giuseppe Grassi, Viet-Thanh Pham. The effect of caputo fractional difference operator on a novel game theory model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4549-4565. doi: 10.3934/dcdsb.2020302

[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic and Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[14]

Yadong Shu, Ying Dai, Zujun Ma. Evolutionary game theory analysis of supply chain with fairness concerns of retailers. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022098

[15]

Julien Arino, Chris Bauch, Fred Brauer, S. Michelle Driedger, Amy L. Greer, S.M. Moghadas, Nick J. Pizzi, Beate Sander, Ashleigh Tuite, P. van den Driessche, James Watmough, Jianhong Wu, Ping Yan. Pandemic influenza: Modelling and public health perspectives. Mathematical Biosciences & Engineering, 2011, 8 (1) : 1-20. doi: 10.3934/mbe.2011.8.1

[16]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics and Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485

[17]

Akio Matsumoto, Ferenc Szidarovszky. Stability switching and its directions in cournot duopoly game with three delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (11) : 5905-5923. doi: 10.3934/dcdsb.2021069

[18]

Jewaidu Rilwan, Poom Kumam, Onésimo Hernández-Lerma. Stability of international pollution control games: A potential game approach. Journal of Dynamics and Games, 2022, 9 (2) : 191-202. doi: 10.3934/jdg.2022003

[19]

Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic mean-field-game of backward stochastic differential systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 653-678. doi: 10.3934/mcrf.2018028

[20]

Georgios Konstantinidis. A game theoretic analysis of the cops and robber game. Journal of Dynamics and Games, 2014, 1 (4) : 599-619. doi: 10.3934/jdg.2014.1.599

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (445)
  • HTML views (219)
  • Cited by (0)

[Back to Top]