Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection

  • Received: 01 March 2017 Accepted: 29 July 2017 Published: 01 June 2018
  • MSC : Primary: 92D30; Secondary: 92C42, 60J20, 91A13

  • We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.

    Citation: Francesco Salvarani, Gabriel Turinici. Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 629-652. doi: 10.3934/mbe.2018028

    Related Papers:

    [1] Yicang Zhou, Zhien Ma . Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences and Engineering, 2009, 6(2): 409-425. doi: 10.3934/mbe.2009.6.409
    [2] Ke Guo, Wanbiao Ma . Global dynamics of an SI epidemic model with nonlinear incidence rate, feedback controls and time delays. Mathematical Biosciences and Engineering, 2021, 18(1): 643-672. doi: 10.3934/mbe.2021035
    [3] Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya . Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences and Engineering, 2010, 7(2): 347-361. doi: 10.3934/mbe.2010.7.347
    [4] Hui Cao, Yicang Zhou, Zhien Ma . Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Mathematical Biosciences and Engineering, 2013, 10(5&6): 1399-1417. doi: 10.3934/mbe.2013.10.1399
    [5] A. Q. Khan, M. Tasneem, M. B. Almatrafi . Discrete-time COVID-19 epidemic model with bifurcation and control. Mathematical Biosciences and Engineering, 2022, 19(2): 1944-1969. doi: 10.3934/mbe.2022092
    [6] Marcin Choiński, Mariusz Bodzioch, Urszula Foryś . A non-standard discretized SIS model of epidemics. Mathematical Biosciences and Engineering, 2022, 19(1): 115-133. doi: 10.3934/mbe.2022006
    [7] John E. Franke, Abdul-Aziz Yakubu . Periodically forced discrete-time SIS epidemic model with disease induced mortality. Mathematical Biosciences and Engineering, 2011, 8(2): 385-408. doi: 10.3934/mbe.2011.8.385
    [8] Zhen Jin, Guiquan Sun, Huaiping Zhu . Epidemic models for complex networks with demographics. Mathematical Biosciences and Engineering, 2014, 11(6): 1295-1317. doi: 10.3934/mbe.2014.11.1295
    [9] Wenzhang Huang, Maoan Han, Kaiyu Liu . Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences and Engineering, 2010, 7(1): 51-66. doi: 10.3934/mbe.2010.7.51
    [10] Peter Rashkov, Ezio Venturino, Maira Aguiar, Nico Stollenwerk, Bob W. Kooi . On the role of vector modeling in a minimalistic epidemic model. Mathematical Biosciences and Engineering, 2019, 16(5): 4314-4338. doi: 10.3934/mbe.2019215
  • We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.


    [1] [ A. Abakuks, Optimal immunisation policies for epidemics, Advances in Appl. Probability, 6 (1974): 494-511.
    [2] [ R. M. Anderson and R. M. May, Infectious Diseases of Humans Dynamics and Control, Oxford University Press, 1992.
    [3] [ J. Appleby, Getting a flu shot? it may be better to wait, CNN, September 15, http://edition.cnn.com/2016/09/26/health/wait-for-flu-shot/index.html, 2016.
    [4] [ N. Bacaër, A Short History of Mathematical Population Dynamics, Springer-Verlag London, Ltd., London, 2011.
    [5] [ Y. Bai,N. Shi,Q. Lu,L. Yang,Z. Wang,L. Li,H. Han,D. Zheng,F. Luo,Z. Zhang,X. Ai, Immunological persistence of a seasonal influenza vaccine in people more than 3 years old, Human Vaccines & Immunotherapeutics, 11 (2015): 1648-1653.
    [6] [ C. T. Bauch,D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. USA, 101 (2004): 13391-13394 (electronic).
    [7] [ C. T. Bauch,A. P. Galvani,D. J. D. Earn, Group interest versus self-interest in smallpox vaccination policy, Proceedings of the National Academy of Sciences, 100 (2003): 10564-10567.
    [8] [ C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc Biol Sci, 272 (2005): 1669-1675.
    [9] [ E. A. Belongia,M. E. Sundaram,D. L. McClure,J. K. Meece,J. Ferdinands,J. J. VanWormer, Waning vaccine protection against influenza a (h3n2) illness in children and older adults during a single season, Vaccine, 33 (2015): 246-251.
    [10] [ Adrien Blanchet and Guillaume Carlier, From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130398, 11pp.
    [11] [ R. Breban, R. Vardavas and S. Blower, Mean-field analysis of an inductive reasoning game: Application to influenza vaccination Phys. Rev. E, 76 (2007), 031127.
    [12] [ D. L. Brito,E. Sheshinski,M. D. Intriligator, Externalities and compulsary vaccinations, Journal of Public Economics, 45 (1991): 69-90.
    [13] [ B. Buonomo,A. d'Onofrio,D. Lacitignola, Global stability of an {SIR} epidemic model with information dependent vaccination, Mathematical Biosciences, 216 (2008): 9-16.
    [14] [ P. Cardaliaguet,S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM Control Optim. Calc. Var., 23 (2017): 569-591.
    [15] [ F. Carrat,A. Flahault, Influenza vaccine: The challenge of antigenic drift, Vaccine, 25 (2007): 6852-6862.
    [16] [ F. H. Chen, A susceptible-infected epidemic model with voluntary vaccinations, Journal of Mathematical Biology, 53 (2006): 253-272.
    [17] [ M. L. Clements,B. R. Murphy, Development and persistence of local and systemic antibody responses in adults given live attenuated or inactivated influenza a virus vaccine, Journal of Clinical Microbiology, 23 (1986): 66-72.
    [18] [ C. T. Codeço,P. M. Luz,F. Coelho,A. P Galvani,C. Struchiner, Vaccinating in disease-free regions: a vaccine model with application to yellow fever, Journal of The Royal Society Interface, 4 (2007): 1119-1125.
    [19] [ F. Coelho and C. T. Codeço, Dynamic modeling of vaccinating behavior as a function of individual beliefs PLoS Comput Biol, 5 (2009), e1000425, 10pp.
    [20] [ M.-G. Cojocaru, Dynamic equilibria of group vaccination strategies in a heterogeneous population, Journal of Global Optimization, 40 (2008): 51-63.
    [21] [ R. B. Couch,J. A. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (1983): 529-549.
    [22] [ N. Cox, Influenza seasonality: Timing and formulation of vaccines, Bulletin of the World Health Organization, 92 (2014): 311-311.
    [23] [ O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.
    [24] [ Josu Doncel, Nicolas Gast, and Bruno Gaujal, Mean-Field Games with Explicit Interactions, working paper or preprint, 2016.
    [25] [ A. d'Onofrio,P. Manfredi,E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71 (2007): 301-317.
    [26] [ A. d'Onofrio,P. Manfredi,E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Mathematical Medicine and Biology, 25 (2008): 337-357.
    [27] [ P. Doutor,P. Rodrigues,M. do Céu Soares,F. A. C. C. Chalub, Optimal vaccination strategies and rational behaviour in seasonal epidemics, Journal of Mathematical Biology, 73 (2016): 1437-1465.
    [28] [ J. Dushoff,J. B Plotkin,C. Viboud,D. J. D. Earn,L. Simonsen, Mortality due to influenza in the United States-an annualized regression approach using multiple-cause mortality data, American journal of epidemiology, 163 (2006): 181-187.
    [29] [ J. M. Ferdinands,A. M. Fry,S. Reynolds,J. G. Petrie,B. Flannery,M. L. Jackson,E. A. Belongia, Intraseason waning of influenza vaccine protection: Evidence from the us influenza vaccine effectiveness network, 2011-2012 through 2014-2015, Clinical Infectious Diseases, 64 (2017): p544.
    [30] [ P. E. M. Fine,J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, American Journal of Epidemiology, 124 (1986): 1012-1020.
    [31] [ P. J. Francis, Optimal tax/subsidy combinations for the flu season, Journal of Economic Dynamics and Control, 28 (2004): 2037-2054.
    [32] [ D. Fudenberg and D. K. Levine, The Theory of Learning in Games volume 2 of MIT Press Series on Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 1998.
    [33] [ S. Funk,M. Salathé,V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of The Royal Society Interface, 7 (2010): 1247-1256.
    [34] [ A. P. Galvani,T. C. Reluga,G. B. Chapman, Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum, Proceedings of the National Academy of Sciences, 104 (2007): 5692-5697.
    [35] [ P.-Y. Geoffard,T. Philipson, Disease eradication: Private versus public vaccination, The American Economic Review, 87 (1997): 222-230.
    [36] [ N. C. Grassly,C. Fraser, Seasonal infectious disease epidemiology, Proceedings of the Royal Society of London B: Biological Sciences, 273 (2006): 2541-2550.
    [37] [ S. Greenland and R. R. Frerichs, On measures and models for the effectiveness of vaccines and vaccination programmes, International Journal of Epidemiology, 17 (1988), p456.
    [38] [ M. E. Halloran, I. M. Longini and C. J. Struchiner, Design and Analysis of Vaccine Studies, Statistics for Biology and Health. Springer New York, 2009.
    [39] [ H. W. Hethcote,P. Waltman, Optimal vaccination schedules in a deterministic epidemic model, Mathematical Biosciences, 18 (1973): 365-381.
    [40] [ M. Huang,R. P. Malhamé,P. E. Caines, Nash equilibria for large-population linear stochastic systems of weakly coupled agents, In Elkébir Boukas and Roland P. Malhamé, editors,, Analysis, Control and Optimization of Complex Dynamic Systems, Springer US,, 4 (2005): 215-252.
    [41] [ M. Huang,R. P. Malhamé,P. E. Caines, Large population stochastic dynamic games: Closed-loop mckean-vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006): 221-252.
    [42] [ R. Jordan,D. Kinderlehrer,F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998): 1-17.
    [43] [ S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J., 8 (1941): 457-459.
    [44] [ E. Kissling, B. Nunes, C. Robertson, M. Valenciano, A. Reuss, A. Larrauri, J. M. Cohen, B. Oroszi, C. Rizzo, A. Machado, D. Pitigoi, L. Domegan, I. Paradowska-Stankiewicz, U. Buchholz, A. Gherasim, I. Daviaud, J. K. Horvath, A. Bella, E. Lupulescu, J. O'Donnell, M. Korczynska, A. Moren and I.-MOVE case-control study team, I-move multicentre casecontrol study 2010/11 to 2014/15: Is there within-season waning of influenza type/subtype vaccine effectiveness with increasing time since vaccination?, Euro Surveill., 21 (2016), 30201.
    [45] [ A. Lachapelle,J. Salomon,G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010): 567-588.
    [46] [ L. Laguzet,G. Turinici, Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis, Mathematical Biosciences, 263 (2015): 180-197.
    [47] [ L. Laguzet,G. Turinici, Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-2010 influenza A (H1N1) epidemic in France, Bulletin of Mathematical Biology, 77 (2015): 1955-1984.
    [48] [ J.-M. Lasry,P.-L. Lions, Lions, Jeux à champ moyen. I: Le cas stationnaire,, C. R., Math., Acad. Sci. Paris, 343 (2006): 619-625.
    [49] [ J.-M. Lasry,P.-L. Lions, Lions, Jeux à champ moyen. II: Horizon fini et contrôle optimal,, C. R., Math., Acad. Sci. Paris, 343 (2006): 679-684.
    [50] [ J.-M. Lasry,P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007): 229-260.
    [51] [ A. S. Monto,S. E. Ohmit,J. G. Petrie,E. Johnson,R. Truscon,E. Teich,J. Rotthoff,M. Boulton,J. C. Victor, Comparative efficacy of inactivated and live attenuated influenza vaccines, New England Journal of Medicine, 361 (2009): 1260-1267.
    [52] [ R. Morton,K. H. Wickwire, On the optimal control of a deterministic epidemic, Advances in Appl. Probability, 6 (1974): 622-635.
    [53] [ J. Müller, Optimal vaccination strategies-for whom?, Mathematical Biosciences, 139 (1997): 133-154.
    [54] [ S. Ng,V. J. Fang,D. K. M. Ip,K.-H. Chan,G. M. Leung,J. S. Malik Peiris,B. J. Cowling, Estimation of the association between antibody titers and protection against confirmed influenza virus infection in children, Journal of Infectious Diseases, 208 (2013): 1320-1324.
    [55] [ K. L. Nichol,A. Lind,K. L. Margolis,M. Murdoch,R. McFadden,M. Hauge,S. Magnan,M. Drake, The effectiveness of vaccination against influenza in healthy, working adults, New England Journal of Medicine, 333 (1995): 889-893.
    [56] [ M. T Osterholm,N. S. Kelley,A. Sommer,E. A. Belongia, Efficacy and effectiveness of influenza vaccines: A systematic review and meta-analysis, The Lancet Infectious Diseases, 12 (2012): 36-44.
    [57] [ T. C. Reluga,C. T. Bauch,A. P. Galvani, Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204 (2006): 185-198.
    [58] [ T. C. Reluga,A. P. Galvani, A general approach for population games with application to vaccination, Mathematical Biosciences, 230 (2011): 67-78.
    [59] [ S. P. Sethi,P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Oper. Res. Soc., 29 (1978): 129-136.
    [60] [ E. Shim,G. B. Chapman,J. P. Townsend,A. P. Galvani, The influence of altruism on influenza vaccination decisions, Journal of The Royal Society Interface, 9 (2012): 2234-2243.
    [61] [ D. M. Skowronski,S. Aleina Tweed,S. Aleina Tweed,G. De Serres, Rapid decline of influenza vaccine-induced antibody in the elderly: Is it real, or is it relevant?, The Journal of Infectious Diseases, 197 (2008): 490-502.
    [62] [ N. M. Smith, J. S. Bresee, D. K. Shay, T. M. Uyeki, N. J. Cox and R. A. Strikas, Prevention and control of influenza: Recommendations of the advisory committee on immunization practices (acip), MMWRRecomm Rep, 55 (2006), 1-42. https://www.cdc.gov/mmwr/preview/mmwrhtml/rr5510a1.htm.
    [63] [ P. G. Smith, L. C. Rodrigues and P. E. M. Fine, Assessment of the protective efficacy of vaccines against common diseases using case-control and cohort studies, International Journal of Epidemiology, 13 (1984), 87-93.
    [64] [ C. J. Struchiner, M. E. Halloran, J. M. Robins and A. Spielman, The behaviour of common measures of association used to assess a vaccination programme under complex disease transmission patterns-a computer simulation study of malaria vaccines, International Journal of Epidemiology, 19 (1990), 187-196.
    [65] [ I. Swiecicki, T. Gobron and D. Ullmo, Schrödinger approach to mean field games, Phys. Rev. Lett., 116(2016), 128701.
    [66] [ J. D Tamerius, J. Shaman, W. J. Alonso, K. Bloom-Feshbach, C. K. Uejio, An. Comrie and C. Viboud, Environmental predictors of seasonal influenza epidemics across temperate and tropical climates, PLoS Pathog, 9 (2013), e1003194.
    [67] [ J. J. Treanor,H. K. Talbot,S. E. Ohmit,L. A. Coleman,M. G. Thompson,P.-Y. Cheng,J. G. Petrie,G. Lofthus,J. K. Meece,J. V. Williams,L. Berman,C. Breese Hall,A. S. Monto,M. R. Griffin,E. Belongia,D. K. Shay, Effectiveness of seasonal influenza vaccines in the United States during a season with circulation of all three vaccine strains, Clinical Infectious Diseases, 55 (2012): 951-959.
    [68] [ G. Turinici, Metric gradient flows with state dependent functionals: the Nash-MFG equilibrium flows and their numerical schemes, Nonlinear Analysis 165 (2017) 163-181.
    [69] [ R. Vardavas, R. Breban and S. Blower, Can influenza epidemics be prevented by voluntary vaccination?, PLoS Comput Biol, 3 (2007), e85.
    [70] [ G. A. Weinberg and P. G. Szilagyi, Vaccine epidemiology: Efficacy, effectiveness, and the translational research roadmap, Journal of Infectious Diseases, 201 (2010), 1607-1610
    [71] [ X. Zhao,V. J. Fang,S. E. Ohmit,A. S. Monto,A. R. Cook,B. J. Cowling, Quantifying protection against influenza virus infection measured by hemagglutination-inhibition assays in vaccine trials,, Epidemiology, 27 (2016): 143-151.
  • This article has been cited by:

    1. Guirong Jiang, Qigui Yang, Periodic solutions and bifurcation in an SIS epidemic model with birth pulses, 2009, 50, 08957177, 498, 10.1016/j.mcm.2009.04.021
    2. Soodeh Hosseini, Mohammad Abdollahi Azgomi, Adel Torkaman Rahmani, Malware propagation modeling considering software diversity and immunization, 2016, 13, 18777503, 49, 10.1016/j.jocs.2016.01.002
    3. Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya, Global stability for a discrete SIS epidemic model with immigration of infectives, 2012, 18, 1023-6198, 1913, 10.1080/10236198.2011.602973
    4. Raul Nistal, Manuel de la Sen, Santiago Alonso-Quesada, Asier Ibeas, 2014, A nonlinear SEIR epidemic model with feedback vaccination control, 978-3-9524269-1-3, 158, 10.1109/ECC.2014.6862291
    5. Jinhu Xu, Yan Geng, Stability preserving NSFD scheme for a delayed viral infection model with cell-to-cell transmission and general nonlinear incidence, 2017, 23, 1023-6198, 893, 10.1080/10236198.2017.1304933
    6. Yuhua Long, Lin Wang, Global dynamics of a delayed two-patch discrete SIR disease model, 2020, 83, 10075704, 105117, 10.1016/j.cnsns.2019.105117
    7. Xiaolin Fan, Lei Wang, Zhidong Teng, Global dynamics for a class of discrete SEIRS epidemic models with general nonlinear incidence, 2016, 2016, 1687-1847, 10.1186/s13662-016-0846-y
    8. Jinhu Xu, Jiangyong Hou, Yan Geng, Suxia Zhang, Dynamic consistent NSFD scheme for a viral infection model with cellular infection and general nonlinear incidence, 2018, 2018, 1687-1847, 10.1186/s13662-018-1560-8
    9. Zohreh Eskandari, Javad Alidousti, Stability and codimension 2 bifurcations of a discrete time SIR model, 2020, 357, 00160032, 10937, 10.1016/j.jfranklin.2020.08.040
    10. Jinhu Xu, Yan Geng, A nonstandard finite difference scheme for a multi-group epidemic model with time delay, 2017, 2017, 1687-1847, 10.1186/s13662-017-1415-8
    11. Zengyun Hu, Zhidong Teng, Chaojun Jia, Chi Zhang, Long Zhang, Dynamical analysis and chaos control of a discrete SIS epidemic model, 2014, 2014, 1687-1847, 10.1186/1687-1847-2014-58
    12. Yueli Luo, Shujing Gao, Dehui Xie, Yanfei Dai, A discrete plant disease model with roguing and replanting, 2015, 2015, 1687-1847, 10.1186/s13662-014-0332-3
    13. Qamar Din, Qualitative behavior of a discrete SIR epidemic model, 2016, 09, 1793-5245, 1650092, 10.1142/S1793524516500923
    14. Tailei Zhang, Junli Liu, Zhidong Teng, Threshold conditions for a discrete nonautonomous SIRS model, 2015, 38, 01704214, 1781, 10.1002/mma.3186
    15. Zengyun Hu, Zhidong Teng, Long Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, 2014, 97, 03784754, 80, 10.1016/j.matcom.2013.08.008
    16. Zhidong Teng, Lei Wang, Linfei Nie, Global attractivity for a class of delayed discrete SIRS epidemic models with general nonlinear incidence, 2015, 38, 01704214, 4741, 10.1002/mma.3389
    17. Zengyun Hu, Zhidong Teng, Haijun Jiang, Stability analysis in a class of discrete SIRS epidemic models, 2012, 13, 14681218, 2017, 10.1016/j.nonrwa.2011.12.024
    18. Jahangir Chowdhury, Sourav Rana, Sabyasachi Bhattacharya, Priti Kumar Roy, 2017, Chapter 23, 978-981-10-3757-3, 319, 10.1007/978-981-10-3758-0_23
    19. Lei Wang, Zhidong Teng, Haijun Jiang, Global attractivity of a discrete SIRS epidemic model with standard incidence rate, 2013, 36, 01704214, 601, 10.1002/mma.2734
    20. Tailei Zhang, Permanence and extinction in a nonautonomous discrete SIRVS epidemic model with vaccination, 2015, 271, 00963003, 716, 10.1016/j.amc.2015.09.071
    21. Zengyun Hu, Linlin Chang, Zhidong Teng, Xi Chen, Bifurcation analysis of a discrete S I R S SIRS epidemic model with standard incidence rate, 2016, 2016, 1687-1847, 10.1186/s13662-016-0874-7
    22. JUPING ZHANG, ZHEN JIN, DISCRETE TIME SI AND SIS EPIDEMIC MODELS WITH VERTICAL TRANSMISSION, 2009, 17, 0218-3390, 201, 10.1142/S0218339009002788
    23. Junhong Li, Ning Cui, Bifurcation and Chaotic Behavior of a Discrete-Time SIS Model, 2013, 2013, 1026-0226, 1, 10.1155/2013/705601
    24. Qiaoling Chen, Zhidong Teng, Lei Wang, Haijun Jiang, The existence of codimension-two bifurcation in a discrete SIS epidemic model with standard incidence, 2013, 71, 0924-090X, 55, 10.1007/s11071-012-0641-6
    25. S.M. Salman, E. Ahmed, A mathematical model for Creutzfeldt Jacob Disease (CJD), 2018, 116, 09600779, 249, 10.1016/j.chaos.2018.09.041
    26. Pengfei Wu, Jingshun Duanmu, Jiyong Du, 2012, Analysis of an SIS Epidemic Model with Disease-Induced Mortality, 978-0-7695-4647-6, 596, 10.1109/ICCSEE.2012.17
    27. Yan Geng, Jinhu Xu, Stability preserving NSFD scheme for a multi-group SVIR epidemic model, 2017, 01704214, 10.1002/mma.4357
    28. Lei Wang, Qianqian Cui, Zhidong Teng, Global dynamics in a class of discrete-time epidemic models with disease courses, 2013, 2013, 1687-1847, 10.1186/1687-1847-2013-57
    29. J. Hallberg Szabadváry, Y. Zhou, On qualitative analysis of a discrete time SIR epidemical model, 2021, 7, 25900544, 100067, 10.1016/j.csfx.2021.100067
    30. Fang Zheng, 2022, Chapter 12, 978-981-19-2447-7, 121, 10.1007/978-981-19-2448-4_12
    31. Zhidong Teng, Linfei Nie, Jiabo Xu, Dynamical behaviors of a discrete SIS epidemic model with standard incidence and stage structure, 2013, 2013, 1687-1847, 10.1186/1687-1847-2013-87
    32. Javier Cifuentes-Faura, Ursula Faura-Martínez, Matilde Lafuente-Lechuga, Mathematical Modeling and the Use of Network Models as Epidemiological Tools, 2022, 10, 2227-7390, 3347, 10.3390/math10183347
    33. Limin Zhang, Jiaxin Gu, Guangyuan Liao, Bifurcations and model fitting of a discrete epidemic system with incubation period and saturated contact rate, 2025, 1023-6198, 1, 10.1080/10236198.2025.2457987
    34. Jahangir Chowdhury, Fahad Al Basir, Anirban Mukherjee, Priti Kumar Roy, A theta logistic model for the dynamics of whitefly borne mosaic disease in Cassava: impact of roguing and insecticide spraying, 2025, 1598-5865, 10.1007/s12190-025-02419-x
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5354) PDF downloads(732) Cited by(9)

Article outline

Figures and Tables

Figures(8)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog