| Citation: |
Pei Yu, Maoan Han, Wenjing Zhang. MULTIPLE RECURRENT OUTBREAK CYCLES IN AN AUTONOMOUS EPIDEMIOLOGICAL MODEL DUE TO MULTIPLE LIMIT CYCLE BIFURCATION[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2278-2298. doi: 10.11948/20200301
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MULTIPLE RECURRENT OUTBREAK CYCLES IN AN AUTONOMOUS EPIDEMIOLOGICAL MODEL DUE TO MULTIPLE LIMIT CYCLE BIFURCATION
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Abstract
Multiple recurrent outbreak cycles have been commonly observed in infectious diseases such as measles and chicken pox. This complex outbreak dynamics in epidemiologicals is rarely captured by deterministic models. In this paper, we investigate a simple 2-dimensional SI epidemiological model and propose that the coexistence of multiple attractors attributes to the complex outbreak patterns. We first determine the conditions on parameters for the existence of an isolated center, then properly perturb the model to generate Hopf bifurcation and obtain limit cycles around the center. We further analytically prove that the maximum number of the coexisting limit cycles is three, and provide a corresponding set of parameters for the existence of the three limit cycles. Simulation results demonstrate the case with the maximum coexisting attractors, which contains one stable disease free equilibrium and two stable endemic periodic solutions separated by one unstable periodic solution. Therefore, different disease outcomes can be predicted by a single nonlinear deterministic model based on different initial data. -
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References
[1] S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, P. Rohani, Seasonality and the dynamics of infectious diseases, Ecol. Lett., 2006, 9(4), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x [2] R. M. Anderson, R. M. May, B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press Inc, New York, 1992. [3] J. L. Aron, Multiple attractors in the response to a vaccination program, Theor. Popul. Biol., 1990, 38(1), 58-67. doi: 10.1016/0040-5809(90)90003-E [4] J. L. Aron, I. B. Schwartz, Seasonality and period-doubling bifurcations in an epidemic model, J. Theor. Biol., 1984, 110(4), 665-679. doi: 10.1016/S0022-5193(84)80150-2 [5] K. M. Bakker, M. E. Martinez-Bakker, B. Helm, T. J. Stevenson, Digital epidemiology reveals global childhood disease seasonality and the effects of immunization, Proc. National Academy of Sciences, 2016, 113(24), 6689-6694. doi: 10.1073/pnas.1523941113 [6] M. S. Bartlett, Measles periodicity and community size, J. Royal Statistical Soc., Series A (General), 1957, 120(1), 48-70. doi: 10.2307/2342553 [7] B. Bolker, B. Grenfell, B. Chaos and biological complexity in measles dynamics, Proc. R. Soc. Lond. B 251, 1993, (1330), 75-81. [8] B. Buonomo, N. Chitnis, A. d'Onofrio, Seasonality in epidemic models: a literature review, Ricerche di Matematica, 2017, 67(1), 1-19. [9] D. J. Earn, P. Rohani, B. M. Bolker, B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 2000, 287(5453), 667-670. doi: 10.1126/science.287.5453.667 [10] S. Ellner, B. Bailey, G. Bobashev, A. Gallant, B. Grenfell, D. Nychka, Noise and nonlinearity in measles epidemics: combining mechanistic and statistical approaches to population modeling, AM Naturalist, 1998, 151(5), 425-440. doi: 10.1086/286130 [11] S. D. Fretwell, Populations in a Seasonal Environment, Princeton University Press, Princeton, 1972. [12] P. Glendinning, L. P. Perry, Melnikov analysis of chaos in a simple epidemiological model, J. Math. Biol., 1997, 35(3), 359-373. [13] N. C. Grassly, C. Fraser, Mathematical models of infectious disease transmission, Nat. Rev. Microbiol., 2008, 6(6), 477-487. doi: 10.1038/nrmicro1845 [14] B. Grenfell, J. Harwood, (Meta)population dynamics of infectious diseases, Trends Ecol. Evol., 1997, 12(10), 395-399. doi: 10.1016/S0169-5347(97)01174-9 [15] D. E. Griffin, Immune Responses During Measles Virus Infection, Springer, Berlin, Heidelberg, 1995, 117-134. [16] M. Han, P. Yu, Normal forms, Melnikov functions and bifurcations of limit cycles, Vol. 181, Springer-Verlag, London, 2012. [17] H. W. Hethcote, P. van Den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 1991, 29(3), 271-287. doi: 10.1007/BF00160539 [18] S. A. Levin, B. Grenfell, A. Hastings, A. S. Perelson, Mathematical and computational challenges in population biology and ecosystems science, Science, 1997, 275(5298), 334-343. doi: 10.1126/science.275.5298.334 [19] W. M. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 1987, 25(4), 359-380. doi: 10.1007/BF00277162 [20] W. M. Liu, S. A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, J. Math. Biol., 1986, 23(2), 187-204. doi: 10.1007/BF00276956 [21] W. P. London, J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps: I. seasonal variation in contact rates, AM J. Epidemiol., 1973, 98(6), 453-468. doi: 10.1093/oxfordjournals.aje.a121575 [22] L. F. Olsen, W. M. Schaffer, Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics, Science, 1990, 249(4968), 499-504. doi: 10.1126/science.2382131 [23] L. F. Olsen, G. L. Truty, W. M. Schaffer, Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark, Theor. Popul. Biol., 1988, 33(3), 344-370. [24] V. E. Pitzer, C. Viboud, W. J. Alonso, T. Wilcox, C. J. Metcalf, C. A. Steiner, A. K. Haynes, B. T. Grenfell, Environmental drivers of the spatiotemporal dynamics of respiratory syncytial virus in the united states, PLoS Pathogens, 2015, 11(1), e1004591. doi: 10.1371/journal.ppat.1004591 [25] W. G. Van Panhuis, J. Grefenstette, S. Y. Jung, N. S. Chok, A. Cross, H. Eng, B. Y. Lee, V. Zadorozhny, S. Brown, D. Cummings, D. S. Burke, Contagious diseases in the united states from 1888 to the present, New Engl. J. Med., 2013, 369(22), 2152-2158. doi: 10.1056/NEJMms1215400 [26] W. M. Schaffer, B. Kendall, C. W. Tidd, L. F. Olsen, Transient periodicity and episodic predictability in biological dynamics, IMA J. Math. Appl. Med. Biol., 1993, 10(4), 227-247. doi: 10.1093/imammb/10.4.227 [27] D. Schenzle, An age-structured model of pre-and post-vaccination measles transmission, IMA J. Math. Appl. Med. Biol., 1984, 1(2), 169-191. doi: 10.1093/imammb/1.2.169 [28] I. B. Schwartz, Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models, J. Math. Biol., 1985, 21(3), 347-361. doi: 10.1007/BF00276232 [29] H. Smith, Subharmonic bifurcation in an sir epidemic model, J. Math. Biol., 1983, 17(2), 163-177. doi: 10.1007/BF00305757 [30] J. Wingfield, G. Kenagy, Natural Regulation of Reproductive Cycles, In Vertebrate Endocrinology: Fundamentals and Biomedical Implications (Eds. P. K. T. Pang and M. P. Schreibman), Vol. 4, Part B, 181-241, 1991. [31] P. Yu, Computation of normal forms via a perturbation technique, J. Sound and Vib., 1998, 211(1), 19-38. doi: 10.1006/jsvi.1997.1347 [32] P. Yu, W. Zhang, L. M. Wahl, Dynamical analysis and simulation of a 2-dimensional disease model with convex incidence, Commun. Nonlinear Sci. Numer. Simulat., 2016, 37, 163-192. doi: 10.1016/j.cnsns.2015.12.022 [33] W. Zhang, L. M. Wahl, P. Yu, Conditions for transient viremia in deterministic in-host models: viral blips need no exogenous trigger, SIAM J. Appl. Math., 2013, 73(2), 853-881. doi: 10.1137/120884535 [34] W. Zhang, L. M. Wahl, P. Yu, Viral blips may not need a trigger: How transient viremia can arise in deterministic in-host models, SIAM Rev., 2014, 56(1), 127-155. doi: 10.1137/130937421 -
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Pei Yu, Maoan Han, Wenjing Zhang. MULTIPLE RECURRENT OUTBREAK CYCLES IN AN AUTONOMOUS EPIDEMIOLOGICAL MODEL DUE TO MULTIPLE LIMIT CYCLE BIFURCATION[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2278-2298. doi: 10.11948/20200301
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Figure 1. Bifurcation diagrams for system (1.4) with the equilibria
$ {\rm E_0} $ and$ {\rm E_1} $ shown in red and blue colors, respectively: (a)$ R_0<1 $ ; (b)$ R_0 = 1 $ ; and (c)$ R_0 > 1 $ . -
Figure 2. The phase portrait of (4.9) when
$ D \! = \! \frac{3}{16} $ . -
Figure 3. Simulation of system (1.4) for
$ D = 0.2 $ ,$ b_{00} = 0 $ ,$ b_{01} = -0.10658741 $ ,$ b_{02} = -2.38197601 $ and$ b_{11} = 0.04474986 $ : (a)$ \epsilon = 0.1 $ , showing two limit cycles with outer stable and inner unstable, both of them enclosing the stable equilibrium$ {\rm E_-} $ ; and (b)$ \epsilon = -0.1 $ , showing no limit cycles, with the trajectories inside the separatrix spiraling from the separatrix and eventually converging to the$ {\rm E_0} $ , while the trajectories outside the separatrix converging directly to the$ {\rm E_0} $ . -
Figure 4. Simulation of three limit cycles in system (1.4) for
$ D = 0.1875 $ ,$ b_{00} = 0.01879970 $ ,$ b_{01} = 0.15844293 $ ,$ b_{11} = -0.09655128 $ and$ b_{12} = -0.40310728 $ : (a) two trajectories starting from two different initial points (plotted in light and dark gray colors) converging towards the large stable limit cycle; and (b) two trajectories starting from two different initial points (plotted in black and gray colors) converging to the small stable limit cycle; an unstable limit cycle between the larger and smaller stable limit cycles is not shown.