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doi: 10.3934/dcdsb.2021097
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## Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases

 Department of Mathematics and Statistics, The University of Toledo, Toledo, OH 43606, USA

Received  October 2020 Revised  February 2021 Early access March 2021

Fund Project: This research was partially supported by Simons Foundation-Collaboration Grants for Mathematicians 523360

Disease transmission can present significantly different cyclic patterns including small fluctuations, regular oscillations, and singular oscillations with short endemic period and long inter-epidemic period. In this paper we consider the slow-fast dynamics and nonlinear oscillations during the transmission of mosquito-borne diseases. Under the assumption that the host population has a small natural death rate, we prove the existence of relaxation oscillation cycles by geometric singular perturbation techniques and the delay of stability loss. Generation and annihilation of periodic orbits are investigated through local, semi-local bifurcations and bifurcation of slow-fast cycles. It turns out that relaxation oscillation cycles occur only if the basic reproduction number $\mathcal{R}_0$ is greater than 1, while small fluctuations and regular oscillations exist under much less restrictive conditions. Our results here provide a sound explanation for different cyclic patterns exhibited in the transmission of mosquito-borne diseases.

Citation: Chunhua Shan. Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021097
##### References:
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Rozov, Asymptotic Methods in Singularly Perturbed Systems, translated from the Russian by I, Aleksanova, Monographs in Contemporary Mathematics, Consultants Bureau, New York, 1994.  Google Scholar [27] N. G. Reich, et al., Interactions between serotypes of dengue highlight epidemiological impact of cross-immunity, J. R. Soc. Interface., 10 (2013), art. no. 0414. doi: 10.1098/rsif.2013.0414.  Google Scholar [28] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar [29] S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation, J. Differential Equations, 60 (1985), 131-141.  doi: 10.1016/0022-0396(85)90124-X.  Google Scholar [30] C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar [31] C. Shan, Y. Yi and H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations, 260 (2016), 4339-4365.  doi: 10.1016/j.jde.2015.11.009.  Google Scholar [32] M. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, Springer Nature Switzerland AG, 2020. doi: 10.1007/978-3-030-36399-4.  Google Scholar

show all references

##### References:
 [1] A. Abdelrazec, J. Belair, C. Shan and H. Zhu, Modeling the spread and control of dengue with limited public health resources, Math. Biosci., 271 (2016), 136-145.  doi: 10.1016/j.mbs.2015.11.004.  Google Scholar [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, Oxford University Press, Oxford, UK, 1992.   Google Scholar [3] E. Benoit, Linear dynamic bifurcation with noise, in: E. Benoit (Ed.), Dynamic Bifurcations, Luminy, 1990, in: Lecture Notes in Math., vol.1493, Springer, Berlin, 1991,131–150. doi: 10.1007/BFb0085028.  Google Scholar [4] CDC, West Nile virus final annual maps & data for 1999-2018, https://www.cdc.gov/westnile/statsmaps/finalmapsdata/index.html. Google Scholar [5] S.-N. Chow, W. Liu and Y. Yi, Center manifold theory for smooth invariant manifolds, Trans. Amer. Math. Soc., 352 (2000), 5179-5211.  doi: 10.1090/S0002-9947-00-02443-0.  Google Scholar [6] S.-N. Chow, W. Liu and Y. Yi, Center manifold theory for invariant sets, J. Differential Equations, 168 (2000), 355-385.  doi: 10.1006/jdeq.2000.3890.  Google Scholar [7] P. De Maesschalck and F. Dumortier, Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), 225-267.  doi: 10.1016/j.jde.2005.01.004.  Google Scholar [8] P. De Maesschalck, F. Dumortier and R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math., 22 (2011), 165-206.  doi: 10.1016/j.indag.2011.09.008.  Google Scholar [9] P. De Maesschalck and S. Schecter, The entry-exit function and geometric singular perturbation theory, J. Differential Equations, 260 (2016), 6697-6715.  doi: 10.1016/j.jde.2016.01.008.  Google Scholar [10] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math J., 21 (1971), 193-226.  doi: 10.1512/iumj.1972.21.21017.  Google Scholar [11] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar [12] H. W. Hethcote, Asymptotic behavior in a deterministic epidemic model, Bull. Math. Biol., 35 (1973), 607-614.   Google Scholar [13] H. W. Hethcote, H. W. Stech and P. Van Den Driessche, Nonlinear oscillations in epidemic models, SIAM J. Appl. Math., 40 (1981), 1-9.  doi: 10.1137/0140001.  Google Scholar [14] T.-H. Hsu, Number and stability of relaxation oscillations for predator-prey systems with small death rates, SIAM J. Appl. Dyn. Syst., 18 (2019), 33-67.  doi: 10.1137/18M1166705.  Google Scholar [15] J. Huang, S. Ruan, P. Yu and Y. Zhang, Bifurcation analysis of a mosquito population model with a saturated release rate of sterile mosquitoes, SIAM J. Appl. Dyn. Syst., 18 (2019), 939-972.  doi: 10.1137/18M1208435.  Google Scholar [16] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.  Google Scholar [17] M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.  Google Scholar [18] J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Appl. Math., 25, SIAM, Philadelphia, 1976.  Google Scholar [19] C. Li, J. Li, Z. Ma and H. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987-1004.  doi: 10.1016/j.jmaa.2014.06.035.  Google Scholar [20] M. Li, W. Liu, C. Shan and Y. Yi, Turning points and relaxation oscillation cycles in epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.  Google Scholar [21] W. Liu, Exchange lemmas for singularly perturbation problems with certain turning points, J. Differential Equations, 167 (2000), 134-180.  doi: 10.1006/jdeq.2000.3778.  Google Scholar [22] W. Liu, Geometric singular perturbations for multiple turning points: Invariant manifolds and exchange lemmas, J. Dynam. Differential Equations, 18 (2006), 667-691.  doi: 10.1007/s10884-006-9020-7.  Google Scholar [23] W. Liu, Simon A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar [24] M. Lu, J. Huang, S. Ruan and P. Yu, Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate, J. Differential Equations, 267 (2019), 1859-1898.  doi: 10.1016/j.jde.2019.03.005.  Google Scholar [25] 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 [26] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov and N. Kh. Rozov, Asymptotic Methods in Singularly Perturbed Systems, translated from the Russian by I, Aleksanova, Monographs in Contemporary Mathematics, Consultants Bureau, New York, 1994.  Google Scholar [27] N. G. Reich, et al., Interactions between serotypes of dengue highlight epidemiological impact of cross-immunity, J. R. Soc. Interface., 10 (2013), art. no. 0414. doi: 10.1098/rsif.2013.0414.  Google Scholar [28] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163.  doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar [29] S. Schecter, Persistent unstable equilibria and closed orbits of a singularly perturbed equation, J. Differential Equations, 60 (1985), 131-141.  doi: 10.1016/0022-0396(85)90124-X.  Google Scholar [30] C. Shan and H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, J. Differential Equations, 257 (2014), 1662-1688.  doi: 10.1016/j.jde.2014.05.030.  Google Scholar [31] C. Shan, Y. Yi and H. Zhu, Nilpotent singularities and dynamics in an SIR type of compartmental model with hospital resources, J. Differential Equations, 260 (2016), 4339-4365.  doi: 10.1016/j.jde.2015.11.009.  Google Scholar [32] M. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, Springer Nature Switzerland AG, 2020. doi: 10.1007/978-3-030-36399-4.  Google Scholar
(a) Bifurcation curves in $(\beta_1, b)$ plane. (b) Pitchfork bifurcation occurs at $E_0$ when $\beta_1 = \hat{\beta}_1$ if $b = \hat{b}$
Dynamics of layer problem (7) with $S_{h}^{0} <N^{0}$
Dynamics of system (7) on $M(\mathcal{Z}_0)$, in which double arrow indicates the fast movement along the regular orbits, and single arrow indicates the slow moment on the slow manifold $\mathcal{Z}_0$. The blue line is the one of a family of slow-fast cycles
Two limit cycles are on the center manifold $M^{\varepsilon}$, and the outer one is a relaxation oscillation cycle
(a) Hopf bifurcation curve in $(\varepsilon, b)$-plane. (b) Hopf bifurcation curve in $(\beta, b)$-plane, where $\varepsilon = 4\times 10^{-5}, d_2 = 0.02, \mu_0 = 0.03, \mu_1 = 0.0305, N = 10000, M = 250000, \beta_2 = 0.025$
Limit cycles generated by Hopf bifurcation. For parameters, $\beta_1 = 0.00115$ and $b = 2$ in (a); $\beta_1 = 0.0009760249$ and $b = 0.05$ in (b)
Bifurcation diagram in $(\beta_1, b)$-plane for $\varepsilon>0$ small
Generation and annihilation of limit cycles. Green curves signify stable limit cycles and pink curves signify unstable limit cycles
Relaxation oscillation cycle (blue curve) coexists with small fluctuation due to the unstable limit cycle (red curve). All parameters are chosen as those in Fig. 4 except $\varepsilon = 10^{-5}$
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