# American Institute of Mathematical Sciences

2014, 11(4): 785-805. doi: 10.3934/mbe.2014.11.785

## Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate

 1 Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan 2 Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1, Hungary

Received  April 2013 Revised  September 2013 Published  March 2014

We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals. Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable. We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.
Citation: Yoichi Enatsu, Yukihiko Nakata. Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences & Engineering, 2014, 11 (4) : 785-805. doi: 10.3934/mbe.2014.11.785
##### References:
 [1] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4. [2] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931. [3] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8. [4] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [5] O. Diekmann and K. Korvasova, A didactical note on the advantage of using two parameters in Hopf bifurcation studies, J. Biological Dynamics, 7 (2013), 21-30. doi: 10.1080/17513758.2012.760758. [6] O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [7] Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. RWA., 13 (2012), 2120-2133. doi: 10.1016/j.nonrwa.2012.01.007. [8] Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 61-74. doi: 10.3934/dcdsb.2011.15.61. [9] F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959 (Translated from Russian). [10] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2. [13] Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcation of an SIRS epidemic model with nonlinear incidence rate, Disc. Cont. Dynam. Sys. B, 15 (2011), 93-112. doi: 10.3934/dcdsb.2011.15.93. [14] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: 10.1093/imammb/dqi001. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. [18] Y. N. Kyrychko and K. B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. RWA., 6 (2005), 495-507. doi: 10.1016/j.nonrwa.2004.10.001. [19] W. M. Liu, S. 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. [20] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. [21] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. [22] Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays, Nonlinear Anal. RWA., 12 (2011), 1897-1910. doi: 10.1016/j.nonrwa.2010.12.002. [23] Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, Disc. Cont. Dynam. Sys. Supplement, II (2011), 1119-1128. [24] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. [25] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [26] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics Vol. 57, Springer, Berlin, 2011. doi: 10.1007/978-1-4419-7646-8. [27] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delayed SIR epidemic model with finite incubation time, Nonlinear Anal. TMA., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8. [28] W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. doi: 10.3934/mbe.2006.3.267. [29] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [30] R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41 (2009), 2319-2325. doi: 10.1016/j.chaos.2008.09.007. [31] Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B, 13 (2010), 195-211. doi: 10.3934/dcdsb.2010.13.195.

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##### References:
 [1] E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Analysis, 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4. [2] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931. [3] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8. [4] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain J. Math., 9 (1979), 31-42. doi: 10.1216/RMJ-1979-9-1-31. [5] O. Diekmann and K. Korvasova, A didactical note on the advantage of using two parameters in Hopf bifurcation studies, J. Biological Dynamics, 7 (2013), 21-30. doi: 10.1080/17513758.2012.760758. [6] O. Diekmann, S. A. van Gils, S. M. V. Lunel and H. O. Walther, Delay Equations Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [7] Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. RWA., 13 (2012), 2120-2133. doi: 10.1016/j.nonrwa.2012.01.007. [8] Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 61-74. doi: 10.3934/dcdsb.2011.15.61. [9] F. R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959 (Translated from Russian). [10] H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539. [11] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [12] G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamics models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2. [13] Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcation of an SIRS epidemic model with nonlinear incidence rate, Disc. Cont. Dynam. Sys. B, 15 (2011), 93-112. doi: 10.3934/dcdsb.2011.15.93. [14] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Biol., 22 (2005), 113-128. doi: 10.1093/imammb/dqi001. [17] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, 1993. [18] Y. N. Kyrychko and K. B. Blyuss, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Anal. RWA., 6 (2005), 495-507. doi: 10.1016/j.nonrwa.2004.10.001. [19] W. M. Liu, S. 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. [20] W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. [21] C. C. McCluskey, Global stability of an SIR epidemic model with delay and general incidence, Math. Biosci. Eng., 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837. [22] Y. Muroya, Y. Enatsu and Y. Nakata, Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays, Nonlinear Anal. RWA., 12 (2011), 1897-1910. doi: 10.1016/j.nonrwa.2010.12.002. [23] Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, Disc. Cont. Dynam. Sys. Supplement, II (2011), 1119-1128. [24] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976. [25] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [26] H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics Vol. 57, Springer, Berlin, 2011. doi: 10.1007/978-1-4419-7646-8. [27] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delayed SIR epidemic model with finite incubation time, Nonlinear Anal. TMA., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8. [28] W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng., 3 (2006), 267-279. doi: 10.3934/mbe.2006.3.267. [29] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [30] R. Xu and Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41 (2009), 2319-2325. doi: 10.1016/j.chaos.2008.09.007. [31] Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B, 13 (2010), 195-211. doi: 10.3934/dcdsb.2010.13.195.
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