# zbMATH — the first resource for mathematics

Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate. (English) Zbl 1445.92273
Summary: In this paper, we study a new SEIRS epidemic model describing nonlinear incidence with a more general form and the transmission of influenza virus with disease resistance. The basic reproductive number $$\operatorname{Re}_{0}$$ is obtained by using the method of next generating matrix. If $$\operatorname{Re}_{0}<1$$, the disease-free equilibrium is globally asymptotically stable, and if $$\operatorname{Re}_{0}>1$$, by using the geometric method, we obtain some sufficient conditions for global stability of the unique endemic equilibrium. Finally, numerical simulations are provided to support our theoretical results.

##### MSC:
 92D30 Epidemiology 34D20 Stability of solutions to ordinary differential equations 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D25 Population dynamics (general) 34D23 Global stability of solutions to ordinary differential equations
Full Text:
##### References:
 [1] Beretta, E; Capasso, V, On the general structure of epidemic systems, global asymptotic stability, Comput. Math. Appl., 12A, 677-694, (1986) · Zbl 0622.92016 [2] Buonomo, B; d’Onofrio, A; Lacitignola, D, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl., 404, 385-398, (2013) · Zbl 1304.92118 [3] Khanh, NH, Stability analysis of an influenza virus model with disease resistance, J. Egypt. Math. Soc., 24, 193-199, (2016) · Zbl 1339.34056 [4] Mateus, JP; Silva, CM, Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Anal., Real World Appl., 34, 379-402, (2017) · Zbl 1354.34085 [5] Nakata, Y; Kuniya, T, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363, 230-237, (2010) · Zbl 1184.34056 [6] Qi, LX; Cui, JA, The stability of an SEIRS model with nonlinear incidence, vertical transmission and time delay, Appl. Math. Comput., 221, 360-366, (2013) · Zbl 1329.92139 [7] Sun, CJ; Lin, YP; Tang, SP, Global stability for a special SEIR epidemic model with nonlinear incidence rates, Chaos Solitons Fractals, 33, 290-297, (2007) · Zbl 1152.34357 [8] Tipsri, S; Chinviriyasit, W, The effect of time delay on the dynamics of an SEIR model with nonlinear incidence, Chaos Solitons Fractals, 75, 153-172, (2015) · Zbl 1352.92172 [9] Wang, WD, Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15, 423-428, (2002) · Zbl 1015.92033 [10] Zhang, TL; Teng, ZD, Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence, Chaos Solitons Fractals, 37, 1456-1468, (2008) · Zbl 1142.34384 [11] Zhou, X; Cui, JA, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simul., 16, 4438-4450, (2011) · Zbl 1219.92060 [12] Dietz, K, Overall population patterns in the transmission cycle of infectious disease agents, (1982), Berlin [13] Heesterbeek, JAP; Metz, JAJ, The saturating contact rate in marriage and epidemic models, J. Math. Biol., 31, 529-539, (1993) · Zbl 0770.92021 [14] Liu, WM; Levin, SA; Iwasa, Y, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23, 187-204, (1986) · Zbl 0582.92023 [15] Li, MY; Muldowney, JS, Global stability for the SEIR model in epidemiology, Math. Biosci., 125, 155-164, (1995) · Zbl 0821.92022 [16] Hethcote, HW; Driessche, P, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29, 271-287, (1991) · Zbl 0722.92015 [17] Mateus, JP; Silva, CM, A non-autonomous SEIRS model with general incidence rate, Appl. Math. Comput., 247, 169-189, (2014) · Zbl 1338.92133 [18] Korobeinikov, A, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69, 1871-1886, (2007) · Zbl 1298.92101 [19] Diekmann, O; Heesterbeek, JAP; Metz, JAJ, 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, 365-382, (1990) · Zbl 0726.92018 [20] Driessche, P; Watmough, J, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48, (2002) · Zbl 1015.92036 [21] Feng, XM; Ruan, SG; Teng, Z; Wang, K, Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Biosci., 266, 52-64, (2015) · Zbl 1356.92081 [22] Li, MY; Muldowney, J, A geometric approach to global stability problems, SIAM J. Math. Anal., 27, 1070-1083, (1996) · Zbl 0873.34041 [23] Butler, GJ; Waltman, P, Persistence in dynamics systems, J. Differ. Equ., 63, 255-263, (1986) · Zbl 0603.58033 [24] Martin, RH, Logarithmic norms and projections applied to linear different differential systems, J. Math. Anal. Appl., 45, 432-454, (1974) · Zbl 0293.34018 [25] Kermack, WO; McKendrick, AG, Contribution to mathematical theory of epidemics, Proc. R. Soc. A, 115, 700-721, (1927) · JFM 53.0517.01 [26] Korobeinikon, A; Wake, GC, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models, Appl. Math. Lett., 15, 955-960, (2002) · Zbl 1022.34044 [27] Li, Y; Muldowney, JS, On bendixsons criterion, J. Differ. Equ., 106, 27-39, (1993) · Zbl 0786.34033 [28] Brown, GC; Hasibuan, R, Conidial discharge and transmission efficiency of neozygites floridana, an entomopathogenic fungus infecting two-spotted spider mites under laboratory conditions, J. Invertebr. Pathol., 65, 10-16, (1995) [29] Li, XB; Yang, LJ, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Anal., Real World Appl., 13, 2671-2679, (2012) · Zbl 1254.92083 [30] Yang, LX; Yang, XF; Zhu, QY; Wen, LS, A computer virus model with graded cure rates, Nonlinear Anal., Real World Appl., 14, 414-422, (2013) · Zbl 1258.68020 [31] Capasso, V; Serio, G, A generalization of the kermack-mckendrick deterministic epidemic model, Math. Biosci., 42, 43-61, (1978) · Zbl 0398.92026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.