Global stability of multi-group SIR epidemic model with group mixing and human movement

Qianqian Cui; Qianqian Cui

doi:10.3934/mbe.2019087

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 4: 1798-1814. doi: 10.3934/mbe.2019087

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Global stability of multi-group SIR epidemic model with group mixing and human movement

Qianqian Cui ^,

School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, P.R. China

Received: 20 November 2018 Accepted: 13 February 2019 Published: 06 March 2019

In this paper, an SIR multi-group epidemic model with group mixing and human movement is investigated. The control reproduction number $\mathfrak{R}_v$ is derived and the global dynamics of the model are completely determined by the value of $\mathfrak{R}_v$. By using the graph-theoretical approach, the results show that the disease-free equilibrium is globally asymptotically stable if $\mathfrak{R}_v < 1$, and the unique endemic equilibrium is globally asymptotically stable if $\mathfrak{R}_v>1$. Two numerical examples are further presented to testify the validity of the theoretical results.
- multi-group epidemic model,
- human movement,
- group mixing,
- global dynamics,
- Lyapunov function
Citation: Qianqian Cui. Global stability of multi-group SIR epidemic model with group mixing and human movement[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 1798-1814. doi: 10.3934/mbe.2019087

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Abstract

In this paper, an SIR multi-group epidemic model with group mixing and human movement is investigated. The control reproduction number $\mathfrak{R}_v$ is derived and the global dynamics of the model are completely determined by the value of $\mathfrak{R}_v$. By using the graph-theoretical approach, the results show that the disease-free equilibrium is globally asymptotically stable if $\mathfrak{R}_v < 1$, and the unique endemic equilibrium is globally asymptotically stable if $\mathfrak{R}_v>1$. Two numerical examples are further presented to testify the validity of the theoretical results.

References

[1]	M. Kermark and A. Mckendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700–721.
[2]	H. Shu, D. Fan and J.Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. Real World Appl., 13 (2012), 1581–1592.
[3]	H. Guo, M. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259–284.
[4]	H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793–2802.
[5]	T. Kuniya, Y. Murya and Y. Enatsu, Threshold dynamics of an SIR epidemic model with hybird of nultigroup and patch structures, Math. Biosci. Eng., 11 (2014), 1357–1393.
[6]	T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model with varying total population size, Appl. Math. Comput., 265 (2015), 785–798.
[7]	A. Lajmanovich and J. A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221–236.
[8]	M. Y. Li and Z. Shuai, Global stability of an epidemic model in a patchy environment, Canadian Appl. Math. Quart., 17 (2009), 175–187.
[9]	M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Eqn., 248 (2010), 1–20.
[10]	Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. Real World Appl, 14 (2013), 1693–1704.
[11]	H. L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University, 1995.
[12]	R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 280–286.
[13]	W. Wang and X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454–1472.
[14]	L.Wang andW. Yang, Global dynamics of a two-patch SIS model with infection during transport, Appl. Math. Comput., 217 (2011), 8458-8467.
[15]	How Many Ebola Patients Have Been Treated Outside of Africa?, Source of New York Times, 2014. Available from: https://ritholtz.com/2014/10/ how-many-ebola-patients-have-been-treated-outside-africa/.
[16]	J. Arino, J. R. Davis, D. Hartley, et al., A multi-species epidemic model with spatial dynamics, Math. Med. Biol., 22 (2005), 129–142.
[17]	X. Liu and Y. Takeuchi, Spread of disease with transport-related infection and entry screening, J Theor. Biol., 242 (2006), 517–528.
[18]	T. Chen, Z. Sun and B. Wu, Stability of multi-group models with cross-dispersal based on graph theory, Appl. Math. Model., 47 (2017), 745–754.
[19]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653.
[20]	T. Kuniya and Y. Muroya, Global stability of a multi-group SIS epidemic model for population migration, Discrete Continuous Dynam. Systems - B, 19 (2014), 1105–1118.
[21]	Y. Muroya, Y. Enatsu and T. Kuniya, Global stability of extended multi-group SIR epidemic models with patches through migration and cross patch infection, Acta Math. Sci., 33 (2013), 341–361.
[22]	A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
[23]	N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Springer, Berlin, 2006.
[24]	C. Castillo-Chavez and H. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Springer, Berlin, 1995, 33–50.
[25]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transimission, Math. Biosci., 180 (2002), 29–48.
[26]	M. C. Eisenberg, Z. Shuai, J. H. Tien, et al., A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105–112.
[27]	H. I. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dyn. Differ. Equ., 6 (1994), 583–600.
[28]	K. Mischaikow, H. L. Smith and H. R. Thieme, Asymptotically autonomous semiflows: Chain recur-rence and Lyapunov functions, Trans. Am. Math. Soc., 347 (1995), 1669–1685.
[29]	J. P. LaSalle and S. Kefscgetz, Stability by Liapunovs Direct Method with Applications, Academic Press, New York, 1961.
[30]	M. Y. Li, J. R. Graef, L. Wang, et al., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160 (1999), 191–213.

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