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Construction of Lyapunov functionals for delay differential equations in virology and epidemiology. (English) Zbl 1257.34053
Summary: We present a method for constructing a Lyapunov functional for some delay differential equations in virology and epidemiology. Here, some delays are incorporated to the original ordinary differential equations, for which a Lyapunov function is already obtained. We present simple and clear explanation of our method using some models whose Lyapunov functionals are already obtained. Moreover, we present several new results for constructing Lyapunov functionals using our method.
MSC:
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
92D30Epidemiology
92C60Medical epidemiology
References:
[1]Korobeinikov, A.: Global properties of basic virus dynamics models, Bull. math. Biol. 66, 879-883 (2004)
[2]Korobeinikov, A.: Lyapunov function and global stability for SIR and SIRS epidemic models with nonlinear transmission, Bull. math. Biol. 30, 615-626 (2006)
[3]Korobeinikov, A.: Global properties of infectious disease model with nonlinear incidence, Bull. math. Biol. 69, 1871-1886 (2007)
[4]Mccluskey, C. C.: Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonl. anal. RWA 11, 55-59 (2010) · Zbl 1185.37209 · doi:10.1016/j.nonrwa.2008.10.014
[5]Mccluskey, C. C.: Global stability of an SIR epidemic model with delay and general non linear incidence, Math. biosc. Eng. 7, 837-857 (2010)
[6]Huang, G.; Takeuchi, Y.; Ma, W.: Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. math. 70, 2693-2708 (2010) · Zbl 1209.92035 · doi:10.1137/090780821
[7]Huang, G.; Takeuchi, Y.: Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. math. Biol. 63, 125-139 (2011) · Zbl 1230.92048 · doi:10.1007/s00285-010-0368-2
[8]Liu, S.; Wang, L.: Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. biosc. Eng. 7, 675-685 (2010)
[9]Xu, R.: Global stability of an HIV-1 infection model with saturated infection and intracellular delay, J. math. Anal. appl. 35, 75-81 (2011) · Zbl 1222.34101 · doi:10.1016/j.jmaa.2010.08.055
[10]Guo, H.; Li, M. Y.; Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. appl. Math. Q. 14, 259-284 (2006) · Zbl 1148.34039
[11]Yuan, Z.; Wang, L.: Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonl. anal. RWA 11, 995-1004 (2010)
[12]Bonzi, B.; Fall, A. A.; Iggidr, A.; Sallet, G.: Stability of differential susceptibility and infectivity epidemic model, J. math. Biol. 62, 39-64 (2011) · Zbl 1232.92055 · doi:10.1007/s00285-010-0327-y
[13]Nowak, M. A.; Bangham, C. R. M.: Population dynamics of immune responses to persistent viruses, Science 272, 74-79 (1996)
[14]Li, M. Y.; Shu, H.: Global stability of an in-hosts viral model with intracellular delay, Bull. math. Biol. 72, 1492-1505 (2010) · Zbl 1198.92034 · doi:10.1007/s11538-010-9503-x
[15]Kajiwara, T.; Sasaki, T.: Global stability of pathogen-immune dynamics with absorption, J. biol. Dyn. 4, 258-269 (2010)
[16]Pang, H.; Wang, W.; Wang, K.: Global properties of virus dynamics with CTL immune response, J. southwest China normal univ. 30, 797-799 (2005)
[17]Huang, G.; Yokoi, H.; Takeuchi, Y.; Kajiwara, T.; Sasaki, T.: Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japanese J. Indust. appl. Math. 28, 383-411 (2011) · Zbl 1226.92049 · doi:10.1007/s13160-011-0045-x
[18]Guo, H.; Li, M. Y.; Shuai, Z.: A graph-theoretic approach to the method of global Lyapunov functions, Proc. amer. Math. soc. 136, 2793-2802 (2008) · Zbl 1155.34028 · doi:10.1090/S0002-9939-08-09341-6
[19]Li, M. Y.; Shuai, Z.; Wang, C.: Global stability of multi-group epidemic models with distributed delays, J. math. Anal. appl. 361, 38-47 (2010) · Zbl 1175.92046 · doi:10.1016/j.jmaa.2009.09.017
[20]Li, M. Y.; Shuai, Z.: Global-stability problem for coupled system of differential equations on networks, J. differential equations 248, 1-20 (2010) · Zbl 1190.34063 · doi:10.1016/j.jde.2009.09.003