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2014, 11(4): 995-1001. doi: 10.3934/mbe.2014.11.995

A note on global stability for malaria infections model with latencies

Jinliang Wang 1, , Jingmei Pang 1, and  Toshikazu Kuniya 2,

1. 

School of Mathematical Science, Heilongjiang University, Harbin 150080, China, China
2. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received  June 2013 Revised  October 2013 Published  March 2014

A recent paper [Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.
Keywords: Global stability, malaria infection, latency distribution, Lyapunov functional..
Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 37G9.
Citation: Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies. Mathematical Biosciences & Engineering, 2014, 11 (4) : 995-1001. doi: 10.3934/mbe.2014.11.995
References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.  Google Scholar

[2]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. Google Scholar

[3]

J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol.99. Applied Mathematical Science, New York, 1993. Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar

[5]

W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.  Google Scholar

[6]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.  Google Scholar

[7]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.  Google Scholar

[8]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57.  Google Scholar

[9]

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.  Google Scholar

[10]

J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.  Google Scholar

[11]

R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., New York, 1971.  Google Scholar

[12]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delaydistributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[13]

C. C. McCluskey, Global stability for an SIER epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603.  Google Scholar

[14]

H. R. Thieme, Mathematics In Population Biology, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[15]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.  Google Scholar

[16]

P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.  Google Scholar

[17]

Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10 (2013), 463-481. doi: 10.3934/mbe.2013.10.463.  Google Scholar

show all references

References:
[1]

F. V. Atkinson and J. R. Haddock, On determining phase spaces for functional differential equations, Funkcial. Ekvac., 31 (1988), 331-347.  Google Scholar

[2]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics (eds. O. Arino et al.), Wuerz, Winnepeg, Canada, (1995), 33-50. Google Scholar

[3]

J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol.99. Applied Mathematical Science, New York, 1993. Google Scholar

[4]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.  Google Scholar

[5]

W. M. Hirsch, H. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753. doi: 10.1002/cpa.3160380607.  Google Scholar

[6]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139. doi: 10.1007/s00285-010-0368-2.  Google Scholar

[7]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for Delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821.  Google Scholar

[8]

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60. doi: 10.3934/mbe.2004.1.57.  Google Scholar

[9]

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.  Google Scholar

[10]

J. P. Lasalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.  Google Scholar

[11]

R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., New York, 1971.  Google Scholar

[12]

C. C. McCluskey, Complete global stability for an SIR epidemic model with delaydistributed or discrete, Nonlinear Anal. RWA., 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[13]

C. C. McCluskey, Global stability for an SIER epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610. doi: 10.3934/mbe.2009.6.603.  Google Scholar

[14]

H. R. Thieme, Mathematics In Population Biology, Princeton University Press, Princeton, NJ, 2003.  Google Scholar

[15]

J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Math. Med. Biol., 29 (2012), 283-300. doi: 10.1093/imammb/dqr009.  Google Scholar

[16]

P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng., 4 (2007), 205-219. doi: 10.3934/mbe.2007.4.205.  Google Scholar

[17]

Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10 (2013), 463-481. doi: 10.3934/mbe.2013.10.463.  Google Scholar

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