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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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

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