2012, 9(1): 165-174. doi: 10.3934/mbe.2012.9.165

Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes

1. 

Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas C.U., Chilpancingo, Guerrero, Mexico

Received  March 2011 Revised  May 2011 Published  December 2011

A delayed vector-bias model for malaria transmission with incubation period in mosquitoes is studied. The delay $\tau$ corresponds to the time necessary for a latently infected vector to become an infectious vector. We prove that the global stability is completely determined by the threshold parameter, $R_0(\tau)$. If $R_0(\tau)\leq1$, the disease-free equilibrium is globally asymptotically stable. If $R_0(\tau)>1$ a unique endemic equilibrium exists and is globally asymptotically stable. We apply our results to Ross-MacDonald malaria models with an incubation period (extrinsic or intrinsic).
Citation: Cruz Vargas-De-León. Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes. Mathematical Biosciences & Engineering, 2012, 9 (1) : 165-174. doi: 10.3934/mbe.2012.9.165
References:
[1]

F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

[2]

G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics,, J. Theoret. Biol., 255 (2008), 16.  doi: 10.1016/j.jtbi.2008.07.033.  Google Scholar

[3]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[4]

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

[5]

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

[6]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811.  doi: 10.1086/284749.  Google Scholar

[7]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.  doi: 10.1007/s11538-005-9037-9.  Google Scholar

[8]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[9]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and nonlinear incidence rate,, Math. Medic. Biol., 26 (2009), 225.  doi: 10.1093/imammb/dqp006.  Google Scholar

[10]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'', Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[11]

R. Lacroix, W. R. Mukabana, L. C. Gouagna and J. C. Koella, Malaria infection increases attractiveness of humans to mosquitoes,, PLOS Biol., 3 (2005).  doi: 10.1371/journal.pbio.0030298.  Google Scholar

[12]

Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population,, J. Math. Biol., 62 (2011).  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[13]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[14]

G. Macdonald, The analysis of equilibrium in malaria,, Trop. Dis. Bull., 49 (1952), 813.   Google Scholar

[15]

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

[16]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[17]

R. Ross, "The Prevention of Malaria,'', Second edition, (1911).   Google Scholar

[18]

C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay,, J. Math. Anal. Appl., 381 (2011), 884.  doi: 10.1016/j.jmaa.2011.04.012.  Google Scholar

[19]

C. Vargas-De-León and G. Gómez-Alcaraz, Global stability conditions of delayed SIRS epidemiological model for vector diseases,, Foro-Red-Mat: Revista Electrónica de Contenido Matemático, 28 (2011).   Google Scholar

show all references

References:
[1]

F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

[2]

G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics,, J. Theoret. Biol., 255 (2008), 16.  doi: 10.1016/j.jtbi.2008.07.033.  Google Scholar

[3]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate,, Bull. Math. Biol., 72 (2010), 1192.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[4]

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

[5]

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

[6]

J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria,, Am. Nat., 130 (1987), 811.  doi: 10.1086/284749.  Google Scholar

[7]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.  doi: 10.1007/s11538-005-9037-9.  Google Scholar

[8]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[9]

A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence, and nonlinear incidence rate,, Math. Medic. Biol., 26 (2009), 225.  doi: 10.1093/imammb/dqp006.  Google Scholar

[10]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'', Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[11]

R. Lacroix, W. R. Mukabana, L. C. Gouagna and J. C. Koella, Malaria infection increases attractiveness of humans to mosquitoes,, PLOS Biol., 3 (2005).  doi: 10.1371/journal.pbio.0030298.  Google Scholar

[12]

Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population,, J. Math. Biol., 62 (2011).  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[13]

M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492.  doi: 10.1007/s11538-010-9503-x.  Google Scholar

[14]

G. Macdonald, The analysis of equilibrium in malaria,, Trop. Dis. Bull., 49 (1952), 813.   Google Scholar

[15]

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

[16]

C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.  doi: 10.1016/j.nonrwa.2009.11.005.  Google Scholar

[17]

R. Ross, "The Prevention of Malaria,'', Second edition, (1911).   Google Scholar

[18]

C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay,, J. Math. Anal. Appl., 381 (2011), 884.  doi: 10.1016/j.jmaa.2011.04.012.  Google Scholar

[19]

C. Vargas-De-León and G. Gómez-Alcaraz, Global stability conditions of delayed SIRS epidemiological model for vector diseases,, Foro-Red-Mat: Revista Electrónica de Contenido Matemático, 28 (2011).   Google Scholar

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