# American Institute of Mathematical Sciences

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

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), 639-657. doi: 10.1007/s11538-010-9545-0. [2] G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics, J. Theoret. Biol., 255 (2008), 16-25. doi: 10.1016/j.jtbi.2008.07.033. [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), 1192-1207. doi: 10.1007/s11538-009-9487-6. [4] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [5] 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. [6] J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria, Am. Nat., 130 (1987), 811-827. doi: 10.1086/284749. [7] 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. [8] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [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), 225-239. doi: 10.1093/imammb/dqp006. [10] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [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), e298. doi: 10.1371/journal.pbio.0030298. [12] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543–-568. doi: 10.1007/s00285-010-0346-8. [13] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [14] G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829. [15] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [16] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [17] R. Ross, "The Prevention of Malaria,'' Second edition, Murray, London, 1911. [18] C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay, J. Math. Anal. Appl., 381 (2011), 884-890. doi: 10.1016/j.jmaa.2011.04.012. [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). Available from: http://www.red-mat.unam.mx/foro/volumenes/vol028.

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##### References:
 [1] F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission, Bull. Math. Biol., 73 (2011), 639-657. doi: 10.1007/s11538-010-9545-0. [2] G. R. Hosack, P. A. Rossignol and P. van den Driessche, The control of vector-borne disease epidemics, J. Theoret. Biol., 255 (2008), 16-25. doi: 10.1016/j.jtbi.2008.07.033. [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), 1192-1207. doi: 10.1007/s11538-009-9487-6. [4] G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708. doi: 10.1137/090780821. [5] 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. [6] J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria, Am. Nat., 130 (1987), 811-827. doi: 10.1086/284749. [7] 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. [8] A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886. doi: 10.1007/s11538-007-9196-y. [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), 225-239. doi: 10.1093/imammb/dqp006. [10] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,'' Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. [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), e298. doi: 10.1371/journal.pbio.0030298. [12] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543–-568. doi: 10.1007/s00285-010-0346-8. [13] M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x. [14] G. Macdonald, The analysis of equilibrium in malaria, Trop. Dis. Bull., 49 (1952), 813-829. [15] C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Anal. RWA, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014. [16] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11 (2010), 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005. [17] R. Ross, "The Prevention of Malaria,'' Second edition, Murray, London, 1911. [18] C. Vargas-De-León, Global properties for virus dynamics model with mitotic transmission and intracellular delay, J. Math. Anal. Appl., 381 (2011), 884-890. doi: 10.1016/j.jmaa.2011.04.012. [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). Available from: http://www.red-mat.unam.mx/foro/volumenes/vol028.
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