# American Institute of Mathematical Sciences

May  2015, 14(3): 923-940. doi: 10.3934/cpaa.2015.14.923

## Traveling wave phenomena of a diffusive and vector-bias malaria model

 1 School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Received  July 2014 Revised  January 2015 Published  March 2015

This paper is devoted to the study of a diffusive and vector-bias malaria model. We first analyze the well-posedness of the initial value problem of the model. Then, according to the basic reproduction ratio $\mathcal{R}_0$, we establish the existence and non-existence of traveling wave solutions for the model. The proof of the main theorems is based on Schauder fixed point theorem and the variation of constants formula of ODEs.
Citation: Zhiting Xu, Yiyi Zhang. Traveling wave phenomena of a diffusive and vector-bias malaria model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 923-940. doi: 10.3934/cpaa.2015.14.923
##### References:
 [1] S. Ai, J. Li and J. Liu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237. doi: 10.1137/110860318. [2] B. Buonomo and C. Vargas-De-León, Stability and bifurcation analysis of a vector-bias model for malaria transmission, Math. Biosci., 242 (2013), 59-67. doi: 10.1016/j.mbs.2012.12.001. [3] 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. [4] D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Research Notes in Mathematics, vol. 279, Harlow, Longman, 1992. [5] T. L. Daniel and J. G. Kingsolver, Feeding strategy and the mechanics of blood sucking in insects, J. Theor. Biol., 105 (1983), 661-672. [6] S. M.-A. S. Elsheihh and K. C. Patidar, Analysis of a malaria model with a distributed delay, IMA J. Appl. Math., 79 (2014), 1139-1160. [7] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Differential Equations, 248 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001. [8] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417. doi: 10.1093/imamat/hxq009. [9] S. I. Hay, C. A. Guerra, A. J. Tatem, A. M. Noor and R. W. Snow, The gobal distribution and population at risk of malria: past, present, and future, Lanct Infect. Dis., 4 (2004), 327-336. [10] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [11] J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria, Am. Nat., 130 (1987), 811-827. [12] J. Li, Malaria model with stage-structured mosquitoes, Math. Biosci. Eng., 8 (2011), 753-768. doi: 10.3934/mbe.2011.8.753. [13] J. Li and X. Zou, Modeling spatial spread of infectious diseases with afixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [14] W. T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003. [15] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflow with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [16] 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. [17] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [18] R. Martin and H. Smith, Absract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [19] J. D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002. [20] G. M. Nayyar, J. G. Breman, P. N. Newton and J. Herrington, Poor-quality antimalarial drugs in southeast Asia and sub-Saharan Africa, Lancet Infectious Diseases, 12 (2012), 488-496. [21] R. Ross, The Prevention of Malaria, 2nd edn. Murray, London, 1911. [22] P. A. Rossignol, M. C. Ribeiro, M. Jungery, M. J. Turell, A. Spielman and C. L. Bailey, Enhanced mosquito blood-finding on parasitemic hosts: evidence for vector-parasite mutualism, Proc. Natl. Acad. Sci. USA., 82 (1985), 7725-7727. [23] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [24] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed Reaction-Diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [25] C. Vargas-De-León, Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes, Math. Biosci. Eng., 9 (2012), 165-174. doi: 10.3934/mbe.2012.9.165. [26] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, in: Translations of Mathematical Monographs, vol. 140, Amer. Math. Soc., Providence, 1994. [27] Y. X. Wang and Z. C. Wang, Monostable waves in a time-delayed and diffusiove epidemic model,, Sciencepaper online, (). [28] Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [29] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmissin, Proc. R. Soc. A., 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. [30] P. Weng and Z. Xu, Wavefronts for a global reaction-diffusion systems with nifinite distributed delay, J. Math. Anal. Appl., 345 (2008), 522-534. doi: 10.1016/j.jmaa.2008.04.039. [31] World Health Organization, http://www.who.int/denguecontrol/en/index.html/2013, ., (). [32] C. Wu and D. Xiao, Travelling wave solutions in anon-local and time-delayed reaction-diffusion model, IMA J. Appl. Math., 78 (2013), 1290-1317. doi: 10.1093/imamat/hxs021. [33] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] J. Wu and X. Zou, Travelling wave fronts of reaction diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. [35] Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and atent period, Nonlinear Analysis, 111 (2014), 66-81 doi: 10.1016/j.na.2014.08.012. [36] Z. Xu and P. Weng, Traveling waves for nonlinear and non-monotone delayed reaction-diffusion equations, Acta. Math. Sinica., English Series, 29 (2013), 2159-2180. doi: 10.1007/s10114-013-1769-0. [37] Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst., Ser.B, 17 (2012), 2615-2634. doi: 10.3934/dcdsb.2012.17.2615. [38] L. Zhang, B. Li and J. Shang, Stablity and travelling waves for a time-delayed population stsyem with stage structure, Nonlinear Analysis: Real World Applications, 13 (2012), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007. [39] Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Analysis: Real World Applications, 15 (2014), 118-139. doi: 10.1016/j.nonrwa.2013.06.005.

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##### References:
 [1] S. Ai, J. Li and J. Liu, Mosquito-stage-structured malaria models and their global dynamics, SIAM J. Appl. Math., 72 (2012), 1213-1237. doi: 10.1137/110860318. [2] B. Buonomo and C. Vargas-De-León, Stability and bifurcation analysis of a vector-bias model for malaria transmission, Math. Biosci., 242 (2013), 59-67. doi: 10.1016/j.mbs.2012.12.001. [3] 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. [4] D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Research Notes in Mathematics, vol. 279, Harlow, Longman, 1992. [5] T. L. Daniel and J. G. Kingsolver, Feeding strategy and the mechanics of blood sucking in insects, J. Theor. Biol., 105 (1983), 661-672. [6] S. M.-A. S. Elsheihh and K. C. Patidar, Analysis of a malaria model with a distributed delay, IMA J. Appl. Math., 79 (2014), 1139-1160. [7] J. Fang, J. Wei and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reaction-diffusion system, J. Differential Equations, 248 (2008), 2749-2770. doi: 10.1016/j.jde.2008.09.001. [8] Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417. doi: 10.1093/imamat/hxq009. [9] S. I. Hay, C. A. Guerra, A. J. Tatem, A. M. Noor and R. W. Snow, The gobal distribution and population at risk of malria: past, present, and future, Lanct Infect. Dis., 4 (2004), 327-336. [10] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction-diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925. [11] J. G. Kingsolver, Mosquito host choice and the epidemiology of malaria, Am. Nat., 130 (1987), 811-827. [12] J. Li, Malaria model with stage-structured mosquitoes, Math. Biosci. Eng., 8 (2011), 753-768. doi: 10.3934/mbe.2011.8.753. [13] J. Li and X. Zou, Modeling spatial spread of infectious diseases with afixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [14] W. T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003. [15] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflow with applications, Comm. Pure Appl. Math., 60 (2007), 1-40. doi: 10.1002/cpa.20154. [16] 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. [17] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846. [18] R. Martin and H. Smith, Absract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590. [19] J. D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002. [20] G. M. Nayyar, J. G. Breman, P. N. Newton and J. Herrington, Poor-quality antimalarial drugs in southeast Asia and sub-Saharan Africa, Lancet Infectious Diseases, 12 (2012), 488-496. [21] R. Ross, The Prevention of Malaria, 2nd edn. Murray, London, 1911. [22] P. A. Rossignol, M. C. Ribeiro, M. Jungery, M. J. Turell, A. Spielman and C. L. Bailey, Enhanced mosquito blood-finding on parasitemic hosts: evidence for vector-parasite mutualism, Proc. Natl. Acad. Sci. USA., 82 (1985), 7725-7727. [23] S. Ruan, D. Xiao and J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114. doi: 10.1007/s11538-007-9292-z. [24] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed Reaction-Diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [25] C. Vargas-De-León, Global analysis of a delayed vector-bias model for malaria transmission with incubation period in mosquitoes, Math. Biosci. Eng., 9 (2012), 165-174. doi: 10.3934/mbe.2012.9.165. [26] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, in: Translations of Mathematical Monographs, vol. 140, Amer. Math. Soc., Providence, 1994. [27] Y. X. Wang and Z. C. Wang, Monostable waves in a time-delayed and diffusiove epidemic model,, Sciencepaper online, (). [28] Z. C. Wang, W. T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010. [29] Z. C. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmissin, Proc. R. Soc. A., 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377. [30] P. Weng and Z. Xu, Wavefronts for a global reaction-diffusion systems with nifinite distributed delay, J. Math. Anal. Appl., 345 (2008), 522-534. doi: 10.1016/j.jmaa.2008.04.039. [31] World Health Organization, http://www.who.int/denguecontrol/en/index.html/2013, ., (). [32] C. Wu and D. Xiao, Travelling wave solutions in anon-local and time-delayed reaction-diffusion model, IMA J. Appl. Math., 78 (2013), 1290-1317. doi: 10.1093/imamat/hxs021. [33] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [34] J. Wu and X. Zou, Travelling wave fronts of reaction diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651-687. doi: 10.1023/A:1016690424892. [35] Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and atent period, Nonlinear Analysis, 111 (2014), 66-81 doi: 10.1016/j.na.2014.08.012. [36] Z. Xu and P. Weng, Traveling waves for nonlinear and non-monotone delayed reaction-diffusion equations, Acta. Math. Sinica., English Series, 29 (2013), 2159-2180. doi: 10.1007/s10114-013-1769-0. [37] Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst., Ser.B, 17 (2012), 2615-2634. doi: 10.3934/dcdsb.2012.17.2615. [38] L. Zhang, B. Li and J. Shang, Stablity and travelling waves for a time-delayed population stsyem with stage structure, Nonlinear Analysis: Real World Applications, 13 (2012), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007. [39] Y. Zhang and Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Analysis: Real World Applications, 15 (2014), 118-139. doi: 10.1016/j.nonrwa.2013.06.005.
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