American Institute of Mathematical Sciences

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

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

Zhiting Xu 1, and  Yiyi Zhang 2,

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.
Keywords: reaction malaria model, Steady states., Traveling wave solutions, time-delayed, vector-bias.
Mathematics Subject Classification: Primary: 34C07, 35A10; Secondary: 35Q92, 92D3.
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.

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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.

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[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.

show all references

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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