# American Institute of Mathematical Sciences

December  2018, 15(6): 1479-1494. doi: 10.3934/mbe.2018068

## Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 2 Department of Mathematics, Faculty of Education, University of Khartoum, Khartoum 321, Sudan 3 School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

* Corresponding author: zglin68@hotmail.com (Z. G. Lin)

Received  June 05, 2018 Revised  August 13, 2018 Published  September 2018

Fund Project: The second author is supported by the NNSF of China (Grant No. 11701206) and the third author is supported by the NNSF of China (Grant No. 11771381)

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.

Citation: Abdelrazig K. Tarboush, Jing Ge, Zhigui Lin. Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1479-1494. doi: 10.3934/mbe.2018068
##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [2] P. Álvarez-Caudevilla and J. López-Gómez, Asymptotic behaviour of principal eigenvalues for a class of cooperative systems, J. Differential Equations, 244 (2008), 1093-1113. doi: 10.1016/j.jde.2007.10.004. [3] D. S. Asnis, R. Conetta, A. A. Teixeira, G. Waldman and B. A. Sampson, The West Nile virus outbreak of 1999 in New York: The flushing hospital experience, Clinical Infect Dis., 30 (2000), 413-418. doi: 10.1086/313737. [4] K. W. Blaynech, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [5] E. Braverman and Md. Kamrujjaman, Competitive-cooperative models with various diffusion strategies, Comput. Math. Appl., 72 (2016), 653-662. doi: 10.1016/j.camwa.2016.05.017. [6] B. Chen and R. Peng, Coexistence states of a strongly coupled prey-predator model, J. Partial Diff. Eqs., 18 (2005), 154-166. [7] V. Chevalier, A. Tran and B. Durand, Predictive modeling of west nile virus transmission risk in the mediterranean basin, Int. J. Environ. Res. Public Health, 11 (2014), 67-90. doi: 10.3390/ijerph110100067. [8] G. Cruz-Pacheco, L. Esteva and C. Vargas, Seasonality and outbreaks in west nile virus infection, Bull. Math. Biol., 71 (2009), 1378-1393. doi: 10.1007/s11538-009-9406-x. [9] D. G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal., 17 (1986), 836-849. doi: 10.1137/0517060. [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. [11] S. Fu, L. Zhang and P. Hu, Global behavior of solutions in a Lotka-Volterra predator-prey model with prey-stage structure, Nonlinear Anal. Real World Appl., 14 (2013), 2027-2045. doi: 10.1016/j.nonrwa.2013.02.007. [12] W. Gan and Z. Lin, Coexistence and asymptotic periodicity in a competitor-competitor-mutualist model, J. Math. Anal. Appl., 337 (2008), 1089-1099. doi: 10.1016/j.jmaa.2007.04.022. [13] D. Horstmann, Remarks on some Lotka-Volterra type cross-diffusion models, Nonlinear Anal. Real World Appl., 8 (2007), 90-117. doi: 10.1016/j.nonrwa.2005.05.008. [14] M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive iteraction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2. [15] D. J. Jamieson, J. E. Ellis, D. B. Jernigan and T. A. Treadwell, Emerging infectious disease outbreaks: Old lessons and new challenges for obstetrician-gynecologists, Am. J. Obstet. Gynecol., 194 (2006), 1546-1555. doi: 10.1016/j.ajog.2005.06.062. [16] Y. Jia, J. Wu and H. Xu, Positive solutions of Lotka-Volterra competition model with cross-diffusion, Comput. Math. Appl., 68 (2014), 1220-1228. doi: 10.1016/j.camwa.2014.08.016. [17] A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model, Math. Models Methods Appl. Sci., 22 (2012), 1250009, 26pp. doi: 10.1142/S0218202512500091. [18] K. I. Kim and Z. G. Lin, Coexistence of three species in a strongly coupled elliptic system, Nonlinear Anal., 55 (2003), 313-333. doi: 10.1016/S0362-546X(03)00242-6. [19] W. Ko and K. Ryu, On a predator-prey system with cross-diffusion representing the tendency of prey to keep away from its predators, Appl. Math. Lett., 21 (2008), 1177-1183. doi: 10.1016/j.aml.2007.12.018. [20] K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348. doi: 10.1016/j.jde.2003.08.003. [21] M. Lewis, J. Renclawowicz and P. Driessche, Travalling waves and spread rates for a west nile virus model, Bull. Math. Biol., 68 (2006), 3-23. doi: 10.1007/s11538-005-9018-z. [22] S. Li, J. Wu and S. Liu, Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone, Calc. Var. Partial Differential Equations, 56 (2017), Art. 82, 35 pp. doi: 10.1007/s00526-017-1159-z. [23] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7. [24] Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Sys A, 4 (1998), 193-203. doi: 10.3934/dcds.1998.4.193. [25] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435. [26] D. Nash, F. Mostashari and A. Fine, etc., The Outbreak of West Nile Virus Infection in New York city area in 1999, N. Engl. Med., 344 (2001), 1807-1814. doi: 10.1056/NEJM200106143442401. [27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [28] C. V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 60 (2005), 1197-1217. doi: 10.1016/j.na.2004.10.008. [29] K. A. Rahman, R. Sudarsan and H. J. Eberl, A mixed-culture biofilm model with cross-diffusion, Bull. Math. Biol., 77 (2015), 2086-2124. doi: 10.1007/s11538-015-0117-1. [30] K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061. doi: 10.3934/dcds.2003.9.1049. [31] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. [32] S. Shim, Long time properties of prey-predator system with cross diffusion, Comm. Korean Math. Soc., 21 (2006), 293-320. doi: 10.4134/CKMS.2006.21.2.293. [33] G. Sweers, Strong positivity in $C(\overline Ω)$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833. [34] A. K. Tarboush, Z. G. Lin and M. Y. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841-860. doi: 10.1007/s11425-016-0367-4. [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [36] H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 272 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006. [37] W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. [38] Z. Wen and S. Fu, Turing instability for a competitor-competitor-mutualist model with nonlinear cross-diffusion effects, Chaos Solitons Fractals, 91 (2016), 379-385. doi: 10.1016/j.chaos.2016.06.019. [39] M. J. Wonham, T. C. Beck and M. A. Lewis, An epidemiology model for West Nile virus: Invansion analysis and control applications, Proc. R. Soc. Lond B, 271 (2004), 501-507. [40] Y. P. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015. [41] X. Q. Zhao, Dynamical Systems in Population Biology, Second edition, CMS Books in Mathematics/Ouvrages de Math$\acute{e}$matiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3. [42] H. Zhou and Z. G. Lin, Coexistence in a stroungly coupled system describing a two-species cooperative model, Appl. Math. Lett., 20 (2007), 1126-1130. doi: 10.1016/j.aml.2006.11.012.

show all references

##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [2] P. Álvarez-Caudevilla and J. López-Gómez, Asymptotic behaviour of principal eigenvalues for a class of cooperative systems, J. Differential Equations, 244 (2008), 1093-1113. doi: 10.1016/j.jde.2007.10.004. [3] D. S. Asnis, R. Conetta, A. A. Teixeira, G. Waldman and B. A. Sampson, The West Nile virus outbreak of 1999 in New York: The flushing hospital experience, Clinical Infect Dis., 30 (2000), 413-418. doi: 10.1086/313737. [4] K. W. Blaynech, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [5] E. Braverman and Md. Kamrujjaman, Competitive-cooperative models with various diffusion strategies, Comput. Math. Appl., 72 (2016), 653-662. doi: 10.1016/j.camwa.2016.05.017. [6] B. Chen and R. Peng, Coexistence states of a strongly coupled prey-predator model, J. Partial Diff. Eqs., 18 (2005), 154-166. [7] V. Chevalier, A. Tran and B. Durand, Predictive modeling of west nile virus transmission risk in the mediterranean basin, Int. J. Environ. Res. Public Health, 11 (2014), 67-90. doi: 10.3390/ijerph110100067. [8] G. Cruz-Pacheco, L. Esteva and C. Vargas, Seasonality and outbreaks in west nile virus infection, Bull. Math. Biol., 71 (2009), 1378-1393. doi: 10.1007/s11538-009-9406-x. [9] D. G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal., 17 (1986), 836-849. doi: 10.1137/0517060. [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. [11] S. Fu, L. Zhang and P. Hu, Global behavior of solutions in a Lotka-Volterra predator-prey model with prey-stage structure, Nonlinear Anal. Real World Appl., 14 (2013), 2027-2045. doi: 10.1016/j.nonrwa.2013.02.007. [12] W. Gan and Z. Lin, Coexistence and asymptotic periodicity in a competitor-competitor-mutualist model, J. Math. Anal. Appl., 337 (2008), 1089-1099. doi: 10.1016/j.jmaa.2007.04.022. [13] D. Horstmann, Remarks on some Lotka-Volterra type cross-diffusion models, Nonlinear Anal. Real World Appl., 8 (2007), 90-117. doi: 10.1016/j.nonrwa.2005.05.008. [14] M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive iteraction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2. [15] D. J. Jamieson, J. E. Ellis, D. B. Jernigan and T. A. Treadwell, Emerging infectious disease outbreaks: Old lessons and new challenges for obstetrician-gynecologists, Am. J. Obstet. Gynecol., 194 (2006), 1546-1555. doi: 10.1016/j.ajog.2005.06.062. [16] Y. Jia, J. Wu and H. Xu, Positive solutions of Lotka-Volterra competition model with cross-diffusion, Comput. Math. Appl., 68 (2014), 1220-1228. doi: 10.1016/j.camwa.2014.08.016. [17] A. Jüngel and I. V. Stelzer, Entropy structure of a cross-diffusion tumor-growth model, Math. Models Methods Appl. Sci., 22 (2012), 1250009, 26pp. doi: 10.1142/S0218202512500091. [18] K. I. Kim and Z. G. Lin, Coexistence of three species in a strongly coupled elliptic system, Nonlinear Anal., 55 (2003), 313-333. doi: 10.1016/S0362-546X(03)00242-6. [19] W. Ko and K. Ryu, On a predator-prey system with cross-diffusion representing the tendency of prey to keep away from its predators, Appl. Math. Lett., 21 (2008), 1177-1183. doi: 10.1016/j.aml.2007.12.018. [20] K. Kuto and Y. Yamada, Multiple coexistence states for a prey-predator system with cross-diffusion, J. Differential Equations, 197 (2004), 315-348. doi: 10.1016/j.jde.2003.08.003. [21] M. Lewis, J. Renclawowicz and P. Driessche, Travalling waves and spread rates for a west nile virus model, Bull. Math. Biol., 68 (2006), 3-23. doi: 10.1007/s11538-005-9018-z. [22] S. Li, J. Wu and S. Liu, Effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone, Calc. Var. Partial Differential Equations, 56 (2017), Art. 82, 35 pp. doi: 10.1007/s00526-017-1159-z. [23] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7. [24] Y. Lou, W. M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dynam. Sys A, 4 (1998), 193-203. doi: 10.3934/dcds.1998.4.193. [25] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435. [26] D. Nash, F. Mostashari and A. Fine, etc., The Outbreak of West Nile Virus Infection in New York city area in 1999, N. Engl. Med., 344 (2001), 1807-1814. doi: 10.1056/NEJM200106143442401. [27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [28] C. V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 60 (2005), 1197-1217. doi: 10.1016/j.na.2004.10.008. [29] K. A. Rahman, R. Sudarsan and H. J. Eberl, A mixed-culture biofilm model with cross-diffusion, Bull. Math. Biol., 77 (2015), 2086-2124. doi: 10.1007/s11538-015-0117-1. [30] K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions, Discrete Contin. Dyn. Syst., 9 (2003), 1049-1061. doi: 10.3934/dcds.2003.9.1049. [31] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. [32] S. Shim, Long time properties of prey-predator system with cross diffusion, Comm. Korean Math. Soc., 21 (2006), 293-320. doi: 10.4134/CKMS.2006.21.2.293. [33] G. Sweers, Strong positivity in $C(\overline Ω)$ for elliptic systems, Math. Z., 209 (1992), 251-271. doi: 10.1007/BF02570833. [34] A. K. Tarboush, Z. G. Lin and M. Y. Zhang, Spreading and vanishing in a West Nile virus model with expanding fronts, Sci. China Math., 60 (2017), 841-860. doi: 10.1007/s11425-016-0367-4. [35] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [36] H. Wan and H. Zhu, The backward bifurcation in compartmental models for West Nile virus, Math. Biosci., 272 (2010), 20-28. doi: 10.1016/j.mbs.2010.05.006. [37] W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942. [38] Z. Wen and S. Fu, Turing instability for a competitor-competitor-mutualist model with nonlinear cross-diffusion effects, Chaos Solitons Fractals, 91 (2016), 379-385. doi: 10.1016/j.chaos.2016.06.019. [39] M. J. Wonham, T. C. Beck and M. A. Lewis, An epidemiology model for West Nile virus: Invansion analysis and control applications, Proc. R. Soc. Lond B, 271 (2004), 501-507. [40] Y. P. Wu, The instability of spiky steady states for a competing species model with cross diffusion, J. Differential Equations, 213 (2005), 289-340. doi: 10.1016/j.jde.2004.08.015. [41] X. Q. Zhao, Dynamical Systems in Population Biology, Second edition, CMS Books in Mathematics/Ouvrages de Math$\acute{e}$matiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3. [42] H. Zhou and Z. G. Lin, Coexistence in a stroungly coupled system describing a two-species cooperative model, Appl. Math. Lett., 20 (2007), 1126-1130. doi: 10.1016/j.aml.2006.11.012.
Phase diagrams of $I_b(x)$ and $I_m(x)$ showing the existence of a positive solution of (4) for small cross-diffusion ($\beta_1 = 0.001$ and $\beta_2 = 0.002$)
Phase diagrams of $I_b(x,t)$ and $I_m(x,t)$ shows that the solution of (3) exists and stabilizes to a positive steady-state for small cross-diffusion ($\beta_1 = 0.132$ and $\beta_2 = 0.11$)
Phase diagrams of $I_b(x)$ and $I_m(x)$ shows that the global solution of (3) does not exist for big cross-diffusion ($\beta_1 = 0.133$ and $\beta_2 = 0.11$)
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