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August  2017, 14(4): 901-931. doi: 10.3934/mbe.2017048

Competitive exclusion in an infection-age structured vector-host epidemic model

 1 Department of Public Education, Zhumadian Vocational and Technical College, Zhumadian 463000, China 2 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China 3 Department of Mathematics and Physics, Anyang Institute of Technology, Anyang 455000, China

* Corresponding author: Zhipeng Qiu

Received  December 2015 Accepted  December 2016 Published  February 2017

The competitive exclusion principle means that the strain with the largest reproduction number persists while eliminating all other strains with suboptimal reproduction numbers. In this paper, we extend the competitive exclusion principle to a multi-strain vector-borne epidemic model with age-since-infection. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts, both of which describe the different removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The formulas for the reproduction numbers $\mathcal R^j_0$ of strain $j,j=1,2,···, n$, are obtained from the biological meanings of the model. The strain $j$ can not invade the system if $\mathcal R^j_0<1$, and the disease free equilibrium is globally asymptotically stable if $\max_j\{\mathcal R^j_0\}<1$. If $\mathcal R^{j_0}_0>1$, then a single-strain equilibrium $\mathcal{E}_{j_0}$ exists, and the single strain equilibrium is locally asymptotically stable when $\mathcal R^{j_0}_0>1$ and $\mathcal R^{j_0}_0>\mathcal R^{j}_0,j≠ j_0$. Finally, by using a Lyapunov function, sufficient conditions are further established for the global asymptotical stability of the single-strain equilibrium corresponding to strain $j_0$, which means strain $j_0$ eliminates all other stains as long as $\mathcal R^{j}_0/\mathcal R^{j_0}_0<b_j/b_{j_0}<1,j≠ j_0$, where $b_j$ denotes the probability of a given susceptible vector being transmitted by an infected host with strain $j$.

Citation: Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048
References:
 [1] F. Brauer, Z. S. Shuai and P. V. D. Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Enger., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [2] H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol, 27 (1989), 179-190.  doi: 10.1007/BF00276102. [3] L. M. Cai, M. Martcheva and X. Z. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, J. Biol. Dynamics, 7 (2013), 47-67.  doi: 10.1080/17513758.2013.772253. [4] Y. Dang, Z. Qiu, X. Li and M. Martcheva, Global dynamics of a vector-host epidemic model with age of infection, submitted, 2015. [5] L. Esteva, A. B. Gumel and C. Vargas-de Leon, Qualitative study of transmission dynamics of drug-resistant malaria, Math. Comput. Modelling, 50 (2009), 611-630.  doi: 10.1016/j.mcm.2009.02.012. [6] Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector host model for the dengue fever, J. Math. Biol, 35 (1997), 523-544.  doi: 10.1007/s002850050064. [7] S. M. Garba and A. B. Gumel, Effect of cross-immunity on the transmission dynamics of two srains of dengue, J. Comput. Math, 87 (2010), 2361-2384.  doi: 10.1080/00207160802660608. [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, 1988. [9] W. O. Kermack and A. G. McKendrick, in Contributions to the Mathematical Theory of Epidemics, Contributions to the Mathematical Theory of Epidemics, Royal Society Lond, 115 (1927), 700-721. [10] A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Royal Society Lond B, 268 (2001), 985-993.  doi: 10.1098/rspb.2001.1599. [11] A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics, Theor. Popul. Biol., 60 (2001), 59-71.  doi: 10.1006/tpbi.2001.1525. [12] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122. [13] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, J. Math. Anal. Appl, 408 (2013), 225-246.  doi: 10.1016/j.jmaa.2013.05.064. [14] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol, 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [15] M. Martcheva, M. Iannelli and X. Z. Li, Subthreshold coexistence of strains: The impact of vaccination and mutation, Math. Biosci. Eng., 4 (2007), 287-317.  doi: 10.3934/mbe.2007.4.287. [16] D. L. Qian, X. Z. Li and M. Ghosh, Coexistence of the strains induced by mutation Int. J. Biomath. 5 (2012), 1260016, 25 pp. doi: 10.1142/S1793524512600169. [17] Z. P. Qiu, Q. K. Kong, X. Z. Li and M. M. Martcheva, The vector host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [18] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics if HIV/AIDs?, SIAM J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068. [19] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci, 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3. [20] J. X. Yang, Z. P. Qiu and X. Z. Li, Global stability of an age-structured cholera model, Math. Biosci. Enger., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641. [21] K. Yosida, Functional Analysis Springer-Verlag, Berlin, Heidelberg, New York, 1968.

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References:
 [1] F. Brauer, Z. S. Shuai and P. V. D. Driessche, Dynamics of an age-of-infection cholera model, Math. Biosci. Enger., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [2] H. J. Bremermann and H. R. Thieme, A competitive exclusion principle for pathogen virulence, J. Math. Biol, 27 (1989), 179-190.  doi: 10.1007/BF00276102. [3] L. M. Cai, M. Martcheva and X. Z. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, J. Biol. Dynamics, 7 (2013), 47-67.  doi: 10.1080/17513758.2013.772253. [4] Y. Dang, Z. Qiu, X. Li and M. Martcheva, Global dynamics of a vector-host epidemic model with age of infection, submitted, 2015. [5] L. Esteva, A. B. Gumel and C. Vargas-de Leon, Qualitative study of transmission dynamics of drug-resistant malaria, Math. Comput. Modelling, 50 (2009), 611-630.  doi: 10.1016/j.mcm.2009.02.012. [6] Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector host model for the dengue fever, J. Math. Biol, 35 (1997), 523-544.  doi: 10.1007/s002850050064. [7] S. M. Garba and A. B. Gumel, Effect of cross-immunity on the transmission dynamics of two srains of dengue, J. Comput. Math, 87 (2010), 2361-2384.  doi: 10.1080/00207160802660608. [8] J. K. Hale, Asymptotic Behavior of Dissipative Systems AMS, Providence, 1988. [9] W. O. Kermack and A. G. McKendrick, in Contributions to the Mathematical Theory of Epidemics, Contributions to the Mathematical Theory of Epidemics, Royal Society Lond, 115 (1927), 700-721. [10] A. L. Lloyd, Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods, Royal Society Lond B, 268 (2001), 985-993.  doi: 10.1098/rspb.2001.1599. [11] A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics, Theor. Popul. Biol., 60 (2001), 59-71.  doi: 10.1006/tpbi.2001.1525. [12] P. Magal, C. C. McCluskey and G. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109-1140.  doi: 10.1080/00036810903208122. [13] M. Martcheva and X. Z. Li, Competitive exclusion in an infection-age structured model with environmental transmission, J. Math. Anal. Appl, 408 (2013), 225-246.  doi: 10.1016/j.jmaa.2013.05.064. [14] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol, 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [15] M. Martcheva, M. Iannelli and X. Z. Li, Subthreshold coexistence of strains: The impact of vaccination and mutation, Math. Biosci. Eng., 4 (2007), 287-317.  doi: 10.3934/mbe.2007.4.287. [16] D. L. Qian, X. Z. Li and M. Ghosh, Coexistence of the strains induced by mutation Int. J. Biomath. 5 (2012), 1260016, 25 pp. doi: 10.1142/S1793524512600169. [17] Z. P. Qiu, Q. K. Kong, X. Z. Li and M. M. Martcheva, The vector host epidemic model with multiple strains in a patchy environment, J. Math. Anal. Appl., 405 (2013), 12-36.  doi: 10.1016/j.jmaa.2013.03.042. [18] H. R. Thieme and C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics if HIV/AIDs?, SIAM J. Appl. Math., 53 (1993), 1447-1479.  doi: 10.1137/0153068. [19] H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci, 166 (2000), 173-201.  doi: 10.1016/S0025-5564(00)00018-3. [20] J. X. Yang, Z. P. Qiu and X. Z. Li, Global stability of an age-structured cholera model, Math. Biosci. Enger., 11 (2014), 641-665.  doi: 10.3934/mbe.2014.11.641. [21] K. Yosida, Functional Analysis Springer-Verlag, Berlin, Heidelberg, New York, 1968.
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