# American Institute of Mathematical Sciences

June  2017, 14(3): 805-820. doi: 10.3934/mbe.2017044

## A note on the global properties of an age-structured viral dynamic model with multiple target cell populations

 1 School of Mathematics and Statistics, Henan University, Kaifeng 475001, Henan, China 2 Centre for Disease Modelling, York University, Toronto, Ontario, M3J 1P3, Canada 3 Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA

Received  January 25, 2016 Accepted  October 30, 2016 Published  December 2016

Fund Project: This work was finished when S. Wang visited York University. Wang is supported by NSFC (No.11326200), Foundation of He’nan Educational Committee (No.15A110015, No.15A110018), and The Grant of China Scholarship Council (No.201408410018). L. Rong is partially supported by the NSF grant DMS-1349939.

Some viruses can infect different classes of cells. The age of infection can affect the dynamics of infected cells and viral production. Here we develop a viral dynamic model with the age of infection and multiple target cell populations. Using the methods of semigroup and Lyapunov function, we study the global asymptotic property of the steady states of the model. The results show that when the basic reproductive number falls below 1, the infection is predicted to die out. When the basic reproductive number exceeds 1, there exists a unique infected steady state which is globally asymptotically stable. The model can be extended to study virus dynamics with multiple compartments or coinfection by multiple types of viruses. We also show that under some scenarios the age-structured model can be reduced to an ordinary differential equation system with or without time delays.

Citation: Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044
##### References:
 [1] A. Alshorman, C. Samarasinghe, W. Lu and L. Rong, An HIV model with age-structured latently infected cells, J. Biol. Dyn., (2015), 1-24.  doi: 10.1080/17513758.2016.1198835. [2] R. P. Beasley, Hepatocellular carcinoma and hepatitis B virus, Lancet, 2 (1981), 1129-1133. [3] F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection choleramodel, Math. Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [4] C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Anal. Real World Appl., 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004. [5] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999. [6] C. A. Carter and L. S. Ehrlich, Cell biology of HIV-1 infection of macrophages, Annu. Rev. Microbiol., 62 (2008), 425-443.  doi: 10.1146/annurev.micro.62.081307.162758. [7] I. Castillo, Hepatitis C virus replicates in peripheral blood mononuclear cells of patients with occult hepatitis C virus infection, Gut., 54 (2005), 682-685.  doi: 10.1136/gut.2004.057281. [8] C. Ferrari, Cellular immune response to hepatitis B virus encoded antigens in a cute and chronic hepatitis B virus infection, J. Immunol., 145 (1990), 3442-3449. [9] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theoret. Biol., 229 (2004), 281-288.  doi: 10.1016/j.jtbi.2004.04.015. [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Mono-graphs, American Mathematical Society, Providence, RI, 1988. [11] E. A. Hernandez-Vargas and R. H. Middleton, Modeling the three stages in HIV infection, J. Theor. Biol., 320 (2013), 33-40.  doi: 10.1016/j.jtbi.2012.11.028. [12] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38.  doi: 10.1137/110826588. [13] S. Koenig, Detection of AIDS virus in macrophages in brain tissue from AIDS patients with encephalopathy, Science, 233 (1986), 1089-1093.  doi: 10.1126/science.3016903. [14] A. Kumar and G. Herbein, The macrophage: A therapeutic target in HIV-1 infection Mol. Cell. Therapies, 2 (2014), 10pp. doi: 10.1186/2052-8426-2-10. [15] M. J. Kuroda, Macrophages: Do they impact AIDS progression more than CD4 T cells?, J. Leukoc. Biol., 87 (2010), 569-573.  doi: 10.1189/jlb.0909626. [16] X. Lai and X. Zou, Dynamics of evolutionary competition between budding and lytic viral release strategies, Math. Biosci. Eng., 11 (2014), 1091-1113.  doi: 10.3934/mbe.2014.11.1091. [17] X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell Infection and cell-to-cell transmission, Math. Biosci. Eng., 11 (2014), 1091-1113.  doi: 10.1137/130930145. [18] 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. [19] P. Magal, Compact attractors for time periodic age-structured population models, Electron J. Differ. Equ., 65 (2001), 1-35. [20] P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.  doi: 10.1137/120882056. [21] P. Magal and H. R. Thieme, Eventual compactness for semi ows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695. [22] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An agestructured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.  doi: 10.3934/mbe.2004.1.267. [23] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [24] M. A. Nowak, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [25] M. Pope, Conjugates of dendritic cells and memory T lymphocytes from skin facilitate productive infection with HIV-1, Cell, 78 (1994), 389-398.  doi: 10.1016/0092-8674(94)90418-9. [26] F. Regenstein, New approaches to the treatment of chronic viral-hepatitis-B and viral-hepatitis C, Am. J. Med., 96 (1994), 47-51. [27] L. Rong, H. Dahari, R. M. Ribeiro and A. S. Perelson, Rapid emergence of protease inhibitor resistance in hepatitis C virus Sci. Transl. Med., 2 (2010), 30ra32. doi: 10.1126/scitranslmed.3000544. [28] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiviral theraphy, SIAM J. Appl. Math., 67 (2007), 731-756.  doi: 10.1137/060663945. [29] L. Rong, J. Guedj, H. Dahari, D. Coffield, M. Levi, P. Smith and A. S. Perelson, Analysis of hepatitis C virus decline during treatment with the protease inhibitor danoprevir using a multiscale model PLoS Comput. Biol., 9 (2013), e1002959, 12pp. doi: 10.1371/journal.pcbi.1002959. [30] M. Shen, Y. Xiao and L. Rong, Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics, Math. Biosci., 263 (2015), 37-50.  doi: 10.1016/j.mbs.2015.02.003. [31] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.  doi: 10.1016/j.jmaa.2006.06.064. [32] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3 (1990), 1035-1066. [33] S. Wang, X. Feng and Y. He, Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1959-1967.  doi: 10.1016/S0252-9602(11)60374-3. [34] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [35] K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36-44.  doi: 10.1016/j.jtbi.2007.11.007. [36] J. Wang, R. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate, Electron J. Differ. Equ., 33 (2015), 1-19. [37] J. Wang, R. Zhang and T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321-343.  doi: 10.1093/imamat/hxv039. [38] J. Wang, J. Lang and F. Li, Constructing Lyapunov functionals for a delayed viral infection model with multitarget cells, nonlinear incidence rate, state-dependent removal rate, J. Nonlinear Sci. Appl., 9 (2016), 524-536. [39] J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.  doi: 10.1016/j.nonrwa.2016.08.001. [40] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313.  doi: 10.1016/j.jmaa.2015.06.040. [41] J. Wang and R. Zhang, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227. [42] J. I. Weissberg, Survival in chronic hepatitis B: An analysis of 379 patients, Ann. Intern. Med., 101 (1984), 613-616.  doi: 10.7326/0003-4819-101-5-613. [43] R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001. [44] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.  doi: 10.1016/j.jmaa.2010.08.055. [45] Y. Yang, S. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Beddington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.  doi: 10.3934/mbe.2015.12.859. [46] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Med. Biol., 25 (2008), 99-112.  doi: 10.1093/imammb/dqm010.

show all references

##### References:
 [1] A. Alshorman, C. Samarasinghe, W. Lu and L. Rong, An HIV model with age-structured latently infected cells, J. Biol. Dyn., (2015), 1-24.  doi: 10.1080/17513758.2016.1198835. [2] R. P. Beasley, Hepatocellular carcinoma and hepatitis B virus, Lancet, 2 (1981), 1129-1133. [3] F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection choleramodel, Math. Biosci. Eng., 10 (2013), 1335-1349.  doi: 10.3934/mbe.2013.10.1335. [4] C. J. Browne, A multi-strain virus model with infected cell age structure: Application to HIV, Nonlinear Anal. Real World Appl., 22 (2015), 354-372.  doi: 10.1016/j.nonrwa.2014.10.004. [5] C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999-2017.  doi: 10.3934/dcdsb.2013.18.1999. [6] C. A. Carter and L. S. Ehrlich, Cell biology of HIV-1 infection of macrophages, Annu. Rev. Microbiol., 62 (2008), 425-443.  doi: 10.1146/annurev.micro.62.081307.162758. [7] I. Castillo, Hepatitis C virus replicates in peripheral blood mononuclear cells of patients with occult hepatitis C virus infection, Gut., 54 (2005), 682-685.  doi: 10.1136/gut.2004.057281. [8] C. Ferrari, Cellular immune response to hepatitis B virus encoded antigens in a cute and chronic hepatitis B virus infection, J. Immunol., 145 (1990), 3442-3449. [9] M. A. Gilchrist, D. Coombs and A. S. Perelson, Optimizing within-host viral fitness: Infected cell lifespan and virion production rate, J. Theoret. Biol., 229 (2004), 281-288.  doi: 10.1016/j.jtbi.2004.04.015. [10] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Mono-graphs, American Mathematical Society, Providence, RI, 1988. [11] E. A. Hernandez-Vargas and R. H. Middleton, Modeling the three stages in HIV infection, J. Theor. Biol., 320 (2013), 33-40.  doi: 10.1016/j.jtbi.2012.11.028. [12] G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model, SIAM J. Appl. Math., 72 (2012), 25-38.  doi: 10.1137/110826588. [13] S. Koenig, Detection of AIDS virus in macrophages in brain tissue from AIDS patients with encephalopathy, Science, 233 (1986), 1089-1093.  doi: 10.1126/science.3016903. [14] A. Kumar and G. Herbein, The macrophage: A therapeutic target in HIV-1 infection Mol. Cell. Therapies, 2 (2014), 10pp. doi: 10.1186/2052-8426-2-10. [15] M. J. Kuroda, Macrophages: Do they impact AIDS progression more than CD4 T cells?, J. Leukoc. Biol., 87 (2010), 569-573.  doi: 10.1189/jlb.0909626. [16] X. Lai and X. Zou, Dynamics of evolutionary competition between budding and lytic viral release strategies, Math. Biosci. Eng., 11 (2014), 1091-1113.  doi: 10.3934/mbe.2014.11.1091. [17] X. Lai and X. Zou, Modeling HIV-1 virus dynamics with both virus-to-cell Infection and cell-to-cell transmission, Math. Biosci. Eng., 11 (2014), 1091-1113.  doi: 10.1137/130930145. [18] 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. [19] P. Magal, Compact attractors for time periodic age-structured population models, Electron J. Differ. Equ., 65 (2001), 1-35. [20] P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.  doi: 10.1137/120882056. [21] P. Magal and H. R. Thieme, Eventual compactness for semi ows generated by nonlinear age-structured models, Commun. Pure Appl. Anal., 3 (2004), 695-727.  doi: 10.3934/cpaa.2004.3.695. [22] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An agestructured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells, Math. Biosci. Eng., 1 (2004), 267-288.  doi: 10.3934/mbe.2004.1.267. [23] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune response to persistent viruses, Science, 272 (1996), 74-79.  doi: 10.1126/science.272.5258.74. [24] M. A. Nowak, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. USA, 93 (1996), 4398-4402.  doi: 10.1073/pnas.93.9.4398. [25] M. Pope, Conjugates of dendritic cells and memory T lymphocytes from skin facilitate productive infection with HIV-1, Cell, 78 (1994), 389-398.  doi: 10.1016/0092-8674(94)90418-9. [26] F. Regenstein, New approaches to the treatment of chronic viral-hepatitis-B and viral-hepatitis C, Am. J. Med., 96 (1994), 47-51. [27] L. Rong, H. Dahari, R. M. Ribeiro and A. S. Perelson, Rapid emergence of protease inhibitor resistance in hepatitis C virus Sci. Transl. Med., 2 (2010), 30ra32. doi: 10.1126/scitranslmed.3000544. [28] L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiviral theraphy, SIAM J. Appl. Math., 67 (2007), 731-756.  doi: 10.1137/060663945. [29] L. Rong, J. Guedj, H. Dahari, D. Coffield, M. Levi, P. Smith and A. S. Perelson, Analysis of hepatitis C virus decline during treatment with the protease inhibitor danoprevir using a multiscale model PLoS Comput. Biol., 9 (2013), e1002959, 12pp. doi: 10.1371/journal.pcbi.1002959. [30] M. Shen, Y. Xiao and L. Rong, Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics, Math. Biosci., 263 (2015), 37-50.  doi: 10.1016/j.mbs.2015.02.003. [31] X. Song and A. U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329 (2007), 281-297.  doi: 10.1016/j.jmaa.2006.06.064. [32] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3 (1990), 1035-1066. [33] S. Wang, X. Feng and Y. He, Global asymptotical properties for a diffused HBV infection model with CTL immune response and nonlinear incidence, Acta Math. Sci. Ser. B Engl. Ed., 31 (2011), 1959-1967.  doi: 10.1016/S0252-9602(11)60374-3. [34] K. Wang and W. Wang, Propagation of HBV with spatial dependence, Math. Biosci., 210 (2007), 78-95.  doi: 10.1016/j.mbs.2007.05.004. [35] K. Wang, W. Wang and S. Song, Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 253 (2008), 36-44.  doi: 10.1016/j.jtbi.2007.11.007. [36] J. Wang, R. Zhang and T. Kuniya, Mathematical analysis for an age-structured HIV infection model with saturation infection rate, Electron J. Differ. Equ., 33 (2015), 1-19. [37] J. Wang, R. Zhang and T. Kuniya, The dynamics of an SVIR epidemiological model with infection age, IMA J. Appl. Math., 81 (2016), 321-343.  doi: 10.1093/imamat/hxv039. [38] J. Wang, J. Lang and F. Li, Constructing Lyapunov functionals for a delayed viral infection model with multitarget cells, nonlinear incidence rate, state-dependent removal rate, J. Nonlinear Sci. Appl., 9 (2016), 524-536. [39] J. Wang, J. Lang and X. Zou, Analysis of an age structured HIV infection model with virus-to-cell infection and cell-to-cell transmission, Nonlinear Anal. Real World Appl., 34 (2017), 75-96.  doi: 10.1016/j.nonrwa.2016.08.001. [40] J. Wang, R. Zhang and T. Kuniya, Global dynamics for a class of age-infection HIV models with nonlinear infection rate, J. Math. Anal. Appl., 432 (2015), 289-313.  doi: 10.1016/j.jmaa.2015.06.040. [41] J. Wang and R. Zhang, A note on dynamics of an age-of-infection cholera model, Math. Biosci. Eng., 13 (2016), 227-247.  doi: 10.3934/mbe.2016.13.227. [42] J. I. Weissberg, Survival in chronic hepatitis B: An analysis of 379 patients, Ann. Intern. Med., 101 (1984), 613-616.  doi: 10.7326/0003-4819-101-5-613. [43] R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.  doi: 10.1016/j.jtbi.2009.01.001. [44] R. Xu, Global stability of an HIV-1 infection model with saturation infection and intracellular delay, J. Math. Anal. Appl., 375 (2011), 75-81.  doi: 10.1016/j.jmaa.2010.08.055. [45] Y. Yang, S. Ruan and D. Xiao, Global stability of an age-structured virus dynamics model with Beddington-Deangelis infection function, Math. Biosci. Eng., 12 (2015), 859-877.  doi: 10.3934/mbe.2015.12.859. [46] H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Math. Med. Biol., 25 (2008), 99-112.  doi: 10.1093/imammb/dqm010.
 [1] Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145 [2] C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008 [3] Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101 [4] Andrei Korobeinikov, Philip K. Maini. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Mathematical Biosciences & Engineering, 2004, 1 (1) : 57-60. doi: 10.3934/mbe.2004.1.57 [5] Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022082 [6] Nguyen Dinh Cong. Semigroup property of fractional differential operators and its applications. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022064 [7] Andrey V. Melnik, Andrei Korobeinikov. Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility. Mathematical Biosciences & Engineering, 2013, 10 (2) : 369-378. doi: 10.3934/mbe.2013.10.369 [8] J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293 [9] Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859 [10] C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819 [11] Andrey V. Melnik, Andrei Korobeinikov. Global asymptotic properties of staged models with multiple progression pathways for infectious diseases. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1019-1034. doi: 10.3934/mbe.2011.8.1019 [12] Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207 [13] Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499 [14] Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure and Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347 [15] Jóhann Björnsson, Peter Giesl, Sigurdur F. Hafstein, Christopher M. Kellett. Computation of Lyapunov functions for systems with multiple local attractors. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4019-4039. doi: 10.3934/dcds.2015.35.4019 [16] Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042 [17] Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589 [18] Sophia R.-J. Jang. Discrete host-parasitoid models with Allee effects and age structure in the host. Mathematical Biosciences & Engineering, 2010, 7 (1) : 67-81. doi: 10.3934/mbe.2010.7.67 [19] Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329 [20] Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109

2018 Impact Factor: 1.313

## Metrics

• PDF downloads (63)
• HTML views (57)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]