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Stability of a two-strain tuberculosis model with general contact rate. (English) Zbl 1217.34082
Summary: A two-strain tuberculosis model with general contact rate which allows tuberculosis patients with the drug-sensitive Mycobacterium tuberculosis strain to be treated is presented. The model includes both drug-sensitive and drug-resistant strains. A detailed qualitative analysis about positivity, boundedness, existence, uniqueness and global stability of the equilibria of this model is carried out. Analytical results of the model show that the quantities \(R_1\) and \(R_2\), which represent the basic reproduction numbers of the sensitive and resistant strains, respectively, provide threshold conditions which determine the competitive outcomes of the two strains. Numerical simulations are also conducted to confirm and extend the analytic results.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
92C50 Medical applications (general)
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References:
[1] C. Dye, S. Scheele, P. Dolin, V. Pathania, and M. C. Raviglione, “Global burden of tuberculosis: estimated incidence, prevalence, and mortality by country,” Journal of the American Medical Association, vol. 282, no. 7, pp. 677-686, 1999. · doi:10.1001/jama.282.7.677
[2] T. R. Frieden, T. R. Sterling, S. S. Munsiff, C. J. Watt, and C. Dye, “Tuberculosis,” Lancet, vol. 362, no. 9387, pp. 887-899, 2003. · doi:10.1016/S0140-6736(03)14333-4
[3] B. Song, C. Castillo-Chavez, and J. P. Aparicio, “Tuberculosis models with fast and slow dynamics: the role of close and casual contacts,” Mathematical Biosciences, vol. 180, pp. 187-205, 2002. · Zbl 1015.92025 · doi:10.1016/S0025-5564(02)00112-8
[4] B. Miller, “Preventive therapy for tuberculosis,” Medical Clinics of North America, vol. 77, no. 6, pp. 1263-1275, 1993.
[5] P. Rodrigues, M. G. M. Gomes, and C. Rebelo, “Drug resistance in tuberculosis-a reinfection model,” Theoretical Population Biology, vol. 71, no. 2, pp. 196-212, 2007. · Zbl 1118.92036 · doi:10.1016/j.tpb.2006.10.004
[6] M. A. Espinal, et al., “Standard short-course chemotherapy for drug-resistant tuberculosis: treatment outcomes in 6 countries,” Journal of the American Medical Association, vol. 283, no. 19, pp. 2537-2545, 2000.
[7] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599-653, 2000. · Zbl 0993.92033 · doi:10.1137/S0036144500371907
[8] F. Brauer and C. Castillo-Chávez, Mathematical models in population biology and epidemiology, vol. 40 of Texts in Applied Mathematics, Springer, New York, NY, USA, 2001. · Zbl 0967.92015
[9] B. M. Murphy, B. H. Singer, S. Anderson, and D. Kirschner, “Comparing epidemic tuberculosis in demographically distinct heterogeneous populations,” Mathematical Biosciences, vol. 180, pp. 161-185, 2002. · Zbl 1015.92024 · doi:10.1016/S0025-5564(02)00133-5
[10] C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361-404, 2004. · Zbl 1060.92041 · doi:10.3934/mbe.2004.1.361
[11] B. M. Murphy, B. H. Singer, and D. Kirschner, “On treatment of tuberculosis in heterogeneous populations,” Journal of Theoretical Biology, vol. 223, no. 4, pp. 391-404, 2003. · doi:10.1016/S0022-5193(03)00038-9
[12] C. Castillo-Chavez and Z. Feng, “Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,” Mathematical Biosciences, vol. 151, no. 2, pp. 135-154, 1998. · Zbl 0981.92029 · doi:10.1016/S0025-5564(98)10016-0
[13] C. C. McCluskey, “Lyapunov functions for tuberculosis models with fast and slow progression,” Mathematical Biosciences and Engineering, vol. 3, no. 4, pp. 603-614, 2006. · Zbl 1113.92057 · doi:10.3934/mbe.2006.3.603
[14] C. Castillo-Chavez and Z. Feng, “To treat or not to treat: the case of tuberculosis,” Journal of Mathematical Biology, vol. 35, no. 6, pp. 629-656, 1997. · Zbl 0895.92024 · doi:10.1007/s002850050069
[15] S. Bowong, “Optimal control of the transmission dynamics of tuberculosis,” Nonlinear Dynamics, vol. 61, no. 4, pp. 729-748, 2010. · Zbl 1204.49044 · doi:10.1007/s11071-010-9683-9
[16] S. Bowong and J. J. Tewa, “Mathematical analysis of a tuberculosis model with differential infectivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 4010-4021, 2009. · Zbl 1221.34128 · doi:10.1016/j.cnsns.2009.02.017
[17] H. McCallum, N. Barlow, and J. Hone, “How should pathogen transmission be modelled?” Trends in Ecology and Evolution, vol. 16, no. 6, pp. 295-300, 2001. · doi:10.1016/S0169-5347(01)02144-9
[18] S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135-163, 2003. · Zbl 1028.34046 · doi:10.1016/S0022-0396(02)00089-X
[19] D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Mathematical Biosciences, vol. 208, no. 2, pp. 419-429, 2007. · Zbl 1119.92042 · doi:10.1016/j.mbs.2006.09.025
[20] Z. Yuan and L. Wang, “Global stability of epidemiological models with group mixing and nonlinear incidence rates,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 995-1004, 2010. · Zbl 1254.34075 · doi:10.1016/j.nonrwa.2009.01.040
[21] Y. Tang, D. Huang, S. Ruan, and W. Zhang, “Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate,” SIAM Journal on Applied Mathematics, vol. 69, no. 2, pp. 621-639, 2008. · Zbl 1171.34033 · doi:10.1137/070700966
[22] M. Y. Li and J. S. Muldowney, “Global stability for the SEIR model in epidemiology,” Mathematical Biosciences, vol. 125, no. 2, pp. 155-164, 1995. · Zbl 0821.92022 · doi:10.1016/0025-5564(95)92756-5
[23] H. R. Thieme and C. Castillo-Chavez, “On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic,” in Mathematical and Statistical Approaches to AIDS Epidemiology, vol. 83 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1989. · Zbl 0687.92009
[24] J. Zhang and Z. Ma, “Global dynamics of an SEIR epidemic model with saturating contact rate,” Mathematical Biosciences, vol. 185, no. 1, pp. 15-32, 2003. · Zbl 1021.92040 · doi:10.1016/S0025-5564(03)00087-7
[25] J. A. P. Heesterbeek and J. A. J. Metz, “The saturating contact rate in marriage- and epidemic models,” Journal of Mathematical Biology, vol. 31, no. 5, pp. 529-539, 1993. · Zbl 0770.92021 · doi:10.1007/BF00173891
[26] S. Bowong and J. J. Tewa, “Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3621-3631, 2010. · Zbl 1222.37094 · doi:10.1016/j.cnsns.2010.01.007
[27] P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29-48, 2002. · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[28] C. Castillo-Chavez and H. R. Thieme, “Asymptotically autonomous epidemic models,” in Mathematical Population Dynamics: Analysis of Heterogeneity, O. Arino, et al., Ed., vol. 1 of Theory of Epidemics, pp. 33-50, Wuetz, 1995.
[29] K. Mischaikow, H. Smith, and H. R. Thieme, “Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions,” Transactions of the American Mathematical Society, vol. 347, no. 5, pp. 1669-1685, 1995. · Zbl 0829.34037 · doi:10.2307/2154964
[30] H. R. Thieme, “Persistence under relaxed point-dissipativity (with application to an endemic model),” SIAM Journal on Mathematical Analysis, vol. 24, no. 2, pp. 407-435, 1993. · Zbl 0774.34030 · doi:10.1137/0524026
[31] J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA, 1976, With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics. · Zbl 0364.93002
[32] J. P. LaSalle, “Stability theory for ordinary differential equations,” Journal of Differential Equations, vol. 4, pp. 57-65, 1968. · Zbl 0159.12002 · doi:10.1016/0022-0396(68)90048-X
[33] A. Korobeinikov and P. K. Maini, “A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,” Mathematical Biosciences and Engineering, vol. 1, no. 1, pp. 57-60, 2004. · Zbl 1062.92061 · doi:10.3934/mbe.2004.1.57
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