## Threshold dynamics for compartmental epidemic models in periodic environments.(English)Zbl 1157.34041

The basic reproduction ratio and its computation formulae are established for a large class of compartmental epidemic models in periodic environments. It is proved that a disease cannot invade the disease-free state if the ratio is less than unity and can invade if it is greater than unity. It is also shown that the basic reproduction number of the time-averaged autonomous system is applicable in the case where both the matrix of new infection rate and the matrix of transition and dissipation within infectious compartments are diagonal, but it may underestimate and overestimate infection risks in other cases. The global dynamics of a periodic epidemic model with patch structure is analyzed in order to study the impact of periodic contacts or periodic migrations on the disease transmission.

### MSC:

 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34D05 Asymptotic properties of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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### References:

 [1] Arino, J.; van den Driessche, P., A multi-city epidemic model, Math. Popul. Stud., 10, 175-193 (2003) · Zbl 1028.92021 [2] Arino, J., van den Driessche, P.: The basic reproduction number in a multi-city compartmental epidemic model, Positive Systems (Rome, 2003) pp. 135-142, Lecture Notes in Control and Information Science, vol. 294. Springer, Berlin (2003) · Zbl 1057.92045 [3] Bacaër, N., Approximation of the basic reproduction number R_0 for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69, 1067-1091 (2007) · Zbl 1298.92093 [4] Bacaër, N.; Guernaoui, S., The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53, 421-436 (2006) · Zbl 1098.92056 [5] Billings, L.; Schwartz, I. B., Exciting chaos with noise: unexpcted dynamics in epidemic outbreaks, J. Math. Biol., 44, 31-48 (2002) · Zbl 0990.92036 [6] Cushing, J. M., A juvenile-adult model with periodic vital rates, J. Math. Biol., 53, 520-539 (2006) · Zbl 1112.92052 [7] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., On the definition and the computation of the basic reproduction ratio R_0 in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28, 365-382 (1990) · Zbl 0726.92018 [8] Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), Chichester: Wiley, Chichester · Zbl 0997.92505 [9] Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture notes in biomath, vol. 11, pp. 1-5. Berlin-Heidelberg-New York: Springer (1976) · Zbl 0333.92014 [10] Earn, D. J.D.; Rohani, P.; Bolker, B. M.; Grenfell, B. T., A simple model for complex dynamical transitions in epidemics, Science, 287, 667-670 (2000) [11] Farrington, C. P., On vaccine efficacy and reproduction numbers, Math. Biosci., 185, 89-109 (2003) · Zbl 1021.92034 [12] Feng, Z.; Velasco-Hernández, J. X., Competitive exclusion in a vector-host model for the Dengue fever, J. Math. Biol., 35, 523-544 (1997) · Zbl 0878.92025 [13] Fulford, G. R.; Roberts, M. G.; Heesterbeek, J. A.P., The metapopulation dynamics of an infectious disease: tuberculosis in possums, Theor. Popul. Biol., 61, 15-29 (2003) · Zbl 1038.92034 [14] Greenhalgh, D.; Moneim, I. A., SIRS epidemic model and simulations using different types of seasonal contact rate, Syst. Anal. Model. Simul., 43, 573-600 (2003) · Zbl 1057.92046 [15] Gumel, A. B.; Ruan, S.; Day, T.; Watmough, J.; Brauer, F.; van den Driessche, P.; Gabrielson, D.; Bowman, C.; Alexander, M. E.; Ardal, S.; Wu, J.; Sahai, B. M., Modeling strategies for controlling SARS outbreaks, Proc. R. Soc. Lond.: Biol. Sci., 271, 2223-2232 (2004) [16] Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Company, INC, Malabar, Florida (1980) [17] Heesterbeek, J. A.P.; Roberts, M. G., Threshold quantities for infectious diseases in periodic environments, J. Biol. Syst., 3, 779-787 (1995) [18] Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Series 247. Longman Scientific and Technical (1991) · Zbl 0731.35050 [19] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev, 42, 599-653 (2000) · Zbl 0993.92033 [20] Hyman, J. M.; Li, J.; Stanley, E. A., The differential infectivity and staged progression models for the transmission of HIV, Math. Biosci., 155, 77-109 (1999) · Zbl 0942.92030 [21] Kato, T., Perturbation Theory for Linear Operators (1976), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg [22] Kuznetsov, Y. A.; Piccardi, C., Bifurcation analysis of periodic SEIR and SIR epidemic models, J. Math. Biol., 32, 109-121 (1984) · Zbl 0786.92022 [23] Ma, J.; Ma, Z., Epidemic threshold conditions for seasonally forced SEIR models, Math. Biosci. Eng., 3, 161-172 (2006) · Zbl 1089.92048 [24] Ruan, S.; Wang, W.; Levin, S. A., The effect of global travel on the spread of SARS, Math. Biosc. Eng., 3, 205-218 (2006) · Zbl 1089.92049 [25] Schenzle, D., An age-structured model of pre- and post-vaccination measles transmissions, IMA J. Math. Appl. Med. Biol., 1, 169-191 (1984) · Zbl 0611.92021 [26] Schwartz, I. B., Small amplitude, long periodic out breaks in seasonally driven epidemics, J. Math. Biol., 30, 473-491 (1992) · Zbl 0745.92026 [27] Schwartz, I. B.; Smith, H. L., Infinite subharmonic bifurcation in an SIER epidemic model, J. Math. Biol., 18, 233-253 (1983) · Zbl 0523.92020 [28] Smith, H. L., Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17, 179-190 (1983) · Zbl 0529.92018 [29] Smith, H.L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press (1995) · Zbl 0860.92031 [30] Thieme, H. R., Renewal theorems for linear periodic Volterra integral equations, J. Integral Equ., 7, 253-277 (1984) · Zbl 0566.45016 [31] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036 [32] Wang, W.; Mulone, G., Threshold of disease transmission on a patch environment, J. Math. Anal. Appl., 285, 321-335 (2003) · Zbl 1021.92039 [33] Wang, W.; Ruan, S., Simulating the SARS outbreak in Beijing with limited data, J. Theor. Biol., 227, 369-379 (2004) · Zbl 1439.92185 [34] Wang, W.; Zhao, X.-Q., An epidemic model in a patchy environment, Math. Biosci., 190, 39-69 (2004) · Zbl 1048.92030 [35] Wang, W.; Zhao, X.-Q., An age-structured epidemic model in a patchy environment, SIAM J. Appl. Math., 65, 1597-1614 (2005) · Zbl 1072.92045 [36] Wang, W.; Zhao, X.-Q., An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66, 1454-1472 (2006) · Zbl 1094.92055 [37] Williams, B. G.; Dye, C., Infectious disease persistence when transmission varies seasonally, Math. Biosci., 145, 77-88 (1997) · Zbl 0896.92024 [38] Zhang, F.; Zhao, X.-Q., A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325, 496-516 (2007) · Zbl 1101.92046 [39] Zhao, X.-Q., Dynamical Systems in Population Biology (2003), New York: Springer-Verlag, New York [40] Zhou, Y.; Ma, Z.; Brauer, F., A discrete epidemic model for SARS transmission and control in China, Math. Comput. Model., 40, 1491-1506 (2004) · Zbl 1066.92046
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