×

Endemic threshold results for an age-structured SIS epidemic model with periodic parameters. (English) Zbl 1263.92041

Summary: The main contribution of this paper is to obtain a threshold value for the existence and uniqueness of a nontrivial endemic periodic solution of an age-structured SIS epidemic model with periodic parameters. Under the assumption of the weak ergodicity of a non-autonomous Lotka-McKendrick system, we formulate a normalized system for an infected population as an initial boundary value problem of a partial differential equation. The existence problem for endemic periodic solutions is reduced to a fixed point problem of a nonlinear integral operator acting on a Banach space of locally integrable periodic \(L^{1}\)-valued functions.
We prove that the spectral radius of the Fréchet derivative of the integral operator at zero plays the role of a threshold for the existence and uniqueness of a nontrivial fixed point of the operator corresponding to a non-trivial periodic solution of the original differential equation in a weak sense. If the Malthusian parameter of the host population is equal to zero, our threshold value is equal to the well-known epidemiological threshold value, the basic reproduction number \(R_{0}\). However, if this is not the case, then two threshold values are different from each other and we have to pay attention on their actual biological implications.

MSC:

92D30 Epidemiology
35B10 Periodic solutions to PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
47G10 Integral operators
35Q92 PDEs in connection with biology, chemistry and other natural sciences
47N60 Applications of operator theory in chemistry and life sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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
[2] Bacaër, N.; Guernaoui, S., The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53, 421-436 (2006) · Zbl 1098.92056
[3] Bacaër, N.; Ouifki, R., Growth rate and basic reproduction number for population models with a simple periodic factor, Math. Biosci., 210, 647-658 (2007) · Zbl 1133.92023
[4] Busenberg, S. N.; Iannelli, M.; Thieme, H. R., Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22, 1065-1080 (1991) · Zbl 0741.92015
[5] Busenberg, S. N.; Iannelli, M.; Thieme, H. R., Dynamics of an age structured epidemic model, (Dynamical Systems (1993), World Scientific: World Scientific Singapore), 1-19 · Zbl 0943.92502
[6] Coale, A. J., The use of Fourier analysis to express the relation between time variations in fertility and the time sequence of births in a closed human population, Demography, 7, 93-120 (1970)
[7] Coale, A. J., The Growth and Structure of Human Populations: A Mathematical Investigation (1972), Princeton UP: Princeton UP Princeton
[8] Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), John Wiley and Sons: John Wiley and Sons Chichester · Zbl 0997.92505
[9] Diekmann, O.; Heesterbeek, J. A.P.; Metz, J. A.J., 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, 365-382 (1990) · Zbl 0726.92018
[10] Diekmann, O.; Heesterbeek, J. A.P.; Roberts, M. G., The construction of next-generation matrices for compartmental epidemic models, J. Roy. Soc. Interface, 7, 873-885 (2010)
[11] Feng, Z.; Huang, W.; Castillo-Chavez, C., Global behavior of a multi-group SIS epidemic model with age structure, J. Differential Equations, 218, 292-324 (2005) · Zbl 1083.35020
[12] Hethcote, H. W., Asymptotic behavior in a deterministic epidemic model, Bull. Math. Biol., 35, 607-614 (1973) · Zbl 0279.92011
[13] Hethcote, H. W.; Levin, S. A., Periodicity in epidemiological models, (Levin, S. A.; Hallam, T. G.; Gross, L. J., Applied Mathematical Ecology (1989), Springer: Springer Berlin), 193-211
[14] Iannelli, M., Mathematical Theory of Age-structured Population Dynamics (1995), Giardini editori e stampatori: Giardini editori e stampatori Pisa
[15] Iannelli, M.; Kim, M. Y.; Park, E. J., Asymptotic behavior for an SIS epidemic model and its approximation, Nonlinear Anal., 35, 797-814 (1999) · Zbl 0921.92029
[16] Iannelli, M.; Milner, F. A.; Pugliese, A., Analytical and numerical results for the age-structured SIS epidemic model with mixed inter-intracohort transmission, SIAM J. Math. Anal., 23, 662-688 (1992) · Zbl 0776.35032
[17] Inaba, H., Endemic threshold results for age-duration-structured population model for HIV infection, Math. Biosci., 201, 15-47 (2006) · Zbl 1094.92053
[18] Inaba, H., Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28, 411-434 (1990) · Zbl 0742.92019
[19] Inaba, H., Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Discrete Contin. Dyn. Syst. Ser. B, 6, 1, 69-96 (2006) · Zbl 1088.92049
[20] Inaba, H., Age-structured homogeneous epidemic systems with application to the MSEIR epidemic model, J. Math. Biol., 54, 101-146 (2007) · Zbl 1116.92054
[21] Inaba, H., On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65, 309-348 (2012) · Zbl 1303.92117
[22] Inaba, H., The Malthusian parameter and \(R_0\) for heterogeneous populations in periodic environments, Math. Biosci. Eng., 9, 313-346 (2012) · Zbl 1260.92098
[23] Inaba, H., Weak ergodicity of population evolution processes, Math. Biosci., 96, 195-219 (1989) · Zbl 0698.92020
[24] Inaba, H., A semigroup approach to the strong ergodic theorem of the multistate stable population process, Math. Popul. Stud., 1, 49-77 (1988) · Zbl 0900.92122
[25] Inaba, H.; Nishiura, H., The basic reproduction number of an infectious disease in a stable population: the impact of population growth rate on the eradication threshold, Math. Model. Nat. Phenom., 3, 7, 194-228 (2008) · Zbl 1337.92208
[26] Inaba, H.; Sekine, H., A mathematical model for Chagas disease with infection-age-dependent infectivity, Math. Biosci., 190, 39-69 (2004) · Zbl 1049.92033
[27] Krein, M. G.; Rutman, M. A., Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Transl., 10, 199-325 (1948) · Zbl 0030.12902
[28] Langlais, M.; Busenberg, S. N., Global behaviour in age structured S.I.S. models with seasonal periodicities and vertical transmission, J. Math. Anal. Appl., 213, 511-533 (1997) · Zbl 1002.92556
[29] Marek, I., Frobenius theory of positive operators: comparison theorems and applications, SIAM J. Appl. Math., 19, 607-628 (1970) · Zbl 0219.47022
[30] Nakata, Y.; Kuniya, T., Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363, 230-237 (2010) · Zbl 1184.34056
[31] Perthame, B., Transport Equations in Biology (2007), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1185.92006
[32] Rundnicki, R.; Mackey, M. C., Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl., 187, 548-566 (1994) · Zbl 0823.92022
[33] Smith, H. L., Multiple stable subharmonics for a periodic epidemic model, J. Math. Biol., 17, 179-190 (1983) · Zbl 0529.92018
[34] Thieme, H. R., Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70, 188-211 (2009) · Zbl 1191.47089
[35] Thieme, H. R., Renewal theorems for linear periodic Volterra integral equations, J. Integral Equations, 7, 253-277 (1984) · Zbl 0566.45016
[36] 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
[37] Wang, W.; Zhao, X.-Q., Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20, 699-717 (2008) · Zbl 1157.34041
[38] Yosida, K., Functional Analysis (1980), Springer: Springer Berlin · Zbl 0152.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.