Endemic threshold results in an age-duration-structured population model for HIV infection. (English) Zbl 1094.92053

Inaba, Hisashi

Math. Biosci. 201, No. 1-2, 15-47 (2006).

Summary: We consider an age-duration-structured population model for HIV infection in a homosexual community. First we investigate the invasion problem to establish the basic reproduction ratio \(R_0\) for the HIV/AIDS epidemic by which we can state the threshold criteria: The disease can invade into the completely susceptible population if \(R_0>1\), whereas it cannot if \(R_0<1\). Subsequently, we examine existence and uniqueness of endemic steady states. We will show sufficient conditions for a backward or a forward bifurcation to occur when the basic reproduction ratio crosses unity. That is, in contrast with classical epidemic models, for our HIV model there could exist multiple endemic steady states even if \(R_0\) is less than one. Finally, we show sufficient conditions for the local stability of the endemic steady states.

MSC:

92D30	Epidemiology
35B32	Bifurcations in context of PDEs
35B35	Stability in context of PDEs
35Q92	PDEs in connection with biology, chemistry and other natural sciences
47N60	Applications of operator theory in chemistry and life sciences

Keywords:

HIV/AIDS epidemic; structured populations; basic reproduction ratio; threshold condition; endemic state; bifurcations

References:

[1]	Britton, N. F., Reaction-Diffusion Equations and their Applications to Biology (1986), Academic Press: Academic Press London · Zbl 0602.92001
[2]	Busenberg, S.; Iannelli, M.; Thieme, H., Global behaviour of an age-structured S-I-S epidemic model, SIAM J. Math. Anal., 22, 1065 (1991) · Zbl 0741.92015
[3]	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 (1990) · Zbl 0726.92018
[4]	Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), Wiley: Wiley Chichester · Zbl 0997.92505
[5]	Hadeler, K. P.; van den Driessche, P., Backward bifurcation in epidemic control, Math. Biosci., 146, 15 (1997) · Zbl 0904.92031
[6]	Heijmans, H. J.A. M., The dynamical behaviour of the age-size-distribution of a cell population, (Metz, J. A.J.; Diekmann, O., The Dynamics of Physiologically Structured Populations. The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68 (1986), Springer: Springer Berlin), 185 · Zbl 0568.92015
[7]	Huang, W.; Cooke, K. L.; Castillo-Chavez, C., Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52, 3, 835 (1992) · Zbl 0769.92023
[8]	Iannelli, M., Mathematical Theory of Age-Structured Population Dynamics (1995), Giardini Editori e Stampatori: Giardini Editori e Stampatori Pisa
[9]	Inaba, H., Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28, 411 (1990) · Zbl 0742.92019
[10]	Inaba, H., Calculating \(R_0\) for HIV infection via pair formation, (Arino, O.; Axelrod, D.; Kimmel, M., Advances in Mathematical Population Dynamics - Molecules, Cells and Man (1997), World Scientific: World Scientific Singapore), 355 · Zbl 0930.92023
[11]	Inaba, H., Persistent age distributions for an age-structured two-sex population model, Math. Popul. Studies, 7, 4, 365 (2000) · Zbl 0962.91074
[12]	Krasnoselskii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen
[13]	Kribs-Zaleta, C. M.; Velasco-Hernández, J. X., A simple vaccination model with multiple endemic states, Math. Biosci., 164, 183 (2000) · Zbl 0954.92023
[14]	Kribs-Zaleta, C. M.; Martcheva, M., Vaccination strategies and backward bifurcation in an age-since-infection structured model, Math. Biosci., 177/178, 317 (2002) · Zbl 0998.92033
[15]	Marek, I., Frobenius theory of positive operators: comparison theorems and applications, SIAM J. Appl. Math., 19, 607 (1970) · Zbl 0219.47022
[16]	Martcheva, M.; Thieme, H. R., Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46, 385 (2003) · Zbl 1097.92046
[17]	Metz, J. A.J.; Diekmann, O., The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, vol. 68 (1986), Springer: Springer Berlin · Zbl 0614.92014
[18]	Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer: Springer Berlin · Zbl 0516.47023
[19]	Sawashima, I., On spectral properties of some positive operators, Nat. Sci. Rep. Ochanomizu Univ., 15, 53 (1964) · Zbl 0138.07801
[20]	(Temme, N. M., Nonlinear Analysis, vol. 2, MC Syllabus 26.2 (1978), Mathematisch Centrum: Mathematisch Centrum Amsterdam)
[21]	Thieme, H. R., Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Integr. Eqs., 3, 6, 1035 (1990) · Zbl 0734.34059
[22]	Thieme, H. R.; Castillo-Chavez, C., How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53, 5, 1447 (1993) · Zbl 0811.92021
[23]	Thieme, H. R., Quasi-compact semigroups via bounded perturbation, (Arino, O.; Axelrod, D.; Kimmel, M., Advances in Mathematical Population Dynamics - Molecules, Cells and Man (1997), World Scientific: World Scientific Singapore), 691 · Zbl 0945.47033
[24]	van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29 (2002) · Zbl 1015.92036
[25]	Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics (1985), Marcel Dekker: Marcel Dekker New York/Basel · Zbl 0555.92014
[26]	Yosida, K., Functional Analysis (1980), Springer: Springer Berlin · Zbl 0217.16001