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Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE). (English) Zbl 1135.92030

Summary: One goal of this paper is to give an algorithm for computing a threshold condition for epidemiological systems arising from compartmental deterministic modeling. We calculate a threshold condition \(\mathcal T_0\) of the parameters of the system such that if \(\mathcal T_0<1\) the disease-free equilibrium (DFE) is locally asymptotically stable (LAS), and if \(\mathcal T_0>1\), the DFE is unstable. The second objective, by adding some reasonable assumptions, is to give, depending on the model, necessary and sufficient conditions for global asymptotic stability (GAS) of the DFE. In many cases, we can prove that a necessary and sufficient condition for the global asymptotic stability of the DFE is \(\mathcal R _0\leqslant 1\), where \(\mathcal R _0\) is the basic reproduction number [O. Diekmann and J. A. P. Heesterbeek, Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. New York: Wiley (1999; Zbl 0997.92505)]. To illustrate our results, we apply our techniques to examples taken from the literature. In these examples we improve the results already obtained for the GAS of the DFE. We show that our algorithm is relevant for high dimensional epidemiological models.

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
93A30 Mathematical modelling of systems (MSC2010)
34D05 Asymptotic properties of solutions to ordinary differential equations

Citations:

Zbl 0997.92505
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[1] Diekmann, O.; Heesterbeek, J. A., Mathematical Epidemiology of Infectious Diseases: Model Building. Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), Wiley: Wiley New York · Zbl 0997.92505
[2] Jacquez, J. A.; Simon, C. P.; Koopman, J. S., Core groups and the \(R_0\) s for sub groups in heterogeneous SIS models, (Mollison, D., Epidemic Models: Their Structure and Relation to Data (1996), Cambridge University Press: Cambridge University Press Cambridge, UK), 279-301 · Zbl 0839.92021
[3] Simon, C. P.; Jacquez, J. A., Reproduction number and the stability of equilibria of SI models for heterogeneous populations, SIAM J. Appl. Math., 52, 541-576 (1992) · Zbl 0765.92019
[4] Simon, Carl P.; Jacquez, John A.; Koopman, James S., A Lyapunov function approach to computing \(R_0\), (Isham, V.; Medley, G., Models for Infectious Human Diseases: Their Structure and Relation to Data (1996), Cambridge University Press), 311-314 · Zbl 0851.92018
[5] Hethcote, H. V., The mathematics of infectious disease, SIAM Review, 42, 4, 599-653 (2000) · Zbl 0993.92033
[6] Macdonald, G., The Epidemiology and Control of Malaria (1957), Oxford University Press
[7] Dietz, K., Models for parasitic disease control, Bull. Int. Statist. Inst., 46, 531-544 (1975)
[8] Dietz, K., Transmission and control of arbovirus diseases, (Cooke, K. L., Epidemiology (1975), SIAM: SIAM Philadelphia), 104-121 · Zbl 0322.92023
[9] Becker, N. G., The use of mathematical models in deterministic vaccination policies, Bull. Int. Statist. Inst., 46, 478-490 (1975)
[10] Hale, J. K., Asymptotic behavior of dissipative systems, Am. Math. Soc. Providence (1988) · Zbl 0642.58013
[11] Hethcote, H. V., Mathematical models for the spread of infectious diseases, (Ludwig, D.; Cooke, K. L., Epidemiology: Proceeding of SIMS Conference (1975), SIAM: SIAM Philadelphia), 122-131
[12] Heesterbeek, J. A., A brief history of \(R_0\) and a recipe for its calculation, Acta Biotheorica, 50, 189-204 (2002), (1969) 448-457
[13] Diekmann, O.; Heesterbeek, J. A.; 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
[14] Mosak, J. L., General equilibrium theory in international trade (1944), The principia press: The principia press Bloomington, Indiana · Zbl 0063.04121
[15] Arrow, J. K.; Hurwicz, L., On the stability of the competitive equilibrium, I, Econometrica, 26, 522-552 (1958) · Zbl 0084.15604
[16] Arrow, J. K.; Hurwicz, L., On the stability of the competitive equilibrium, II, Econometrica, 27, 82-109 (1959) · Zbl 0095.34301
[17] Jacquez, J. A.; Simon, C. P., Qualitative theory of compartmental systems, SIAM Rev., 35, 1, 43-79 (1993) · Zbl 0776.92001
[18] LaSalle, J. P., The stability of dynamical systems (1976), SIAM: SIAM Philadelphia · Zbl 0364.93002
[19] Luenberger, D. G., Introduction to Dynamic Systems (1979), John Wiley: John Wiley New York · Zbl 0458.93001
[20] Takayama, A., Mathematical Economics, Dryden Press Hinsdale, III (1985), Cambridge University Press: Cambridge University Press Cambridge
[21] Smith, H. L., Monotone Dynamics Systems. Monotone Dynamics Systems, An Introduction to the Theory of Competitive and Cooperative Systems (1995), AMS
[22] H.R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton, NJ: Princeton University Press, 2003.; H.R. Thieme, Mathematics in population biology, Princeton Series in Theoretical and Computational Biology, Princeton, NJ: Princeton University Press, 2003. · Zbl 1054.92042
[23] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1994), SIAM Philadelphia · Zbl 0815.15016
[24] Varga, R. S., Factorization and normalized iterative methods, (Langer, R. E., Boundary Problems in Differential Equations (1960), University of Press), 121-142
[25] Seibert, P.; Florio, J. S., On the reduction to a subspace of stability properties of systems in metric spaces, Ann. Matematica. Pura Applic., 4169, 291-320 (1995) · Zbl 0853.54038
[26] Van der Driessche, P.; Wattmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180, 29-48 (2002) · Zbl 1015.92036
[27] Castillo-Chavez, C.; Feng, Z.; Huang, W., On the computation of \(R_0\) and its role on global stability, (Castillo-Chavez, C.; Blower, S.; Van der Driessche, P.; Kirschner, D.; Yakubu, A. A., Mathematical Approaches for Emerging and Reemerging Infectious Diseases, An Introduction (1998), Springer), 229-250 · Zbl 1021.92032
[28] Van Loan, C., A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl., 61, 233-251 (1984) · Zbl 0565.65018
[29] R.S. Varga, Matrix Iterative Analysis Prentice Hall Series in Automatic computation (1962).; R.S. Varga, Matrix Iterative Analysis Prentice Hall Series in Automatic computation (1962).
[30] Nold, A., Heterogeneity in disease-transmission modeling, Math. Biosci., 52, 227-240 (1980) · Zbl 0454.92020
[31] Vidyasagar, M., Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Trans. Automat. Control, 4, 25, 773-779 (1980) · Zbl 0478.93044
[32] Seibert, P.; Suarez, R., Global stabilization of nonlinear cascade systems, Syst. Contr. Lett., 14, 347-352 (1990) · Zbl 0699.93073
[33] Seibert, P., On relative stability to a set and the whole space, Kacestv. Met. Teor. nelin. Koleb. 2. Trudy 5 mezdunarod. Konf., 448-457 (1969)
[34] Seibert, P., Relative stability and stability of closed set, Semin. Differ. Equat. Dynam. Syst. II, Maryland 1969, Lect. Notes Math., 144, 185-189 (1970)
[35] LaSalle, J. P., Stability theory for ordinary differential equations, J. Differ. Equations, 4, 57-65 (1968) · Zbl 0159.12002
[36] Bhatia, N. P.; Szego, G. P., Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, vol. 35 (1967), Springer Verlag · Zbl 0155.42201
[37] Cherry, B. R.; Reeves, M. J.; Smith, G., Evaluation of bovine diarrhea virus control using a mathematical model of infection dynamics, Prev. Vet. Med., 33, 91-108 (1998)
[38] Feng, Z.; Castillo-Chavez, C.; Capurro, A. F., A model for tuberculosis with exogenous reinfection, Theor. Pop. Biol., 57, 235-247 (2000) · Zbl 0972.92016
[39] Perelson, A. S.; Kirschner, D. E.; De Boer, R., Dynamics of HIV infection of CD \(4^+\) T cells, Math. Biosci., 114, 81-125 (1993) · Zbl 0796.92016
[40] Castillo-Chavez, C.; Feng, Z., To treat or not to treat: the case of tuberculosis, J. Math. Biol., 35, 629-656 (1997) · Zbl 0895.92024
[41] Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160, 191-213 (1999) · Zbl 0974.92029
[42] Muldowney, J. S.; Li, M. Y.; Van Den Driessche, P., Global stability of the SEIR models in epidemiology, Can. Appl. Math. Quart., 7, 155-164 (1999)
[43] Li, M. Y.; Fan, M.; Wang, K., Global stability of an SEIS model with recruitment and varying total population size, Math. Biosci., 170, 199-208 (2001) · Zbl 1005.92030
[44] Li, M. Y.; Smith, H. L.; Wang, L., Global dynamic of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62, 58-69 (2001) · Zbl 0991.92029
[45] Li, M. Y.; Muldowney, J. S., Global stability for the SEIR model in epidemiology, Math. Biosci., 125, 155-164 (1995) · Zbl 0821.92022
[46] Liu, W. M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359-380 (1987) · Zbl 0621.92014
[47] Ngwa, G. A.; Shu, W. S., A mathematical model for endemic malaria with variable human and mosquito populations, Math. Comp. Model., 32, 747-763 (2000) · Zbl 0998.92035
[48] Hethcote, H. V.; Zhien, M.; Shengbing, L., Effects of quarantine in six endemic models for infectious diseases, Math. Biosci., 180, 141-160 (2002) · Zbl 1019.92030
[49] Esteva, L.; Varga, R. S., A model for dengue disease with variable human population, J. Math. Biol., 38, 220-240 (1999) · Zbl 0981.92016
[50] Esteva, L.; Varga, R. S., Analysis of dengue transmission model, Math. Biosci., 150, 131-151 (1998) · Zbl 0930.92020
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