An SIR epidemic model with partial temporary immunity modeled with delay

Journal of Mathematical Biology

Volumn 59, Issue 6, 2009, Pages 841-880

  Taylor, Michael L ^a   Carr, Thomas W ^a  

^a SOUTHERN METHODIST UNIVERSITY   (United States)

Author keywords

Delay; Epidemiology; Immunity; Oscillations; Resusceptible

Indexed keywords

ALGORITHM; ARTICLE; BASIC REPRODUCTION NUMBER; BIOLOGICAL MODEL; COMMUNICABLE DISEASE; COMPUTER SIMULATION; DISEASE TRANSMISSION; HUMAN; IMMUNITY; IMMUNOLOGY; PERIODICITY; POPULATION DYNAMICS; TIME;

ALGORITHMS; BASIC REPRODUCTION NUMBER; COMMUNICABLE DISEASES; COMPUTER SIMULATION; HUMANS; IMMUNITY; MODELS, BIOLOGICAL; PERIODICITY; POPULATION DYNAMICS; TIME FACTORS;

EID: 70350647741     PISSN: 03036812     EISSN: 14321416     Source Type: Journal    
DOI: 10.1007/s00285-009-0256-9     Document Type: Article

References (28)

1. Anderson RM, May RM (1991) (eds)Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford, UK.
2. Brauer F, Castillo-Chavez C (2001) Mathematical models in population biology and epidemiology. Springer, New York.
3. Bestehorn M, Grigorieva EV, Haken H, Kaschenko SA (2000) Order parameters for class-B lasers with a long time delayed feedback. Phys D 145: 110-129.
4. Carr TW (2003) Period locking due to delayed feedback in a laser with saturable absorber. Phys Rev E 68: 026212.
5. Carr TW, Billings L, Schwartz IB, Triandaf I (2000) Bi-instability and the global role of unstable resonant orbits in a driven laser. Phys D 247: 59-82.
6. Carr TW, Schwartz IB, Kim M-Y, Roy R (2006) Delayed-mutual coupling dynamics of lasers: scaling laws and resonances. SIAM J Appl Dyn Syst 5: 699-725.
7. Chow S-N, Diekmann O, Mallet-Paret J (1985) Stability, multiplicity and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation. Jpn J Appl Math 2: 433-469.
8. Cooke K, van Den Driessche P (1996) Analysis of an SEIRS epidemic model with two delays. J Math Biol 35: 240-260.
9. Cooke K, van Den Driessche P, Zou X (1999) Interaction of maturation delay and nonlinear birth in population and epidemic models. J Math Biol 39: 332-352.
10. Diekmann O, Montijn R (1982) Prelude to Hopf in an epidemic model: analysis of a characteristic equation associated with a nonlinear volterra integral equation. J Math Biol 14: 117-127.
11. Doedel EJ, Oldeman BE (2007) AUTO-07P: Continuation and bifurcation software for ordinary differential equations. California Institute of Technology and Concordia University.
12. Driver RD (1977) Ordinary and delay differential equations. Springer, New York.
13. Driver RD, Sasser DW, Slater ML (1973) The equation x′(t) = ax(t) + b(t - τ) with 'small' delay. Am Math Mon 80: 990-995.
14. Dykman MI, Schwartz IB, Landsman AS (2008) Disease extinction in the presence of random vaccination. Phys Rev Lett 101: 078101.
15. El'sgol'ts LE, Norkin SB (1973) Introduction to the theory and application of differential equations with deviating arguments (translated by J.L.Casti). Academic Press, New York.
16. Engelborghs K, Luzyanina T, Samaey G (2001) DDE-BIFTOOL v. 2.00 user manual: a Matlab package for bifurcation analysis of delay differential equations, Technical Report TW-330, Department of Computer Science, K.E.Leuven, Leuven, Belgium.
17. Grigorieva EV, Kahchenko SA, Loika NA, Samson AM (1992) Nonlinear dynamics in a laser with a negative delayed feedback. Phys D 59: 297.
18. Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42: 599-653.
19. Hethcote HW, Stech HW, van Den Driessche P (1981) Nonlinear oscillations in epidemic models. SIAM J Appl Math 40: 1-9.
20. Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods. Springer, New York.
21. Kim MY, Roy R, Aron JL, Carr TW, Schwartz IB (2005) Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment. Phys Rev Lett 94: 088101.
22. Luzyanina T, Roose D, Bocharov G (2005) Numerical bifurcation analysis of immunological models with time delays. J Comput Appl Math 184: 165-176.
23. Pieroux D, Erneux T (1996) Strongly pulsating lasers with delay. Phys Rev A 53: 2765-2771.
24. Pieroux D, Erneux T, Otsuka K (1994) Minimal model of a class-B lasers with delayed feedback: Cascading branching of periodic solutions and period doubling bifurcation. Phys Rev A 50: 1822-1829.
25. Pieroux D, Erneux T, Gavrielides A, Kovanis V (2000) Hopf bifurcation subject to a large delay in a laser system. SIAM J Appl Math 61: 966-982.
26. Schwartz IB, Erneux T (1994) Subharmonic hysteresis and period doubling bifurcations for a periodically driven laser. SIAM J Appl Math 54: 1083-1100.
27. Schwartz IB, Smith HL (1983) Infinite subharmonic bifurcation in an SEIR epidemic model. J Math Biol 18: 233-253.
28. Thompson S, Shampine LF (2006) A friendly Fortran DDE solver. Appl Numer Math 56: 503-516.