Bulletin of Mathematical Biology

, Volume 71, Issue 1, pp 75–83

# Global Properties of SIR and SEIR Epidemic Models with Multiple Parallel Infectious Stages

• Andrei Korobeinikov
Original Article

## Abstract

We consider global properties of compartment SIR and SEIR models of infectious diseases, where there are several parallel infective stages. For instance, such a situation may arise if a fraction of the infected are detected and treated, while the rest of the infected remains undetected and untreated. We assume that the horizontal transmission is governed by the standard bilinear incidence rate. The direct Lyapunov method enables us to prove that the considered models are globally stable: There is always a globally asymptotically stable equilibrium state. Depending on the value of the basic reproduction number R 0, this state can be either endemic (R 0>1), or infection-free (R 0≤1).

## Keywords

Infectious disease Mass-action Endemic equilibrium state Global stability Direct Lyapunov method Lyapunov function

## References

1. Barbashin, E.A., 1970. Introduction to the Theory of Stability. Wolters-Noordhoff, Groningen.
2. Georgescu, P., Hsieh, Y.-H., 2006. Global stability for a virus dynamics model with nonlinaer incidence of infection and removal. SIAM J. Appl. Math. 67(2), 337–353.
3. Guo, H., Li, M.Y., 2006. Global dynamics of a staged progression model for infectious diseases. Math. Biosci. Eng. 3(3), 513–525.
4. Korobeinikov, A., 2004a. Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. J. IMA 21(2), 75–83.
5. Korobeinikov, A., 2004b. Global properties of basic virus dynamics models. Bull. Math. Biol. 66(4), 879–883.
6. Korobeinikov, A., 2006. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615–626.
7. Korobeinikov, A., 2007. Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871–1886.
8. Korobeinikov, A., 2008. Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate. Math. Med. Biol. J. IMA, to appear. Google Scholar
9. Korobeinikov, A., Maini, P.K., 2004. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1(1), 57–60.
10. Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. J. IMA 22, 113–128.
11. Korobeinikov, A., Petrovskii, S.V., 2008. Toward a general theory of ecosystem stability: plankton-nutrient interaction as a paradigm. In: Hosking, R.J., Venturino, E. (Eds.), Aspects of Mathematical Modelling, pp. 27–40. Birkhäuser, Basel.
12. Korobeinikov, A., Wake, G.C., 2002. Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models. Appl. Math. Lett. 15(8), 955–961.
13. La Salle, J., Lefschetz, S., 1961. Stability by Liapunov’s Direct Method. Academic, New York.
14. Okuonghae, D., Korobeinikov, A., 2006. Dynamics of tuberculosis: the effect of direct observation therapy strategy (DOTS) in Nageria. Math. Model. Nat. Phenom. Epidemiol. 2(1), 99–111.
15. van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.