Further Notes on the Basic Reproduction Number
6.6 R o and Global Stability of the Disease-Free Equilibrium
The change of local stability at the threshold Ro = 1, corresponds to a transcritical bifurcation in the solutions to (6.1). It can be shown that there is a branch of endemic equilibria emanating from the bifurcation point at Ro= 1, (x, y) = (xo, yo). For an introduction to the general theory of these bifurcations in the context of differential equations, see Wiggins [19].
For the simple example of Sect. 6.4.1, there is an equilibrium with Se= So
Ro
,
Ie= Πκ
(µ+α)(µ+κ)
1− 1 Ro
, Ee= (µ+α)Ie
κ ,
Re= αIe
µ ,
defined for Ro > 1. Since the endemic equilibria only exist for Ro > 1, the bifurcation is said to be in the forward direction, and the disease-free equilibrium is the only equilibrium of the system whenRo<1. In models for which endemic equilibria exist near the disease-free equilibrium for Ro <1 the bifurcation is called a backward bifurcation.
Castillo-Chavez et al. [6] use a comparison theorem to derive sufficient conditions for the global asymptotic stability of the disease-free equilibrium of a general disease transmission model whenRo<1. Clearly, in the case of a backward bifurcation this equilibrium can not be globally asymptotically stable whenever Ro < 1. In most models, however, one expects a second threshold for global stability. A slight change to the argument of [6] gives a sufficient condition for global stability in this case as well.
Consider the disease transmission model (6.1) written in the form
x =−Ax−fˆ(x, y), (6.20a)
y =g(x, y). (6.20b)
Theorem 2.IfAis a nonsingular M-matrix andfˆ≥0, then the disease-free equilibrium of (6.20) is globally asymptotically stable.
Proof. Integrating (6.20a) leads to x(t) =e−tAx(0)−
t
0
e−(t−s)Af(x(s), y(s))ˆ ds. (6.21) It can be shown thate−tA ≥0 wheneverA is an M-matrix. Since solutions of (6.20) remain nonnegative, it follows that
0≤x(t)≤e−tAx(0). (6.22) Finally, since e−tA→0 ast→ ∞, it follows thatx(t)→0 ast→ ∞.
For the SEIR model of Sect. 6.4.1, take A = V −F and write (6.4) as follows
x=−(V −F)x−
β(So−S)I 0
, (6.23a)
S=Π−µS−βSI, (6.23b)
R=αI−µR. (6.23c)
From the previous section, we know that (V −F) is a nonsingular M-matrix wheneverRo <1. Hence, to show that the disease-free equilibrium is glob- ally asymptotically stable for Ro <1, it is sufficient to show that S ≤ So. The total population N = S +E +I +R satisfies N = Π −µN, so that N(t) = So−(So−N(0))e−µt, with So = Π/µ. If N(0) ≤ So, then S(t)≤N(t)≤So for all time. If, on the other hand,N(0)> So, thenN(t) decays exponentially to So, and either S(t) →So, or there is some time T after which S(t)< So. Thus, from timeT onward,x(t) is bounded above, in each component, by e−(t−T)(V−F)x(T), which decays exponentially to zero.
Note that for the argument of global stability we are not concerned with the size ofx(T). In fact, ifN(0)> So,x(T) may be much larger thanx(0). In this case the exponential bound on x(t) concerns a decay following an epidemic, not an immediate elimination of the disease. In contrast, if N(0)< So, then the bound onx(t) ise−(t−T)(V−F)x(0), and no epidemic occurs.
A simple model with a backward bifurcation is the vaccination model proposed by Kribs-Zaleta and Velasco-Hern´andez [15].
S=Π−(µ+ξ)S+θSv−βSI+γI, (6.24a) Sv =ξS−(µ+θ)Sv−β(1−r)SvI, (6.24b) I=β(S+ (1−r)Sv)I−(γ+µ)I. (6.24c) As with the model of Sect. 6.4.4, vaccination reduces the force of infection by a factorr. However, in this model, susceptible individuals are vaccinated continuously at a rateξ, and the protection acquired from vaccination wanes at a rateθ. Additionally, individuals recover from infection with no immunity, regardless of their vaccination history. The model has a unique disease-free equilibrium, where So = (1−p)No and Svo = pNo with No = Π/µ and p=ξ/(µ+θ+ξ). In keeping with the conventions of the literature, we denote byRo, the basic reproduction number in the absence of vaccination, and by Rc, the control reproduction number, or the basic reproduction number in the presence of vaccination. For the model above we have
Ro= βNo
γ+µ,
Rc= β(So+ (1−r)Svo) γ+µ <Ro.
The disease-free equilibrium is locally asymptotically stable if Rc < 1 and unstable ifRc >1.
Equation (6.24c) can be written as I = (γ+µ)(Ro−1)I−β
No−S−(1−r)Sv
I. (6.25)
The matrixAin this example is simply (γ+µ)(1−Ro). As with our previous example, we can assume thatS+Sv≤No, sinceN approachesNoasymptot- ically. Hence, the second term in the above equation eventually becomes and remains negative and the disease-free equilibrium is globally asymptotically stable ifRo<1.
The disease-free equilibrium may be locally asymptotically stable, but not globally asymptotically stable ifRc<1<Ro. It is known that there may be multiple endemic equilibria for parameter values in this range; further details can be found in Kribs-Zaleta and Velasco-Hern´andez [15].
References
1. R. M. Anderson and R. M. May,Infectious Diseases of Humans, Oxford University Press, Oxford, 1991.
2. V. Andreasen, J. Lin, and S. A. Levin,The dynamics of cocirculating influenza strains conferring partial cross-immunity, J. Math. Biol., 35 (1997), pp. 825–842.
3. A. Berman and R. J. Plemmons,Nonnegative Matrices in the Mathematical Sciences, Academic, New York, 1970.
4. S. M. Blower, P. M. Small, and P. C. Hopewell, Control strategies for tuberculosis epidemics: new models for old problems, Science, 273 (1996), pp. 497–500.
5. C. Castillo-Chavez and Z. Feng,To treat or not to treat: the case of tuberculosis, J. Math. Biol., 35 (1997), pp. 629–656.
6. C. Castillo-Chavez, Z. Feng, and W. Huang,On the computation ofRo and its role on global stability, in Mathematical approaches for emerging and reemerging infec- tious diseases: models, methods and theory, C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu, eds., Springer, Berlin Heidelberg New York, 2002, pp. 229–250.
7. B. R. Cherry, M. J. Reeves, and G. Smith,Evaluation of bovine viral diarrhea virus control using a mathematical model of infection dynamics, Prev. Vet. Med., 33 (1998), pp. 91–108.
8. O. Diekmann and J. A. P. Heesterbeek,Mathematical epidemiology of infectious dis- eases, Wiley series in mathematical and computational biology, Wiley, West Sussex, England, 2000.
9. Z. Feng and J. X. Velasco-Hern´andez,Competitive exclusion in a vector-host model for the Dengue fever, J. Math. Biol., 35 (1997), pp. 523–544.
10. S. Gandon, M. Mackinnon, S. Nee, and A. Read,Imperfect vaccination: some epidemi- ological and evolutionary consequences, Proc. R. Soc. Lond. B., 270 (2003), pp. 1129–
1136.
11. J. M. Heffernan, R. J. Smith, and L. M. Wahl,Perspectives on the basic reproductive ratio, J. R. Soc. Interface, 2 (2005), pp. 281–293.
12. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), pp. 599–653.
13. M. W. Hirsch and S. Smale,Differential Equations, Dynamical Systems, and Linear Algebra, Academic, Orlando, 1974.
14. Y.-H. Hsieh and C. H. Chen, Modelling the social dynamics of a sex industry: its implications for spread of HIV/AIDS, Bull. Math. Biol., 66 (2004), pp. 143–166.
15. C. M. Kribs-Zaleta and J. X. Velasco-Hern´andez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), pp. 183–201.
16. M. Nu˜no, Z. Feng, M. Martcheva, and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), pp. 964–982.
17. M. G. Roberts and J. A. P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond., 270 (2003), pp. 1359–
1364.
18. P. van den Driessche and J. Watmough,Reproduction numbers and sub-threshold en- demic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp. 29–48.
19. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, Berlin Heidelberg New York, 1990.
20. M. J. Wonham, T. de Camino-Beck, and M. Lewis, An epidemiological model for West Nile virus: Invasion, analysis and control applications, Proc. R. Soc. Lond. B, 271 (2004), pp. 501–507.