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Review
. 2011:2011:527610.
doi: 10.1155/2011/527610. Epub 2011 Aug 16.

The failure of R0

Jing Li  1 , Daniel Blakeley, Robert J Smith
Affiliations

Affiliation

  • 1 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA.
Review

The failure of R0

Jing Li et al. Comput Math Methods Med. 2011.
. 2011:2011:527610.
doi: 10.1155/2011/527610. Epub 2011 Aug 16.

Authors

Jing Li  1 , Daniel Blakeley, Robert J Smith

Affiliation

  • 1 Department of Mathematics, Pennsylvania State University, University Park, State College, PA 16802, USA.

Abstract

The basic reproductive ratio, R(0), is one of the fundamental concepts in mathematical biology. It is a threshold parameter, intended to quantify the spread of disease by estimating the average number of secondary infections in a wholly susceptible population, giving an indication of the invasion strength of an epidemic: if R(0) < 1, the disease dies out, whereas if R(0) > 1, the disease persists. R(0) has been widely used as a measure of disease strength to estimate the effectiveness of control measures and to form the backbone of disease-management policy. However, in almost every aspect that matters, R(0) is flawed. Diseases can persist with R(0) < 1, while diseases with R(0) > 1 can die out. We show that the same model of malaria gives many different values of R(0), depending on the method used, with the sole common property that they have a threshold at 1. We also survey estimated values of R(0) for a variety of diseases, and examine some of the alternatives that have been proposed. If R(0) is to be used, it must be accompanied by caveats about the method of calculation, underlying model assumptions and evidence that it is actually a threshold. Otherwise, the concept is meaningless.

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Figures

Figure 1

Figure 1

A example of a two-stage…

Figure 1

A example of a two-stage basic reproductive ratio. A single human infects R …

Figure 1
A example of a two-stage basic reproductive ratio. A single human infects R H = 2 mosquitoes, each of whom subsequently infects R M = 3 humans. Thus, a single human results in R 0 = 6 infected humans. The next-generation method instead estimates infected individuals in the subsequent generation, regardless of whether that generation is human or mosquito; R 0,N is the geometric mean of R H = 2 and R M = 3. (Reproduced with permission from Smith? [7]).
Figure 2

Figure 2

The graph-theoretical method of de…

Figure 2

The graph-theoretical method of de Camino-Beck et al. [14] applied to a vector-host…

Figure 2
The graph-theoretical method of de Camino-Beck et al. [14] applied to a vector-host model may not produce a unique R 0.
Figure 3

Figure 3

A standard model of malaria.…

Figure 3

A standard model of malaria. Humans can be susceptible or infected, with birth…

Figure 3
A standard model of malaria. Humans can be susceptible or infected, with birth rate ΛH, background death rate μ H and disease-specific death rate σ. Mosquitoes can be susceptible or infected, with birth rate ΛM and background death rate μ M.
Figure 4

Figure 4

The changing face of R …

Figure 4

The changing face of R 0 . All expressions for R …

Figure 4
The changing face of R 0. All expressions for R 0 were generated from the same model, using standard techniques. As the death rate varies, all have the property that if R 0 > 1 then an outbreak will occur, whereas if R 0 < 1, then the disease will be eradicated. However, aside from the common threshold at R 0 = 1, none of these expressions are equal to each other. The curve labelled “Jacobian” illustrates R 0,J, as given by (18). The curve labelled “Adding and subtracting 9” illustrates the value R 0,9 as given by (20). The curve labelled “Next generation” illustrates R 0,N, as given by (25). The curve labelled “Constant term of the characteristic polynomial” illustrates R 0,C, as given by (19). The curve labelled “Exponential” illustrates R 0,e, as given by (21). In this case, the values are , , β MH = 0.05, β HM = 0.03, μ H = 0.15 and μ M = 0.6.
Figure 5

Figure 5

The effects of backward bifurcations.…

Figure 5

The effects of backward bifurcations. Solid curves indicate stable equilibria, while dashed curves…

Figure 5
The effects of backward bifurcations. Solid curves indicate stable equilibria, while dashed curves indicate unstable equilibria. (a) A backward bifurcation at R 0 = 1 may result in persistence of the disease when R 0 < 1. There is a point R a < 1 such that the endemic equilibrium exists for R a < R 0 < 1 and a third, unstable, equilibrium also exists. Hence, the disease-free equilibrium is only globally stable if 0 < R 0 < R a. (b) Backward bifurcations at other points may also affect the outcome. Although the disease persists for all R 0 > 1 and is eradicated when R 0 < 1 (due to the transcritical bifurcation at R 0 = 1), there is a region R m < R 0 < R n, where three equilibria coexist. In this region, the outcome depends on the initial conditions.
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References

    1. Heesterbeek JAP, Dietz K. The concept of R 0 in epidemic theory. Statistica Neerlandica. 1996;50(1):89–110.
    1. Heffernan JM, Smith RJ, Wahl LM. Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface. 2005;2(4):281–293. - PMC - PubMed
    1. Roberts MG. The pluses and minuses of R 0 . Journal of the Royal Society Interface. 2007;4(16):949–961. - PMC - PubMed
    1. Nishiura H. Correcting the actual reproduction number: a simple method to estimate R 0 from early epidemic growth data. International Journal of Environmental Research and Public Health. 2010;7(1):291–302. - PMC - PubMed
    1. Anderson R, May R. Infectious Diseases of Humans. Oxford, UK: Oxford University Press; 1992.

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