The Failure of 𝑅₀

Figure 4

The changing face of 𝑅₀. All expressions for 𝑅₀ were generated from the same model, using standard techniques. As the death rate varies, all have the property that if 𝑅₀>1 then an outbreak will occur, whereas if 𝑅₀<1, then the disease will be eradicated. However, aside from the common threshold at 𝑅₀=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 𝐻_𝑆=45, 𝑀_𝑆=30, 𝛽_𝑀𝐻=0.05, 𝛽_𝐻𝑀=0.03, 𝜇_𝐻=0.15 and 𝜇_𝑀=0.6.