On the Calculation of $R_0$ Using Submodels

Abstract

A set of epidemiological models and their basic reproductive numbers are presented. For some models, the expression of the basic reproductive number can be very complex to determine. In this work, we present some techniques to find the expressions of these basic reproductive numbers starting from submodels of the original one. Some particular models with their respective $R_0$, which can be applied to the study of some important diseases, are also presented.

Appendix

Next-Generation Matrix and Its Characteristic Polynomial

We constructed the next-generation matrix for the system (7), following the technique in [12–15]. Such matrix is given by

$$\begin{aligned} K =\left( \begin{array}{ccc} \frac{c_1N_1}{{\mathcal {V}}_1(I_{10})}&{} \quad \frac{c_2N_1}{{\mathcal {V}} _2(I_{20})}&{} \quad \frac{c_3N_1}{{\mathcal {V}}_3(I_{30})}\\ \frac{c_4N_2}{{\mathcal {V}}_1(I_{10})}&{} \quad \frac{c_5N_2}{{\mathcal {V}} _2(I_{20})}&{} \quad \frac{c_6N_2}{{\mathcal {V}}_3(I_{30})}\\ \frac{c_7}{{\mathcal {V}}_1(I_{10})}&{} \quad \frac{c_8}{{\mathcal {V}} _2(I_{20})}&{} \quad 0 \end{array} \right) , \end{aligned}$$

(19)

where $(I_{10},I_{20},I_{30})=(0,0,0)$ is the free disease equilibrium. To facilitate the writing, we use $({\mathcal {V}}_1,{\mathcal {V}}_2,{\mathcal {V}}_3)$ instead of $({\mathcal {V}}_1(I_{10}),{\mathcal {V}}_2(I_{20}),{\mathcal {V}}_3(I_{30}))$. Then the characteristic polynomial of the K is

$$\begin{aligned} P(\lambda )=-\lambda ^3+(R_1+R_3)\lambda ^2+(R_2^2+R_4^2 +R_6^2-R_1R_3)\lambda +(\bar{R}^3+\hat{R}^3-R_1R_4^2-R_3R_6^2), \end{aligned}$$

(20)

where

$$\begin{aligned} \begin{array}{c} R_1=\frac{c_1N_1}{{\mathcal {V}}_1}, \quad R_2 =\sqrt{\frac{c_2c_4N_1N_2}{{\mathcal {V}}_1{\mathcal {V}}_2}}, \quad R_3=\frac{c_5 N_2}{{\mathcal {V}}_2}, \quad R_4=\sqrt{\frac{c_6c_8N_2}{{\mathcal {V}}_2{\mathcal {V}}_3} },\\ R_6=\sqrt{\frac{c_3c_7N_1}{{\mathcal {V}}_1{\mathcal {V}}_3}}, \quad \bar{R}=\root 3 \of {\frac{c_2c_6c_7N_1N_2}{{\mathcal {V}}_1{\mathcal {V}}_2{\mathcal {V}}_3}},\quad \hat{R}=\root 3 \of {\frac{c_3c_4c_8N_1N_2}{{\mathcal {V}}_1{\mathcal {V}} _2{\mathcal {V}}_3}} \end{array} \end{aligned}$$

are the basic reproductive numbers of independent cycles of the system (7).

Observe that the coefficients in $P(\lambda )$ are completely related. The coefficient of the quadratic term is the sum of the $R_0$’s of all the 1-cycles involved. The coefficient of the linear term has the sum of the $R_0$’s of all the 2-cycles raised to the square, minus the total sum of the products of the $R_0$’s of the 1-cycles of the system. Finally, the constant term is the sum of the $R_0$’s of the 3-cycles raised to the cube, minus the products of 2-cycles raised to the square with 1-cycles.

This structure of $P(\lambda )$, becomes very useful in order to study subsystems of system (7). Clearly, in order to obtain the characteristic polynomial of any subsystem of (7), what proceeds is to take as zero the $R_0$’s of the cycles that are not present in the subsystem.

In order to prove Lemmas 1–5 and Theorems 1 and 2, it is necessary to find the largest eigenvalue of the associated next-generation matrix (19), which in turn is the largest root in norm of the characteristic polynomial $P(\lambda )$. Because the next-generation matrix is non negative, from the Perron–Frobenius theorem, the spectral radius of it is real and positive [9]. As $P(\lambda )$ is of degree three, it was possible to find the roots explicitly using Maple. In the most difficult case, two of the roots were complex and one real, the latter being $R_0$.

For example, for Lemma 1, the characteristic polynomial is

$$\begin{aligned} P(\lambda )=-\lambda ^3+(R_i^2+R_j^2)\lambda \end{aligned}$$

and the roots are $\lambda _1=0$, $\lambda _2=-\sqrt{R_i^2+R_j^2}$ and $\lambda _3=\sqrt{R_i^2+R_j^2}$. Therefore the basic reproductive number is $\lambda _3$.

Similarly, we proceed to the proofs of the others lemmas and theorems.