Reproduction Number and Asymptotic Stability for the Dynamics of a Honey Bee Colony with Continuous Age Structure

Abstract

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case, we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number $R_0$ of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for $R_0$ has potential uses in the tracking and control of an infectious disease within a bee colony.

Appendix

1.1 Test: Uniform Age Distribution

We test the validity of this bifurcation parameter by reducing Eqs. (70) and (71) to a system in which all parameters are constant with respect to age. In doing so, we find from Eq. (89) that

$$\begin{aligned} \hat{R_0}=\beta \dfrac{H_S^*}{u+d}+\beta \dfrac{uH_S^*}{(\mu +d)(u+d)}+\beta \dfrac{F_S^*}{\mu +d}. \end{aligned}$$

(90)

This can be verified by using the next-generation matrix on the infected classes of the following reduced model

$$\begin{aligned} \dfrac{\mathrm {d}{H_S}}{\mathrm {d}{t}}= & {} -uH_S+\beta (H_I+F_I)H_S \end{aligned}$$

(91)

$$\begin{aligned} \dfrac{\mathrm {d}{F_S}}{\mathrm {d}{t}}= & {} uH_S+\beta (H_I+F_I)F_S-\mu F_S \end{aligned}$$

(92)

$$\begin{aligned} \dfrac{\mathrm {d}{H_I}}{\mathrm {d}{t}}= & {} -uH_I+\beta (H_I+F_I)H_S-dH_I \end{aligned}$$

(93)

$$\begin{aligned} \dfrac{\mathrm {d}{F_I}}{\mathrm {d}{t}}= & {} uH_I+\beta (H_I+F_I)F_S-(d+\mu )F_I \end{aligned}$$

(94)

which are a reduced form of Eqs. (5) and (7). The ratio of the disease-free equilibrium values of $F_S,H_S$ will always be such that

$$\begin{aligned} \dfrac{H_S^*}{F_S^*}=\dfrac{\mu }{u}. \end{aligned}$$

(95)

This ratio is found by setting $H_I=F_I=\dfrac{\mathrm {d}{H_S}}{\mathrm {d}{t}}=\dfrac{\mathrm {d}{F_S}}{\mathrm {d}{t}}=0$ in Eqs. (91), (92), (93) and (94).

From these reduced equations, we find the matrices,

$$\begin{aligned} F&=\left[ \begin{array}{cc} \beta H^*_S&{}\beta H^*_S\\ \beta F^*_S&{}\beta F^*_S \end{array}\right] \end{aligned}$$

(96)

$$\begin{aligned} V&=\left[ \begin{array}{cc} d+u&{}0\\ -u&{}d+\mu \end{array}\right] \end{aligned}$$

(97)

which yield the next-generation matrix

$$\begin{aligned} FV^{-1}=\left[ \begin{array}{cc} \dfrac{\beta H^*_S}{u+d}+\dfrac{\beta uH^*_S}{(u+d)(\mu +d)}&{}\dfrac{\beta H^*_S}{\mu +d}\\ \dfrac{\beta F^*_S}{u+d}+\dfrac{\beta uF^*_S}{(u+d)(\mu +d)}&{}\dfrac{\beta F^*_S}{\mu +d} \end{array}\right] \end{aligned}$$

(98)

Each term in this matrix has a biological interpretation which is the expected number of infections in each class (H or F) caused by a single infected individual in each class. For example, the term

$$\begin{aligned} \dfrac{\beta H^*_S}{u+d} \end{aligned}$$

(99)

gives the expected number of susceptible hive bees that an infected hive bee will infect while it is still a hive bee. The term

$$\begin{aligned} \dfrac{\beta uH^*_S}{(u+d)(\mu +d)} \end{aligned}$$

(100)

represents the probability that an infected hive bee will be recruited to foraging duties during its life time, multiplied by the expected number of susceptible hive bees that would then become infected. The expected number of susceptible hive bees infected by a single forager is given by

$$\begin{aligned} \dfrac{\beta H_S^*}{\mu +d} \end{aligned}$$

(101)

The interpretations for the second row of matrix (98) are similar, but give the expected numbers of susceptible foragers that will become infected.

The basic reproduction number for this uniform age distribution model is then determined by the largest eigenvalue of the matrix $FV^{-1}$. Since we have the relation (95), matrix (98) is rank 1. Therefore, one of its eigenvalues is zero and the other is given by its trace. We can see that the trace of matrix (98) gives the same expression for the basic reproduction number as (90).

The three terms that appear in (89) are analogous to the three terms that appear in Eq. (90). This suggests that (89) correctly determines not only the threshold for disease persistence, but also correctly estimates the number of secondary infections subsequent to one primary infection (Heffernan et al. 2005).