Modelling the Use of Insecticide-Treated Cattle to Control Tsetse and Trypanosoma brucei rhodesiense in a Multi-host Population

Abstract

We present a mathematical model for the transmission of Trypanosoma brucei rhodesiense by tsetse vectors to a multi-host population. To control tsetse and T. b. rhodesiense, a proportion, ψ, of cattle (one of the hosts considered in the model) is taken to be kept on treatment with insecticides. Analytical expressions are obtained for the basic reproduction number, R _0n in the absence, and $R_{0n}^{T}$ in the presence of insecticide-treated cattle (ITC). Stability analysis of the disease-free equilibrium was carried out for the case when there is one vertebrate host untreated with insecticide. By considering three vertebrate hosts (cattle, humans and wildlife) the sensitivity analysis was carried out on the basic reproduction number ($R_{03}^{T}$) in the absence and presence of ITC. The results show that $R_{03}^{T}$ is more sensitive to changes in the tsetse mortality. The model is then used to study the control of tsetse and T. b. rhodesiense in humans through application insecticides to cattle either over the whole-body or to restricted areas of the body known to be favoured tsetse feeding sites. Numerical results show that while both ITC strategies result in decreases in tsetse density and in the incidence of T. b. rhodesiense in humans, the restricted application technique results in improved cost-effectiveness, providing a cheap, safe, environmentally friendly and farmer based strategy for the control of vectors and T. b. rhodesiense in humans.

Appendix

1.1 A.1 Derivation of the Tsetse Recruitment Rate, Λ _V

The recruitment rate for the tsetse flies is based on newly emerged flies that are feeding for the first time (Hargrove et al. 2012; Rogers 1988; Welburn et al. 2006). These newly emerged flies have to survive death due to natural mortality and the effects of insecticides, and should not have fed for them to be susceptible to T. b. rhodesiense infection. Assuming a constant birth rate, B _V , the number of newly born flies that survive and have not fed in one unit of time (that is, between t−1 and 1) is given by

$$ \varLambda_V=B_V\int _{t-1}^te^{-\int_{s}^t(a+m_V(p(\xi)))\,d\xi}\,ds. $$

(26)

Changing variables by letting u=t−s, Eq. (26) becomes

$$ \varLambda_V=B_V\int _{0}^1e^{-\int_{t-u}^t(a+m_V(p(\xi)))d\xi}du. $$

(27)

Changing variables again by letting x=ξ−t, Eq. (27) becomes

$$ \varLambda_V=B_V\int _{0}^1e^{-\int_{-u}^0(a+m_V(p(t+x)))\,dx}\,du. $$

(28)

Equation (28) can be solved by using the the trapezium rule given by,

$$ \int_{x_1}^{x_2}f(x)\,dx= \frac{(x_2-x_1)}{2n}\Biggl[f(x_1)+f(x_2)+2\sum _{k=1}^{n-1}f\biggl(x_1+k\biggl( \frac{x_2-x_1}{n}\biggr)\biggr)\Biggr]. $$

(29)

Using Eq. (29), the exponential part of Eq. (28) becomes

$$ \frac{-u}{2n_1}\Biggl[2a+m_V\bigl(p(t-u) \bigr)+m_V\bigl(p(t)\bigr)+2\sum_{k_1=1}^{n_1-1} \biggl(a+m_V\biggl(p\biggl(t+u\biggl(\frac{k_1}{n_1}-1\biggr) \biggr)\biggr)\biggr)\Biggr]. $$

(30)

Substituting Eq. (30) into (28), we obtain

$$ \varLambda_V=B_V\int _{0}^1e^{\frac{-u}{2n_1}[2a+m_V(p(t-u))+m_V(p(t))+2\sum _{k_1=1}^{n_1-1}(a+m_V(p(t+u(\frac{k_1}{n_1}-1))))]}\,du. $$

(31)

Again, using Eq. (29), (31) becomes

$$\begin{aligned} \varLambda_V&= \frac{B_V}{2n_2}\Biggl( 1+ e^{\frac {-1}{2n_1}[2a+m_V(p(t-1))+m_V(p(t))+2\sum _{k_1=1}^{n_1-1}(a+m_V(p(t+\frac{k_1}{n_1}-1)))]} \\ &\quad {} +2\sum_{k_2=1}^{n_2-1}e^{-\frac{1}{2n_1}(\frac {k_2}{n_2})[2a+m_V(p(t-\frac{k_2}{n_2}))+m_V(p(t))+2\sum _{k_1=1}^{n_1-1}(a+m_V(p(t+\frac{k_2}{n_2}(\frac{k_1}{n_1}-1))))]}\Biggr). \end{aligned}$$

(32)

Taking n ₁=n ₂=2, Eq. (32) reduces to

$$ \varLambda_V=\frac{B_V}{4}\bigl[1+3e^{-\frac{1}{4}(4a+m_V(p(t-1))+3m_V(p(t))} \bigr]. $$

(33)

Due to the delay terms in Eq. (32), cases where n ₁=n ₂>2 reduce to a Λ _V that is complex to be solved with system (3) given in Sect. 2.

At the DFE, the proportion of cattle on insecticides is constant and given by $\pi=\frac{\psi}{\psi+d+\mu_{1}}$. The solution for the tsetse recruitment rate at the DFE can thus be obtained by integrating Eq. (28) and is given by

$$ \tilde{\varLambda}_V=\frac{B_V}{(a+\tilde{m}_V(\pi))} \bigl(1-e^{-(a+\tilde {m}_V(\pi))}\bigr). $$

(34)