Individual-based modelling and control of bovine brucellosis

Abstract

We present a theoretical approach to control bovine brucellosis. We have used individual-based modelling, which is a network-type alternative to compartmental models. Our model thus considers heterogeneous populations, and spatial aspects such as migration among herds and control actions described as pulse interventions are also easily implemented. We show that individual-based modelling reproduces the mean field behaviour of an equivalent compartmental model. Details of this process, as well as flowcharts, are provided to facilitate the reproduction of the presented results. We further investigate three numerical examples using real parameters of herds in the São Paulo state of Brazil, in scenarios which explore eradication, continuous and pulsed vaccination and meta-population effects. The obtained results are in good agreement with the expected behaviour of this disease, which ultimately showcases the effectiveness of our theory.

Keywords: bovine brucellosis; individual-based model; mathematical epidemiology.

Figure 1.

Dynamics of brucellosis in cattle populations described in block diagram. The authors in [13] have used six compartments to describe the dynamics of bovine brucellosis in the population. These compartments are: females susceptible (S), vaccinated females (V), latent carriers primiparous (L₁), infectious primiparous females (I₁), latent carriers multiparous (L₂) and female infectious multiparous (I₂).

Figure 2.

Flowchart of the IBM for bovine brucellosis. The initial population is determined at random. Each individual is evaluated according to formulation described in §(b). The algorithm is ended when the time t reaches the maximum value t_f. The transitions indicate a change into different category.

Figure 3.

Spatial mobility of individuals. Interaction between animals of different areas. This flowchart shows that after t=K₁ all animals were transferred to the corral. After t=K₂ the animals return to their pastures. See more details in §(b).

Figure 4.

Simulation of IBM for bovine brucellosis. One run of the IBM is shown. It was considered a vaccination rate of p=95%. The initial condition was S₀=0.8N, V ₀=0, L₁₀=0.05N, I₁₀=0.05N, L₂₀=0.05N, I₂₀=0.05N. Other parameters were defined as: μ=8, $\gamma =\frac{5}{3}$, β=7.98×10⁻²; $\delta =\frac{1}{6}$. At t=0, C₂ has been obtained with 0.25μ, C₄ with 0.25γ and C₆ 0.25δ. It is possible to see that with this vaccination rate the number of infected animals (infected 1) approaches to zero. The number of infected female multiparous (infected 2) approaches zero in a longer time due to life expectancy.

Figure 5.

Simulation of IBM for bovine brucellosis. The mean value of 40 runs and standard deviation error are shown. It was considered a vaccination rate of p=5%. Initial conditions and other parameters are the same as described in figure 4. This small rate of vaccination is not sufficient to eradicate the disease. The number of infected animals (infected 1) fluctuates around 10.

Figure 6.

Simulation of spatial IBM for bovine brucellosis. It has been considered a continuous rate of vaccination p=75%. The mean value of 40 runs and standard deviation error are shown. The other parameters were: μ=8, $\gamma =\frac{5}{3}$ years, β=7.98×10⁻², $\delta =\frac{1}{6}$ years. The initial condition was S₀=0.8N, V ₀=0, L₁₀=0.05N, I₁₀=0.05N, L₂₀= 0.05N, I₂₀=0.05N, and each pasture started with 400 animals and empty barn. The infection rate in the corral was arbitrarily set to β_barn=10β at t=6 years. This explains the peak of the number of infected.

Figure 7.

Number of vaccinated females using two different control approaches: continuous (a) and pulsed (b). The mean value of 40 runs and standard deviation error are shown. The pulses occur in intervals of 0.5 year. Both strategies are efficient to eradicate the diseases, but pulsed control presents a lower financial cost.