A Deterministic Model for Q Fever Transmission Dynamics within Dairy Cattle Herds: Using Sensitivity Analysis and Optimal Controls

Abstract

This paper presents a differential equation model which describes a possible transmission route for Q fever dynamics in cattle herds. The model seeks to ascertain epidemiological and theoretical inferences in understanding how to avert an outbreak of Q fever in dairy cattle herds (livestock). To prove the stability of the model's equilibria, we use a matrix-theoretic method and a Lyapunov function which establishes the local and global asymptotic behaviour of the model. We introduce time-dependent vaccination, environmental hygiene, and culling and then solve for optimal strategies. The optimal control strategies are necessary management practices that may increase animal health in a Coxiella burnetii-induced environment and may also reduce the transmission of the disease from livestock into the human population. The sensitivity analysis presents the relative importance of the various generic parameters in the model. We hope that the description of the results and the optimality trajectories provides some guidelines for animal owners and veterinary officers on how to effectively minimize the bacteria and control cost before/during an outbreak.

Figure 1

A flowchart of Q fever transmission dynamics within dairy cattle herds. The light blue box = susceptible cattle (S); red box = asymptomatic cattle (A); green box = symptomatic cattle (I); white box = vaccinated cattle (V); the presence of Coxiella burnetii in the environment (B) is encircled in blue.

Figure 2

(a) The scatter diagrams for some selected parameters in ℛ_C (scatter plots of β, Λ, η, ρ, α, θ, μ, and ν in ℛ_C). (b) Perfect rank relations of the parameters in ℛ_C (Tornado plot of parameters in ℛ_C), (c) The coupling effect of both vaccination and environmental hygiene on ℛ_C (3D plot of ε and vℛ_C).

Figure 3

Optimal trajectory results with and without controls. All initial state values are assumed for illustrative purposes. (a) Control profile. (b) Susceptible cattle. (c) Vaccinated cattle. (d) Asymptomatic cattle. (e) Symptomatic cattle. (f) Bacterial load.

Figure 4

Optimal trajectory results with and without controls, with a change in initial state value data. All initial state values are assumed for illustrative purposes. (a) Control profile. (b) Susceptible cattle. (c) Vaccinated cattle. (d) Asymptomatic cattle. (e) Symptomatic cattle. (f) Bacterial load.

Figure 5

Optimal trajectory results marked by diversity in control procedures. All initial state values are assumed for illustrative purposes. (a) Without culling control profile. (b) Without culling in the objective function. (c) Without vaccinated control profile. (d) Without vaccination in the objective function. (e) Asymptomatic cattle (without vaccination in the objective function). (f) Bacterial load (without vaccination in the objective function).

Figure 6

Optimal trajectory results marked by diversity in control procedures. All initial state values are assumed for illustrative purposes. (a) Control values (no environmental hygiene). (b) Bacterial load (no environmental hygiene). (c) Asymptomatic cattle (no environmental hygiene). (d) Symptomatic cattle (culling only).