STL and SARIMA findings concurred and were accurate. Endemic PTB seems to be slowly declining and case diagnosis is likely seasonal, which can be expected to persist if past conditions continue.

We study a stochastic differential equation growth model to describe individual growth in random environments. In particular, in this paper, we discuss the estimation of the drift and the diffusion coefficients using nonparametric methods for the case of nonequidistant data for several trajectories. We illustrate the methodology by using bovine growth data. Our goal is to assess: (i) if the parametric models (with specific functional forms for the drift and the diffusion coefficients) previously used by us to describe the evolution of bovine weight were adequate choices; (ii) whether some alternative specific parameterized functional forms of these coefficients might be suggested for further parametric analysis of this data. Modelling animal growth in random environments

The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY t = β(α − Y t )dt + σdW t , where Y t = h(X t ), X t is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W t is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal's life. For simplicity, we consider two phases with growth coefficients β 1 and β 2. Results and methods are illustrated using bovine growth data.

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A class of stochastic differential equations models was applied to describe the evolution of the weight of mertolengo cattle. We have determined the optimal mean profit obtained by selling an animal at the cattle market, using two approaches. One consists in determining the optimal selling age (independently of the weight) and the other consists in selling the animal when a fixed optimal weight is achieved for the first time (independently of the age). The profit probability distribution can be computed for such optimal age/weight. For typical market values, we observed that the second approach achieves a higher optimal mean profit compared with the first one, and, in most cases, even provides a lower standard deviation.

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We apply a class of stochastic differential equations to model the growth of individual animals in randomly fluctuating environments using real weight data of males of the Mertolengo cattle breed. The use of these more realistic models can help farmers to optimize their profit. To this end we obtain the probability distribution, the first two moments and others quantities of interest of the profit obtained by raising and selling an animal under the more general, and more realistic, situation where the raising costs and the price per kg paid to the farmers depends on the animal's age and weight category. We also present sensitivity results on how the expected profit and the optimal selling age vary with small changes on the estimates of the model parameters. We conclude that farmers are selling the animals a little earlier than the optimal selling age, which results in a lower profit per animal. The sensitivity analysis of the parameters shows that small changes on the parameters result in very small effects on the optimal profit and negligible effects on the optimal selling age.

Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails.