A stochastic model for Bovine Babesiosis (BB) including ticks, and both juvenile and adult cattle is developed. This model is formulated by a system of continuous-time Markov chains (CTMCs) that is derived based on an extension of the deterministic ordinary differential equation model developed by Saad-Roy et al. (Bull Math Biol 77:514–547, 2015). The nonlinear CTMC model is approximated by a multitype branching process, giving a theoretical estimate of the probability of an outbreak of BB. Unlike the deterministic dynamics where the basic reproduction number is a sharp threshold parameter, the stochastic model indicates that there is always a positive probability of disease extinction within the cattle population. For parameter values from Colombia data, conditional probability distributions are numerically obtained for the time to disease extinction or outbreak, and are found to depend on the host type at the initiation of infection. The models with and without the inclusion of juvenile cattle are compared, and our result highlights that neglecting juvenile bovine in the models may lead to faulty predictions of critical disease statistics: particularly, it may underestimate the risk of infection. Endemic disease prevalence in adult cattle is examined for certain parameter values in the corresponding deterministic model. Notably, with long-lasting immunity, increased tick to juvenile infectivity decreases the proportion of infectious adults.

中文翻译：

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牛巴贝虫病与幼年和成年牛的随机模型

开发了牛巴贝斯虫病 (BB) 包括蜱以及幼牛和成年牛的随机模型。该模型由连续时间马尔可夫链 (CTMC) 系统制定，该系统基于 Saad-Roy 等人开发的确定性常微分方程模型的扩展。(Bull Math Biol 77:514–547, 2015)。非线性 CTMC 模型由多类型分支过程近似，给出了 BB 爆发概率的理论估计。与基本繁殖数是一个尖锐的阈值参数的确定性动力学不同，随机模型表明牛群中疾病灭绝的概率始终为正。对于来自哥伦比亚数据的参数值，条件概率分布是从疾病灭绝或爆发时间的数值上获得的，并且发现取决于感染开始时的宿主类型。比较了包含和不包含幼牛的模型，我们的结果强调，在模型中忽略幼牛可能会导致对关键疾病统计数据的错误预测：特别是，它可能会低估感染风险。在相应的确定性模型中针对某些参数值检查成年牛的地方病流行率。值得注意的是，由于具有持久的免疫力，增加对青少年传染性的蜱虫会降低传染性成人的比例。比较了包含和不包含幼牛的模型，我们的结果强调，在模型中忽略幼牛可能会导致对关键疾病统计数据的错误预测：特别是，它可能会低估感染风险。在相应的确定性模型中针对某些参数值检查成年牛的地方病流行率。值得注意的是，由于具有持久的免疫力，增加对青少年传染性的蜱虫会降低传染性成人的比例。比较了包含和不包含幼牛的模型，我们的结果强调，在模型中忽略幼牛可能会导致对关键疾病统计数据的错误预测：特别是，它可能会低估感染风险。在相应的确定性模型中针对某些参数值检查成年牛的地方病流行率。值得注意的是，由于具有持久的免疫力，增加对青少年传染性的蜱虫会降低传染性成人的比例。