A comparison of stochastic and deterministic dynamics of tuberculosis model

Tuberculosis is a deadly infectious disease leading to a major health concern in Southern India while there are major obstacles to controlling its spread. Here, we present a mathematical model with five compartments. We categorize the infected compartment into two subcategories: latent TB-infected individuals and active TB-infected individuals. By introducing white noise in a deterministic system, we formulate a stochastic system. We investigate the model in deterministic and stochastic framework, followed by data calibration using TB infection data from Kanyakumari District in South India from 2019 to 2023. In the deterministic model, we derive the disease-free and endemic equilibrium points, compute the basic reproduction number, and examine their stability. Moreover, we perform sensitivity analysis to evaluate how variations in model parameters affect TB prevalence. In addition, we study the uniqueness of solutions of stochastic model. After that we derive the conditions for disease extinction and stochastic permanence, and execute extensive simulations to capture the variability and randomness in TB transmission dynamics. This study signifies that stochastic dynamics is richer than deterministic one in the way of mitigation TB transmission.结核病（Tuberculosis）是一种致命的传染病，在印度南部引发重大健康问题，同时其传播控制面临诸多障碍。在此，我们提出一个包含五个仓室的数学模型。我们将感染仓室分为两个子类别：潜伏性结核感染者和活动性结核感染者。通过在确定性系统中引入白噪声，我们构建了一个随机系统。我们在确定性和随机框架下对该模型进行研究，随后利用2019-2023年印度南部Kanyakumari地区的结核感染数据进行数据校准。在确定性模型中，我们推导了无病平衡点和地方病平衡点，计算了基本再生数（basic reproduction number），并分析了它们的稳定性。此外，我们进行敏感性分析，以评估模型参数的变化如何影响结核病患病率。另外，我们研究了随机模型解的唯一性。之后，我们推导了疾病灭绝和随机持久性的条件，并进行了大量模拟以捕捉结核病传播动力学中的变异性和随机性。本研究表明，在缓解结核病传播方面，随机动力学比确定性动力学更为丰富。