Many chemical and biological systems involve reacting species with vastly different numbers of molecules/agents. Hybrid simulations model such phenomena by combining discrete (e.g., agent-based) and continuous (e.g., partial differential equation- or PDE-based) descriptors of the dynamics of reactants with small and large numbers of molecules/agents, respectively. We present a stochastic hybrid algorithm to model a stage of the immune response to inflammation, during which leukocytes reach a pathogen via chemotaxis. While large numbers of chemoattractant molecules justify the use of a PDE-based model to describe the spatiotemporal evolution of its concentration, relatively small numbers of leukocytes and bacteria involved in the process undermine the veracity of their continuum treatment by masking the effects of stochasticity and have to be treated discretely. Motility and interactions between leukocytes and bacteria are modeled via random walk and a stochastic simulation algorithm, respectively. Since the latter assumes the reacting species to be well mixed, the discrete component of our hybrid algorithm deploys stochastic operator splitting, in which the sequence of the diffusion and reaction operations is determined autonomously during each simulation step. We conduct a series of numerical experiments to ascertain the accuracy and computational efficiency of our hybrid simulations and, then, to demonstrate the importance of randomness for predicting leukocyte migration and fate during the immune response to inflammation.

许多化学和生物系统都涉及使物种与数量极大不同的分子/试剂发生反应。混合仿真通过分别结合具有少量和大量分子/试剂的反应物动力学的离散（例如基于试剂）和连续（例如基于偏微分方程或基于PDE）的描述符对这种现象进行建模。我们提出了一种随机混合算法来模拟炎症反应的免疫阶段，在此期间白细胞通过趋化作用到达病原体。尽管大量趋化因子分子证明使用基于PDE的模型来描述其浓度的时空演变是合理的，参与该过程的白细胞和细菌数量相对较少，掩盖了随机性的影响，破坏了其连续介质处理的准确性，因此必须进行单独处理。分别通过随机行走和随机模拟算法对白细胞和细菌之间的运动性和相互作用进行建模。由于后者假定反应物种充分混合，因此我们的混合算法的离散组件采用随机算子分裂，其中扩散和反应操作的顺序在每个模拟步骤中自动确定。我们进行了一系列数值实验，以确定混合仿真的准确性和计算效率，然后，