There, ordinary differential equations (ODEs) describing spatially during homogeneous population dynamics are commonly used (see [13]�C[15] and references therein). Indeed, the model proposed here, as well as Li et al. [11], [12] linear model, belong to the class of predator-prey models in which the predator population is fixed. The linear version is just the prey equation of the Lotka-Volterra system (see e.g., [16, p.79]). Non-linear effects and the natural emergence of bistability have been widely studied in this context, see e.g., [17], [ 18, p.74]. Moreover, it has been observed that bistability can be experimentally verified by taking advantage of the hysteresis effect [19]�C[21]. Models for studying the innate immune response in general and the phagocyte-bacterium interactions in particular, are scarce.
Most studies have concentrated on various in vivo medical conditions, hence these are inherently of higher dimension, and always include an equation that describes the bacteria dynamics coupled to the phagocyte concentrations. Most commonly, the normal to hyper-response of the phagocytes to the invasion of bacteria is studied. In Kumar et al., Chow et al. and Reynolds et al. [22]�C[25], the focus is on the relation between the inflammatory response, anti-inflammation mediators and sepsis. In Herald [26], the macrophage dynamics is related to the development of chronic inflammation after eradication of a pathogen. In Pugliese and Gandolfi [27], the in-vivo pathogens and specific and non-specific immunity dynamics are shown to be potentially quite rich: bistable regions and oscillatory regimes appear as the model parameters are varied.
Imran and Smith [28] consider the influence of bacterial nutrients on the innate immune response to bacterial infection and identify locally stable disease-free region, which is used to design a successful antibiotics treatment. Finally, models concentrating on in vivo blood-tissue dynamics showed good fit to experimental data of Escherichia coli concentration in milk produced from infected cows [29]. Early modelling efforts of bacterium-phagocyte dynamics in vitro [30], [31] focused on probabilistic modelling of the number of digested particles per neutrophil. These issues were further explored experimentally, and led to the proposal of a three-compartment linear ODE model in which the dynamics of viable, phagocytosed and perforated bacteria are presented [32].
The works of Li et al. [11], [12] were the first to relate the implication of such experiments and models to the observed in-vivo critical value of neutrophils. Notably, in mathematics, as in biology, characterization of a simplified system (such as an in-vitro AV-951 experiment) serves as a building block for more complex systems. The mathematical building block of Li et al. [11] was utilized in several models of in-vivo dynamics, e.g.