# Understanding Neutrophil Dynamics during COVID-19 Infection
## 2 Materials and Methods  
### Mathematical Model  
To determine the mechanistic interactions that lead to increased inflammation in severe COVID-19 infections, we develop a mathematical model of immune responses which extends on a previous model of nonspecific inflammation published in [30]. Briefly, we model the interaction between active and apoptotic neutrophils \(n(t)\) and \(a(t)\), macrophages \(m(t)\), proinflammatory mediators \(c(t)\), anti-inflammatory mediators \(g(t)\), and CD8 T-cells \(e(t)\) over time \(t\), as follows. We assume that, following SARS-CoV-2 infection, epithelial tissue gets damaged according to function  
\[f(t)=H(A\pi-t)\sin^{2}(pt), \tag{1}\]  
where  
\[H(A\pi-t)= \begin{cases}1&\text{if }t<A\pi/p\\ 0&\text{if }t>A\pi/p,\end{cases}  \tag{2}\]  
as in [30]. Function Equation (1) accounts for \(A\) repeated cycles of tissue damage occurring every \(2\pi/p\) days as a result of inflammatory processes triggered by SARS-CoV-2 infection (see Figure 1A for a schematic of a single \(A=1\) tissue damage cycle and Figure 1B for a schematic of four \(A=4\) tissue damage cycles for \(p=1\)). The physical damage to tissue results in release of cytokines that further enhance an inflammatory response. While there are a number of proinflammatory cytokines and mediators produced in response to tissue damage or viral infection (IL-1, IL-6, IL-8, TNF-\(\alpha\), among others [57]) we model them through a single proinflammatory population \(c(t)\), which is produced at rate \(\alpha f(t)\) and decays at per capita rate \(\gamma_{c}\). This proinflammatory population recruits neutrophils and macrophages at the site of the inflammation. As in [30], we ignore neutrophil activation and differentiation and assume that activated neutrophils \(n(t)\) are recruited at rate \(\alpha_{c}\) and macrophages are recruited at rate \(\alpha_{m}\), respectively. Active neutrophils die and transform into apoptotic neutrophils \(a(t)\) at rate \(vg(t)\), proportional to regulatory factors \(g(t)\) (in our model, the anti-inflammatory cytokine IL-10). Both active and apoptotic neutrophils contribute to increased production of proinflammatory mediators. Here, however, we only incorporate the contribution of apoptotic neutrophils, who undergo necrosis at rate \(\gamma_{a}\), leading to damage of healthy tissue and hence further accumulation of proinflammatory factors \(c(t)\)[11; 58], as necrosis leads to release of toxic chemicals from internal granules [11; 58; 59; 60]. We model this by adding a Hill-type production term with a Hill coefficient of two, \(k_{a}\gamma_{a}a(t)^{2}/(a(t)^{2}+\beta_{a}^{2})\), into the proinflammatory population \(c(t)\), which saturates at high \(a(t)\) levels. Here, \(\beta_{a}\) is the concentration of neutrophils required for half-maximal release of proinflammatory mediators and \(k_{a}\) is the maximal level of proinflammatory population recruited after tissue damage. We assume that macrophages \(m(t)\) are recruited in response to proinflammatory mediators \(c(t)\) at rate \(\alpha_{m}\), die at per capita rate \(\gamma_{m}\), and contribute to the removal of apoptotic neutrophils through phagocytosis at rate \(\phi\). Recognition of apoptotic neutrophils by macrophages is thought to reprogram them to release anti-inflammatory signals, which recruit anti-inflammatory populations \(g(t)\) at rate \(k_{g}\). We assume that the anti-inflammatory population is present at low levels \(s_{g}/\gamma_{g}\) at the time of the infection, with \(s_{g}\) being the production rate and \(\gamma_{g}\) the per capita decay rate. Lastly, we consider that the CD8 T-cell population \(e(t)\) is present in the epithelium at base level \(s_{e}/\gamma_{e}\), where \(s_{e}\) is the production and \(\gamma_{e}\) is the per capita death rate. Following viral infection, CD8 T-cells expand to become cytotoxic T-cells at rate \(\alpha_{e}\), proportional to the antigen-presenting cell concentration (in this case macrophages, \(m(t)\)).