Table 1.

PDEs generated from cellularization of the influenza ODE model for virus $\stackrel{\sim }{v}$ , type I IFN $\stackrel{\sim }{f}$ , chemokines $\stackrel{}{c}$ and IL-10 $\stackrel{}{l}$ according to a general form for reaction–diffusion transport with diffusion coefficient. All symbols with subscripts are parameters from the ODE model. $\eta$ and $\theta$ are the global and local scaling coefficients, respectively, according to cellularization. $\mathcal{B}\left(x,y\right)$ is a binary function equal to one when $x=y$ and zero otherwise. $\Vert \mathcal{V}\left(\sigma ,t\right)\Vert$ is the volume of a cell $\sigma =\sigma \left(x_i,t\right)$ at $x_i$ and time $t$ with type $\tau \left(\sigma ,t\right)$ (e.g., uninfected $\widehat{H}$, infected $\widehat{I}$, macrophage $\widehat{M}$). $a$, $d$, $p$ and $T^{\ast }$ are the total antibodies, dead cells, APCs and TNF in the spatial domain. $\stackrel{-}{z}$ is a mean cellular measurement of $\stackrel{}{z}$.

General form	${\partial }_t\stackrel{\sim }{z}=\left(D_Z{\partial }_i^2-q\right)\stackrel{\sim }{z}+r$
Field	Decay rate $q$	Source rate $r$
Virus $\stackrel{}{v}$	${\mu }_v+\frac{g_{\mathit{va}}a}{\eta }+\frac{\theta g_v}{a_v\stackrel{}{v}+\theta }+\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{H}\right)g_{\mathit{vh}}}{\theta \Vert \mathcal{V}\Vert }$	$\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{I}\right)g_{\mathit{vi}}{\rho }^{’}}{\Vert \mathcal{V}\Vert }$
Type I IFN $\stackrel{}{f}$	${\mu }_f+\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{I}\right)g_{\mathit{fi}}}{\theta \Vert \mathcal{V}\Vert }$	$\frac{\theta b_{\mathit{fp}}p}{\eta }+\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{I}\right)b_{\mathit{fi}}{\rho }^{’}}{\Vert \mathcal{V}\Vert }$
Chemokines $\stackrel{}{c}$	${\mu }_c$	$\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{M}\right)b_c}{\Vert \mathcal{V}\Vert \zeta }$
IL-10 $\stackrel{}{l}$	${\mu }_l$	$\frac{1}{\Vert \mathcal{V}\Vert }\left(\frac{\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{M}\right)b_l}{\zeta }+\mathcal{B}\left(\tau \left(\sigma ,t\right),\widehat{H}\right){\mu }_l{b}_{\mathit{lh}}{\rho }^{’}\right)$
Auxiliary forms	$\zeta =1+\frac{\eta \left(g_1\stackrel{-}{l}+\theta g_2\right)}{\left(a_{11}{T}^{\ast }+a_{12}d\right)\left(\stackrel{-}{l}+\theta d_2\right)}$	$\overline{z}=\frac{1}{\mathcal{V}\left(\sigma ,\mathrm{t}\right)}{\int }_{\mathcal{V}\left(\sigma ,\mathrm{t}\right)}\stackrel{\sim }{z}dV$