Exact description of SIR-Bass epidemics on 1D lattices

Figure 1.  Convergence to the (integrodifferential) ODE limitk (75) under Rescaling Assumptions 1. Snapshot at $ t=2 $ of $ [S_k(t)] $ on a segment $ k=0,\dots, \lfloor\frac{5}{ {\Delta x}}\rfloor $ so that $ x\in(0,5) $. Blue solid line represents the explicit solutionk (76) of the limiting ODEk (75); Green dotted line and orange dashed line represent the solution $ [S_k](t) $ of (73) for $ {\Delta x}=0.5 $ and $ {\Delta x}=0.1 $ respectively. Choice of parameters and initial conditions : $ p(x)=1-\frac{x}{5} $, $ q(x)= 5+x $, $ r(x)=2-\frac{2x}{5} $, $ [ S^0](x)=0.2-\frac{x}{25} $, $ [ R^0](x)=0.2+\frac{3x}{50} $

Figure 2.  Convergence towards PDE limitk (79) under Rescaling Assumptions 2. Snapshot at $ t=2 $ of $ [S_k(t)] $ on a segment $ k=0,\dots, \lfloor\frac{10}{ {\Delta x}}\rfloor $, with $ {\Delta x}>0 $, so that $ x\in(0,10) $. Blue solid line represents the solution $ S(t,x) $ of the limiting PDEk (79); Green dotted line and orange dashed line represent the solution $ [S_k](t) $ of (78) for $ {\Delta x}=0.1 $ and $ {\Delta x}=0.01 $ respectively. Choice of parameters and initial conditions : $ \tilde p(x)=0.1+\frac{0.2x}{10} $, $ \tilde q(x)=1+\frac{x}{10} $, $ \tilde r(x)=0.3+\frac{0.5x}{10} $, $ [\tilde I^0](x)=0.2+\frac{0.5x}{10} $, $ [\tilde R^0](x)=0.5-\frac{0.3x}{10} $. The parameters $ p $, $ q $, $ r $ and initial conditions $ [S^0] $, $ [I^0] $, $ [R^0] $ are then defined by the rescalingk (77)