# linear pde – Problem of Epidemic Spatial Diffusion .. how to draw a second-order differential system graph with partial derivative

I am currently working on simulating the spread of an epidemic in a population. If we do not consider that the population moves in space, the differential system is as follows: Differential SIS system without spatial diffusion. You will find the corresponding program below. If we consider that the population (the infected persons are more specific) has anarchic mobility, the differential system will contain a partial derivative of the second order with respect to the spatial variable x (the problem is considered only in a spatial dimension), as indicated the following image: SIS differential system with spatial diffusion included. The problem I have is how to implement this second-order partial derivative in the program, and how to plot the graphs of such a differential system .. (even if there is a spatial variable, the axes remain the same: population and time) If you delete the partial derivative, it becomes a simple differential system and the program I created is as follows:

from scipy import arange
since scipy.integrate import odeint
import matplotlib.pyplot as plt
import numpy as np

N0 = 1000000 # Initial population number
I0 = 100 # Initial infected number
PopuIni = [N0,I0]

b = 1/3000000 # infection rate β
g = 1/20 # Healing rate of infected persons
d = 5/1000 # Mortality rate, not related to the epidemic (common between infected and susceptible)
n = 4/1000% of the birth rate (vertical infection of parents to offspring is not
considered here)

def EpidEvol (N, t):
S, I = N
derS = n * (S + I) + g * I - d * S - b * S * I
derI = b * S * I - d * I - g * I
return [derS,derI]

tmax = 157
t = arange (0, tmax, 0,1)
N = odeint (EpidEvol, PopuIni, t)

Thank you for your attention .. I sincerely hope to receive help from you