# illegal use of ODEINT

given the following system: $$\frac{dP}{dt} = \alpha P(1-\frac{P}{K}) - \beta P I$$ $$\frac{dI}{dt} = \beta P I - \rho I$$

how do I solve the system numerically. as when I attempt to solve this is the error I get.

/home/gideon/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:19: RuntimeWarning: >divide by zero encountered in double_scalars

/home/gideon/anaconda3/lib/python3.7/site-packages/scipy/integrate/odepack.py:247: >ODEintWarning: Illegal input detected (internal error). Run with full_output = 1 to get >quantitative information. warnings.warn(warning_msg, ODEintWarning)

here is my code

'''
x[0]  : P(t). population not infected
x[1]  : I(t). population Infected
k     : population carrying capacity
alpha : growth rate
rho   : death rate
beta  : infection rate
t     : time
'''

t = np.linspace(0,20,50)
fig,ax = plt.subplots(1,figsize = (10,4))
plt.suptitle('Infection Model')

def update_plot(k,alpha,rho,beta):
ax.clear()

xprime = lambda x,k,alpha,rho,beta,t: np.array([alpha*x[0]*(1-(x[0]/k))-beta*x[0]*x[1],
beta*x[0]*x[1]-rho*x[1]])
x0 = np.array([(4/5)*k,k/5])
x = odeint(xprime,x0,t,args=(k,alpha,rho,beta,))
y1 = x[:,0]
y2 = x[:,1]
ax.plot(t,y1,'b',label = 'P(t)')
ax.plot(t,y2,'g--',label = 'I(t)')
plt.legend(loc = 'best')
plt.show()

k = widgets.FloatSlider(min=1,max = 10 , value =1, description = 'K :')
alpha = widgets.FloatSlider(min=1,max = 10 , value =1, description = r'$$\alpha$$ :')
rho = widgets.FloatSlider(min=1,max = 10 , value =1, description = r'$$\rho$$ :')
beta = widgets.FloatSlider(min=1,max = 10 , value =1, description = r'$$\beta$$ :')

widgets.interactive(update_plot,k=k,alpha=alpha,rho=rho,beta=beta)


I'm trying to plot the system for varying parameters but the graphs I am getting are incorrect. i think it is due to the nan values I'm getting in the solution array $$x$$

xprime = lambda x,k,alpha,rho,beta,t:

xprime = lambda x,t,k,alpha,rho,beta: