R0= 1.1;  (* Infection rate  *)
R1 = 1.0;  (* Recovery rate *) 
x0=1.0;  (*Starting Point at t=0*)
P = 300000000;  (* Population size *) 
T0 = 365; (* days *) 
Remove[x]
(* analsol=DSolve[{x'[t]==R0*x[t],x[0]==x0},x,{t,0, T0}] *)
(* Plot[Evaluate[x[t]/.analsol],{t,0, 10}, AxesLabel -> {Style["t",Bold, 16] , Style["Total Cases", Bold, 16]}, TicksStyle->Directive[Black, 12],  PlotRange->All]  *) 
numsol=NDSolve[{x'[t]== R0*(1 - x[t]/P) x[t]  - R1*x[t],x[0]==x0},x, {t,0, T0}];
Plot[Evaluate[x[t]/.numsol],{t,0,T0}, AxesLabel -> {Style["t",Bold, 16] , Style["Total Cases", Bold, 16]}, TicksStyle->Directive[Black, 12],  PlotRange->All]
deriv = D[Evaluate[x[t]/.numsol], t];
Plot [deriv, {t,0,T0}, AxesLabel ->{Style["t",Bold, 16] , Style["Cases per day", Bold, 16]}, TicksStyle->Directive[Black, 12], PlotRange->All]