Numerical Solutions of Vaccine Evolution

Pars={μ->0.00000,σ -> .000027,
(Subscript[k, X])-> .000001,(Subscript[k, Y])-> .000001,(Subscript[k, Z])-> .03,
(Subscript[d, Z])->1,
(Subscript[ϕ, X])->1000,(Subscript[ϕ, Y])->1000,(Subscript[ϕ, R])->1,
a->0,b->0,c->0.0,d->1,r -> 3,s-> 3};

(*Cp[t]=(  1- (((Subscript[H, u])[t]+  (Subscript[H, vs])[t] + (Subscript[H, vr])[t] + (Subscript[H, vms])[t] + (Subscript[H, vmr])[t] + (Subscript[H, m])[t])/ X)   );*)

Sols=NDSolve[

{
R'[t]==(
(-(r V[t] R[t]/((Subscript[ϕ, R]) +R[t])) 
- (r  (1+c)  W[t] R[t]/((Subscript[ϕ, R])+R[t])) 
+a -b R[t]
)/.Pars), 

V'[t]==(
((r V[t] R[t]/((Subscript[ϕ, R]) +R[t])) -d V[t] - μ V[t] 
-(Subscript[k, Z]) Z[t] V[t] -(Subscript[k, Y]) Y[t] V[t] -(Subscript[k, X]) X[t] V[t]
)/.Pars),

W'[t]==(
((r (1 + c) W[t] R[t]/((Subscript[ϕ, R]) +R[t])) -d W[t] + μ V[t] 
-(Subscript[k, Z]) Z[t] W[t] -(Subscript[k, Y]) Y[t] W[t] 
)/.Pars),

X'[t]==(
(( s X[t] V[t]/((Subscript[ϕ, X]) +V[t])) 
)/.Pars),

Y'[t]==(
( s Y[t] (V[t] + W[t])/((Subscript[ϕ, Y]) +V[t] + W[t]) 
)/.Pars),

Z'[t]==(
( σ (V[t] + W[t])(100 - Z[t]) - (Subscript[d, Z]) Z[t]) 
/.Pars),


V[0]==1.0,W[0]==0.0, X[0]==1,Y[0]==1,Z[0]==0, R[0]== 100000000000000000000
},

{V[t],W[t],X[t],Y[t],Z[t],R[t]},{t,0,20}   ];

data0 = Table[{t,X[t],Y[t]}/.Sols,{t,20,20,1}]
Export["Dropbox/Antia-Bull/Vaccines/out",data0]

LogPlot[{Evaluate [V[t]/. Sols],Evaluate [W[t]/. Sols],Evaluate [X[t]/. Sols],Evaluate [Y[t]/. Sols],Evaluate[Z[t]/.Sols]},{t,0,20},PlotRange->{1,10000000},Frame->True,PlotStyle->{Red,Orange,Blue,Dashed,Brown,Gray},PlotLegends->{"V","W","Immunity to V","Immunity to W","Innate Immunity"}]

Show[%17,Axes->False]

The plot above places all variables on a single plot. The command “PlotRange” sets the Y axis. The {t,0,900} sets the x axis.