data=Import["/Users/troyday/Desktop/Plague.txt","Table"]

This is the mortality data and KM say it can be interp as dz/dt, which equals l y(t). If infection lasts ~2 weeks, then l = 1/2 and thus the data presented in KM can be multiplied by 2 to get y(t).

datai={{1.139`,10.523`},{2.054`,15.555`},{2.86`,17.659`},{4.052`,24.808`},{6.093`,54.377`},{7.112`,54.368`},{9.19`,130.432`},{10.053`,183.033`},{10.917`,295.846`},{11.997`,391.624`},{13.05`,445.782`},{14.017`,643.642`},{14.789`,768.804`},{15.105`,777.338`},{16.118`,705.297`},{17.107`,694.384`},{19.118`,927.901`},{18.068`,867.871`},{20.044`,805.883`},{18.917`,757.303`},{21.094`,583.136`},{22.034`,403.298`},{23.078`,348.561`},{23.898`,209.685`},{25.025`,112.988`},{26.075`,65.114`},{27.038`,49.244`},{27.324`,51.353`},{27.893`,38.703`},{28.322`,38.699`},{28.792`,27.057`},{29.225`,27.053`},{30.076`,31.351`}};

Length[datai]

data=Table[{datai[[i,1]],2 datai[[i,2]]},{i,1,Length[datai]}]

Export["Plague.tsv",data,"TSV"]

ListPlot[data,PlotRange->{0,2200},AspectRatio->1,Frame->True,FrameLabel->{"Time (weeks)","Number of infected people at time t"},LabelStyle->Directive[FontSize->14],PlotStyle->{Red,PointSize[.015]}]

Export["1.eps",%16,"EPS"]

Manipulate[tmax=30;sol[β_,μ_,sinit_,iinit_]:=NDSolve[{s'[t]==-β s[t] i[t],i'[t]==β s[t] i[t]-μ i[t],s[0]==sinit,i[0]==iinit},{s[t],i[t]},{t,0,tmax}];f[q_]:=Quiet[a/.Solve[Exp[-q a]==1-a,a][[2]]//N];osize=(Evaluate[{10000.0-s[t]}/.sol[(q/10000.0),1,10000,1]]/.t->tmax)[[1]][[1]];Plot[Evaluate[{s[t],i[t]}/.sol[(q/10000.0),1,10000,1]],{t,0,tmax},PlotRange->{{0,tmax},{0,10000}},AspectRatio->1,Frame->True,FrameLabel->{"Time (weeks)","Number of people"},LabelStyle->Directive[FontSize->14],PlotStyle->{Blue,Red},Epilog->{Arrow[{{29,8000},{29,(1-f[q]) 10000}}],Arrow[{{29,8000},{29,10000}}],Text[Style["A = ",Black,FontSize->16],{22,6000}],Text[Style[Round[(osize/10000),.001],Black,FontSize->16],{26,6000}],Text[Style["Total number infected = ",Black,FontSize->16],{15.5,7000}],Text[Style[Round[f[q] 10000,1],Black,FontSize->16],{26,7000}]}],{q,1.5,3,.01}]

f[q_]:=Quiet[a/.Solve[Exp[-q a]==1-a,a][[2]]//N]

f[1.5]

((.822-.797)/(2.1-2))

((.797)/(Exp[2 (.797)]-2))

.21-.2

Text[Style["Outbreak Size = ",Black,FontSize->16],{18,1200}]

Epilog->{PointSize->Large,Point[b,f[b]]}

Manipulate[Quiet[Plot[{a/.Solve[Exp[-q a]==1-a,a][[2]]//N},{q,1.1,4},PlotRange->{{0,4},{0,1}},Epilog->{{Red,Thick,Line[{{b-.1,((f[b])/(Exp[b f[b]]-b)) (-.1)+f[b]},{b+.1,((f[b])/(Exp[b f[b]]-b)) (.1)+f[b]}}]},{PointSize->Large,Point[{b,f[b]}]}}]],{b,1.1,4,.01}]

((Exp[x])/3)

(a/(Exp[q a]-q)) (x-q)+a

{Blue,Line[{{b-.1,((f[b])/(Exp[b f[b]]-b)) (-.1)+f[b]},{b+.1,((f[b])/(Exp[b f[b]]-b)) (.1)+f[b]}}]}

Graphics[{PointSize->Large,Point[{2.5,.5}]}]

Plot[{a/.Solve[Exp[-q a]==1-a,a][[2]]//N,{PointSize->Large,Point[{1,.5}]}},{q,1.1,4},PlotRange->{0,1},AxesOrigin->{0,0},AxesLabel->{q,"Outbreak size, A"},LabelStyle->Directive[FontSize->14]]

Manipulate[Quiet[Plot[a/.Solve[Exp[-q a]==1-a,a][[2]]//N,{q,1.1,4},PlotRange->{0,1},AxesOrigin->{0,0},AxesLabel->{q,"Outbreak size, A"},LabelStyle->Directive[FontSize->14]]],{b,1.1,4}]

Manipulate[tmax=30;sol[β_,μ_,sinit_,iinit_]:=NDSolve[{s'[t]==-β s[t] i[t],i'[t]==β s[t] i[t]-μ i[t],s[0]==sinit,i[0]==iinit},{s[t],i[t]},{t,0,tmax}];f[q_]:=Quiet[a/.Solve[Exp[-q a]==1-a,a][[2]]//N];Plot[Evaluate[{s[t],i[t]}/.sol[(q/10000.0),1,10000,1]],{t,0,tmax},PlotRange->{{0,tmax},{0,10000}},AspectRatio->1,Frame->True,FrameLabel->{"Time (weeks)","Number of people"},LabelStyle->Directive[FontSize->14],PlotStyle->{Blue,Red},Epilog->{Arrow[{{29,8000},{29,(1-f[q]) 10000}}],Arrow[{{29,8000},{29,10000}}],Text[Style["A = ",Black,FontSize->16],{22,6000}],Text[Style[Round[f[q],.001],Black,FontSize->16],{26,6000}],Text[Style["Total number infected = ",Black,FontSize->16],{15.5,7000}],Text[Style[Round[f[q] 10000,1],Black,FontSize->16],{26,7000}]}],{q,1.5,3,.01}]