SIR models

If you click on !!! (the menu bar at the top) maple will execute all commands.

You can also enter each command as you go along reading this document. You may enter your own commands at any time and place in this document.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=

-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6gcl-I#miGF$6%Q(restartF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q":F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%'lspaceGQ,0.2777778emF'/%'rspaceGFL-I'mspaceGF$6&/%'heightGQ&0.0exF'/%&widthGQ&0.0emF'/%&depthGFT/%*linebreakGQ(newlineF'-FP6&FRFUFX/FenQ%autoF'-F,6%Q%withF'F/F2-I(mfencedGF$6$-F#6%-F,6%Q.LinearAlgebraF'F/F2/%+executableGF=F9F9F5FOFgn-F66-Q"~F'F9F;F>F@FBFDFFFH/FKQ&0.0emF'/FNF\pFhoF[o-F_o6$-F#6%-F,6%Q&plotsF'F/F2FfoF9F9F5FOFgnFhoFhoFhoFhoF[o-F_o6$-F#6%-F,6%Q/VectorCalculusF'F/F2FfoF9F9F5FOFgnFOFgnFhoFhoFhoFhoFhoFho-F,6%Q$SIRF'F/F2Fho-F66-Q*&coloneq;F'F9F;F>F@FBFDFFFHFJFMFho-F66/Q%procF'/%%boldGF1/F3Q%boldF'/%+fontweightGFhqF;F>F@FBFDFFFHF[pF]p-F_o6$-F#6'-F,6%Q'&#945;F'/F0F=F9-F66-Q",F'F9F;/F?F1F@FBFDFFFHF[p/FNQ,0.3333333emF'-F,6%Q'&#947;F'FbrF9FfoF9F9-F66-Q";F'F9F;FfrF@FBFDFFFHF[pFMFOFgn-F66/Q,descriptionF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFho-I#msGF$6#Q*SIR~modelF'F\sFOFgn-F66/Q'returnF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFhoFOFgn-F,6%Q"PF'F/F2-F66-Q(&srarr;F'F9F;F>F@FBFDFFFHFJFM-F_o6&-F#6JFis-F_o6&-F#6%-I#mnGF$6$Q"1F'F9FfoF9F9/%%openGQ"[F'/%&closeGQ"]F'-F66-Q*&uminus0;F'F9F;F>F@FBFDFFFH/FKQ,0.2222222emF'/FNFeuF_r-F66-Q"*F'F9F;F>F@FBFDFFFH/FKQ,0.1666667emF'/FNF[vFisFctFguFis-F_o6&-F#6%-Fht6$Q"2F'F9FfoF9F9F[uF^uFcrFhoFisF]v-F66-Q"+F'F9F;F>F@FBFDFFFHFduFfuF_rFguFisFctFguFisF]vFhoFauFhoFirFguFisF]vFcrFhoFis-F_o6&-F#6%-Fht6$Q"3F'F9FfoF9F9F[uF^uFdvFirFguFisF]vFfoF9F9/F\uQ'&lang;F'/F_uQ'&rang;F'F\sFOFgn-F66/Q$endF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFhoFbqF5FOFgnFOFgn-F66-Q"#F'F9F;F>F@FBFDFFFHF[pF]p-F,6%Q4F:=SIR(0.001,~0.1);F'F/F2FOFgnFOFgn-F,6%Q'&#961;F'FbrF9F_qFbq-F_o6$-F#6%-F,6%Q"FF'F/F2FfoF9F9F\sFOFgnF_sFho-Fcs6#QERelative~removal~rate~of~SIR~model~FF'F\sFOFgnFfsFhoFbx-F_o6$-F#6%-F_o6&-F#6)-Fht6$Q"0F'F9FcrFgtFcrF`yFfoF9F9F^wF`wFfoF9F9Fgv-F66-Q"/F'F9F;F>/FAF1FBFDFFFHFjuF\v-F_o6$-F#6)FgtFauFbx-F_o6$-F#6%-F_o6&-F#6)FgtFcrFgtFcrF`yFfoF9F9F^wF`wFfoF9F9FctFfoF9F9F\sFOFgnFbwFhoFbqF5FOFgnFOFgn-F,6%Q(iterateF'F/F2FhoF_qFhoFbqFho-F_o6$-F#6+FbxFcrFho-F,6%Q#P0F'F/F2FcrFho-F,6%Q"tF'F/F2FfoF9F9-F66-Q!F'F9F;F>F@FBFDFFFHFJFMFOFgnFhoFhoF_sFho-Fcs6#Q@compute~the~tth~iterate~F^t(P0)F'F\sFhoFOFgn-F66/Q'optionF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFho-F,6%Q)rememberF'F/F2F\sFhoFOFgn-F66/Q#ifF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFhoF][lFho-F66-Q&&leq;F'F9F;F>F@FBFDFFFHFJFMFhoF`yFho-F66/Q%thenF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFhoFfsFhoFjzFho-F66/Q%elseF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFhoFfsFhoFbx-F_o6$-F#6&Fcz-F_o6$-F#6-FbxFcrFhoFjzFcrFhoF][lFauFgtFfoF9F9FfoF9F9FhoFbwFhoF\\lF\sFOFgnFbwFhoFbqF5FOFgnFOFgnFew-F,6%Q9iterate(F,<990,10,0>,5);F'F/F2FOFgnFOFgn-F,6%Q&orbitF'F/F2FhoF_qFhoFbqFho-F_o6$-F#6+FbxFcrFhoFjzFcrFho-F,6%Q"TF'F/F2FfoF9F9Fho-F66/Q&localF'FeqFgqFiqF;F>F@FBFDFFFHF[pF]pFho-F,6%Q"iF'F/F2F\sFhoFOFgnF_sFho-Fcs6#QPthe~orbit~of~the~point~P0~under~T~iterates~of~FF'F\sFOFgn-F,6%Q$seqF'F/F2-F_o6$-F#61Fcz-F_o6$-F#6+FbxFcrFhoFjzFcrFhoF`^lFfoF9F9FcrFhoF`^lFho-F66-Q"=F'F9F;F>F@FBFDFFFHFJFMFhoF`yFho-F66-Q#..F'F9F;F>F@FBFDFFFHFduF]pFhoFj]lFfoF9F9FhoFOFgnFhoFhoFbwFhoFbqF5FhoFOFgnFOFgnFew-F,6%QS~the~timeplot,~multigraph,~of~a~3-population~modelF'F/F2FOFgn-F,6%Q)timeplotF'F/F2F_qFbq-F_o6$-F#6)FbxFcrFjzFcrFj]lFfoF9F9FhoF]^lFho-F,6%Q$orbF'F/F2F\sFhoFOFgnF_sFho-Fcs6#QFTimeplot~of~the~three~components~in~FF'F\sFhoFOFgnFa`lF_qFc]lF]`lF\sFhoFOFgn-F,6%Q%plotF'F/F2-F_o6$-F#6M-F_o6&-F#67-F_o6&-F#6&Ff^l-F_o6$-F#6+-F_o6&-F#6)-F,6%Q"nF'F/F2FcrFa`l-F_o6&-F#6%F^blFfoF9F9F[uF^uFctFfoF9F9F[uF^uFcrF^blFa_lFgtFd_lFj]lFfoF9F9FfoF9F9F[uF^uFcr-F_o6&-F#6&Ff^l-F_o6$-F#6+-F_o6&-F#6)F^blFcrFa`lFablF]vFfoF9F9F[uF^uFcrF^blFa_lFgtFd_lFj]lFfoF9F9FfoF9F9F[uF^uFcrFhoFOFgn-F_o6&-F#6&Ff^l-F_o6$-F#6+-F_o6&-F#6)F^blFcrFa`lFablFgvFfoF9F9F[uF^uFcrF^blFa_lFgtFd_lFj]lFfoF9F9FfoF9F9F[uF^uFcrFhoFOFgnFbalFcrFeblFcrFOFgnFaclFfoF9F9F[uF^uFcrF^blFa_lFgtFd_lFj]lFcrFOFgnFhoFhoFhoFhoFho-F,6%Q&styleF'F/F2Fa_l-F_o6&-F#6/-F,6%Q&pointF'F/F2FcrFddlFcrFddlFcr-F,6%Q%lineF'F/F2FcrFgdlFcrFgdlFfoF9F9F[uF^uFcrFOFgnFhoFhoFhoFhoFho-F,6%Q&colorF'F/F2Fa_l-F_o6&-F#6--F,6%Q&blackF'F/F2FcrFaelFcr-F,6%Q%blueF'F/F2Fcr-F,6%Q&greenF'F/F2Fcr-F,6%Q$redF'F/F2FfoF9F9F[uF^uFcr-F,6%Q*thicknessF'F/F2Fa_lFavFcr-F,6%Q+symbolsizeF'F/F2Fa_l-Fht6$Q#14F'F9Fcr-F,6%Q'labelsF'F/F2Fa_l-F_o6&-F#6)F`[l-Fcs6#Q%timeF'Fcr-Fcs6#Q+PopulationF'F`[lFfoF9F9F[uF^uFfoF9F9F`[lFOFgnFhoFhoFbwFhoFbqF5FOFgnFOFhoFew-F,6%Q:~timeplot(F,<99,1,0>,40);F'F/F2FOFgnFOFgnFOFgnFOFgn-F,6%Q)phasediaF'F/F2F_qFbqF]`lFhoF]^lFhoFa`lF\sFOFgnFhoFhoFhoF_sFho-Fcs6#Q@The~orbit~in~the~SI-phase~planeF'F\sFOFgnFa`lF_qFf^l-F_o6$-F#6,-F_o6&-F#6)FisFctFcrFisF]vFfoF9F9F^wF`wFcrFhoFisFa_lFhoFc]lF]`lFfoF9F9F\sFOFgn-F,6%Q&arrowF'F/F2-F_o6$-F#6@-F_o6&-F#6&Ff^l-F_o6$-F#6--F_o6&-F#6)Fa`l-F_o6&-F#6%F`^lFfoF9F9F[uF^uFcrFa`l-F_o6&-F#6'F`^lFdvFgtFfoF9F9F[uF^uFfoF9F9F[uF^uFcrF`^lFa_lFgtFd_lFj]lFauFgtFfoF9F9FfoF9F9/F\uQ"|frF'/F_uQ"|hrF'Fcr-F,6%Q+differenceF'F/F2Fa_l-F,6%F1F/F2FcrFho-F,6%Q&shapeF'F/F2Fa_lFdhlFcrFhoFOFgnFhoFhoFhoFhoFhoFhoFjdlFa_lFjelFcrFhoFfflFa_l-F_o6&-F#6'-F,6%Q$P_1F'F/F2Fcr-F,6%Q$P_2F'F/F2FfoF9F9F[uF^uFfoF9F9F\sFhoFOFgnFbwFhoFbqF5FOFgnFOFgnFew-F,6%QM~phasedia(SIR(0.001,~0.1),~<990,~10,~0>,~20)F'F/F2FOFgnFew-F,6%Q0~several~orbitsF'F/F2FOFgnFew-F,6%QO~display(seq(phasedia(F,SIR(0.001,0.1),P0,40),F'F/F2FOFgnFew-F,6%QM~P0=[<990,10,0>,<900,100,0>,~<800,200,0>]));F'F/F2FOFgnFOFgnFew-F,6%Q8~animation~of~one~orbitF'F/F2FOFgnFew-F,6%Qhn~display(seq(phasedia(F,<1,0.1>,T),T=1..75),insequence=true);F'F/F2FOFgnFOFgn-F,6%Q'growthF'F/F2F_qFbq-F_o6$-F#6+FbxFcr-F,6%Q%PmaxF'F/F2Fcr-F,6%Q%QmaxF'F/F2Fcr-F,6%Q"NF'F/F2FfoF9F9F`[lFOFgnFhoFhoF_sFho-Fcs6#Q\pThe~~SI-growth~function~|+~~~~~~~~~~~~~~~for~a~SIR~model~in~a~population~of~size~NF'F\sFOFgnFdhl-F_o6$-F#6K-F_o6&-F#6&Ff^l-F_o6$-F#61Ff^l-F_o6$-F#62-F_o6&-F#6(-F_o6&-F#6'F`^lFcr-F,6%Q"jF'F/F2FfoF9F9F^wF`wFcrFho-F_o6&-F#6,Fbx-F_o6$-F#6%-F_o6&-F#6-F`^lFcrFb_mFcrFd]mFauF`^lFauFb_mFfoF9F9F^wF`wFfoF9F9FctFcrFhoFbxFi_mF]vFfoF9F9F^wF`wFfoF9F9F[uF^uFcrFhoFOFgnF`^lFhoFa_lFhoF`yFhoFd_lFhoF^]mFfoF9F9FcrFhoFb_mFhoFa_lFhoF`yFhoFd_lFhoFa]mFfoF9F9FfoF9F9F_jlFajlFcrFhoFOFgnFhoFhoFhoFhoFhoFhoFho-F,6%Q(scalingF'F/F2FhoFa_lFho-F,6%Q.UNCONSTRAINEDF'F/F2FcrFhoFcjlFa_lFfjlFcrFhjlFa_lFdhlFcrFhoFOFgnFhoFhoFhoFhoFhoFhoFjdlFa_lFdelFfoF9F9F\sFOFgnFbwFhoFbqF5FOFgnFOFgnFew-F,6%QG~growth(SIR(0.005,~1/4),~80,~10,~100);F'F/F2FOFgnFOFgn-F,6%Q)dirfieldF'F/F2F_qFbqFj\mF`[lFOFgnFhoFhoF_sFho-Fcs6#QhpThe~~SI-directional~field~function~|+~~~~~~~~~~~~~~~for~a~SIR~model~in~~a~population~of~size~NF'F\sFOFgnFdhl-F_o6$-F#6OF^^mFcrFhoFOFgnFhoFhoFhoFhoFhoFhoFhoFa`mFhoFa_lFhoFd`mFcrFhoFcjlFa_lFfjlFcrFhjlFa_lFdhlFcrFhoFOFgnFhoFhoFhoFhoFhoFhoFjdlFa_lFdelFcr-F,6%Q'lengthF'F/F2Fa_lFgtFfoF9F9F\sFOFgnFbwFhoFbqF5FOFgnF`[lFfoF9

We consider SIR models for a population of total size

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkjbW5HRiQ2JFEkMTAwRidGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnRk0vJStleGVjdXRhYmxlR0Y9Rjk=

SIR epidemic

We will go through three examples of epidemic SIR models with diminishing transmission coefficients \316\261=0.01, 0.005, 0.003 and fixed removal rate \316\263=0.25. The fixed removal rate of 0.25 means that the infected individuals are infected for 4 days.

Example 1: Consider the SIR model with transmission coefficient \316\261=0.01 and removal rate \316\263=0.25

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

has relative removal rate

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

and contact number

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

This disease needs at least 25 susceptibles in order to spread. An invidual in this population has 4 potentially contagious contacts over 4 days.

Here is the time plot for the population orbit through P0=<95,5,0>

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

The maximal number of infected occurs when S=\317\201 after 6-7 time steps.

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

and there remain no

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

susceptibles in the population after the epidemic.

Here is the directional field in the SI-phase plane

and the orbits through various points

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

The maximal number of infected lies in the range 45-62 and there are no susceptibles left after the epidemic.

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

Example 2: The SIR model with transmission coefficient \316\261=0.005 and removal rate \317\201=0.25 models an epidemic that is less contagious.

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

has relative removal rate

and contact number

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

This disease needs at least 50 susceptibles in order to spread. An invidual in this population has 2 potentially contagious contacts over 4 days.

Here is the time plot for the population orbit through <95,5,0>

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

The maximal number of infected occurs when S=\317\261

and there remain

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

susceptibles in the population after the epidemic.

Here are several population orbits in the SI-phase plane

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

The maximal number of infected lies in the range 19-50 and there are 6-17 susceptibles left after the epidemic.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJG1heEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y/LyUmZmVuY2VHRj4vJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGWC1GLzYlUSRzZXFGJ0YyRjUtSShtZmVuY2VkR0YkNiQtRiM2KS1GLDYlLUYvNiVRJmV2YWxmRidGMkY1LUYjNiUtSSNtbkdGJDYkUSIzRidGP0YyRjVGQS1GaW42JC1GIzYlRjotRmluNiQtRiM2JUZlbi1GaW42JC1GIzYqLUYvNiVRIlBGJ0YyRjUtRmluNiYtRiM2JC1GZW82JFEiMkYnRj9GP0Y/LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUZFNi1RIixGJ0Y/RkgvRktGNEZMRk5GUEZSRlQvRldRJjAuMGVtRicvRlpRLDAuMzMzMzMzM2VtRidGZHBGRC1GLzYlUSZvcmJpdEYnRjJGNS1GaW42JC1GIzYoLUYvNiVRIkZGJ0YyRjVGZHEtRi82JVEjUDBGJ0YyRjVGZHEtRmVvNiRRIzUwRidGP0Y/Rj9GP0Y/Rj9GP0Y/Rj9GZHFGZnJGRC1GaW42Ji1GIzYlRmVuLUZpbjYkLUYjNiotRmluNiYtRiM2Li1GZW82JFEkMTAwRidGPy1GRTYtUSgmbWludXM7RidGP0ZIRkpGTEZORlBGUkZUL0ZXUSwwLjIyMjIyMjJlbUYnL0ZaRl90LUZlbzYkUSI1RidGPy1GRTYtUScmc2RvdDtGJ0Y/RkhGSkZMRk5GUEZSRlRGaHEvRlpGaXEtRi82JVEia0YnRjJGNUZkcUZhdEZkdEZodEZkcS1GZW82JFEiMEYnRj9GP0Y/L0ZfcVEnJmxhbmc7RicvRmJxUScmcmFuZztGJ0ZkcUZodEZELUZlbzYkUSIxRidGPy1GRTYtUSMuLkYnRj9GSEZKRkxGTkZQRlJGVEZedEZndC1GZW82JFEjMTBGJ0Y/Rj9GP0Y/Rj9GXnFGYXFGP0Y/LUZFNi1RIjtGJ0Y/RkhGZ3FGTEZORlBGUkZURmhxRlkvJStleGVjdXRhYmxlR0Y+Rj8=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEhRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkMvJStleGVjdXRhYmxlR0Y0Ri8=

Example 3:The SIR model with transmission coefficient \316\261=0.003 and removal rate \316\263=0.25

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

has relative removal rate

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

and contact number

This disease needs at least 83 susceptibles in order to spread. An invidual in this population has 1.2 potentially contagious contacts over 4 days.

Here is the time plot for the population orbit through <95,5,0>

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

The maximal number of infected occurs when S=\317\261

and there remain

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

susceptibles in the population after the epidemic.

Here are several population orbits in the SI-phase plane

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

The maximal number of infected lies in the range 5-50 and there remain 18-58 susceptibles in the population after the epidemic.

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

SIR endemic

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

General theory

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEncmhvZW5kRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiRkYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA=

Example 1

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

Example 2

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

Example 3

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