A new respiratory virus has begun circulating in another country. So far, epidemiological investigations have yielded the data below . Write down a model for this disease and use the information below to determine the values of all model parameters . If a vaccine conferring 'all - or - nothing' immunity with 80% effectiveness becomes available before an epidemic begins in your own country, what is the minimum coverage of this vaccine, to prevent an epidemic?
◼
  • All infected individuals go through an incubation period, which lasts on average for 4 days. During this time, individuals are not infectious, nor do they have any excess mortality risk.
    • ◼
  • All infected individuals eventually develop symptoms, and the mean duration of symptoms before either recovery or death is 5 days. Symptomatic individuals are infectious, as well as having a 3% case fatality rate. Those who survive the infection are thought to have long-term immunity.
    • ◼
  • In the source country, the peak prevalence (i.e. maximum number of symptomatic people during the epidemic) was observed to be 8% of the population.
    • dS
    dt
    -β·S(t)·
    I(t)
    N
    -μ·S(t)
    dI
    dt
    β·S(t)·
    I(t)
    N
    -γ·I(t)-μ·I(t)
    dR
    dt
    γ·I(t)-μ·R(t)
    Given information :
    1
    .
    Case Fatality Rate (CFR)} = 3% = 0.03
    2
    .
    Mean duration of symptoms (D} &= 5 days
    The rate of recovery (
    γ
    ) can be calculated using the formula :
    γ
    1-CFR
    D
    So, the rate of recovery (
    γ
    ) from symptomatic infection is approximately 0.194 per day .
    ​
    Given that the rate of recovery (
    γ
    ) is 0.194 per day, and the formula for the Case Fatality Rate (CFR) is :
    CFR
    μ
    μ+γ
    μ≈0.006
    Total population size :
    N11000000
    The estimated value of
    I
    peak
    can be calculated as :
    I
    peak
    Peak PrevalenceN0.0811000000880000
    In[]:=
    Clear["Global`*"]
    n=11000000;​​times=1000;​​reportedData=n*0.08;​​model[β_?NumberQ]:=(model[β]=First[i/.NDSolve[{s'[t]==-β*s[t]*i[t]/n-μ*s[t],i'[t]==β*s[t]*i[t]/n-γ*i[t]-μ*i[t],r'[t]==γ*i[t]-μ*r[t],s[0]==n,i[0]==1,r[0]==0}/.{γ->0.194,μ->0.006},i,{t,times}]]);sol=NMinimize[{(Max[model[b][#]&/@Range[0,times]]-reportedData)^2,{b>0}},{b},MaxIterations->1000]
    Out[]=
    {6.64074×
    -19
    10
    ,{b0.510274}}
    Parameters
    (*Recoveryrate*)​​γ=0.194;​​(*infectionrate*)​​β=0.51;​​(*FoI*)​​λ[t_]:=β*i[t]/n​​(*mortalityrate*)​​μ=0.006;
    Days
    tMax1=200;
    Define the system of differential equations
    sireqns={s'[t]==-β*s[t]*i[t]/n-μ*s[t],​​i'[t]==β*s[t]*i[t]/n-γ*i[t]-μ*i[t],​​r'[t]==γ*i[t]-μ*r[t],​​s[0]==n-1,​​i[0]==1,​​r[0]==0};
    In[]:=
    sol=NDSolve[sireqns,{s,i,r},{t,0,tMax1}];
    In[]:=
    Plot[Evaluate[{i[t]}/.sol],{t,0,tMax1},PlotLegends->{"Infected"},AxesLabel->{"Time","Population"},PlotStyle->{Red},PlotRange->All]
    Out[]=
    Infected

    A separate compartment for vaccination with coverage 62%

    In[]:=
    Clear["Global`*"]
    Parameters
    (*Recoveryrate*)​​γ=0.194;​​(*infectionrate*)​​β=0.51;​​(*FoI*)​​λ[t_]:=β*i[t]/n​​(*TotalPopulationSize*)​​n=11000000;
    Mortality rate
    μ=0.006;
    Days
    tMax1=1000;
    Vaccination coverage
    p=0.62;
    Define the system of differential equations
    eqns=​​{s'[t]==-β*s[t]*i[t]/n-μ*s[t],​​i'[t]==β*s[t]*i[t]/n-γ*i[t]-μ*i[t],​​r'[t]==γ*i[t]-μ*r[t],​​v'[t]==0,​​s[0]==(1-p)*(n),​​i[0]==1,​​r[0]==0,​​v[0]==p*(n-1)};
    Solve the system
    sol=NDSolve[eqns,{s,i,r,v},{t,0,tMax1}];
    In[]:=
    Plot[Evaluate[{s[t],i[t],r[t],v[t]}/.sol],{t,0,tMax1},PlotLegends->{"Susceptible","Infected","Recovered","Vaccinated"},AxesLabel->{"Time","Population"},AxesLabel->{"Time","Population"},PlotStyle->{Blue,Red,Green,{Black,Dashed}},PlotRange->All]
    Out[]=
    Susceptible
    Infected
    Recovered
    Vaccinated
    In[]:=
    Plot[Evaluate[{i[t]}/.sol],{t,0,tMax1},PlotLegends->{"Infected"},AxesLabel->{"Time","Population"},PlotStyle->{Red},PlotRange->All]
    Out[]=
    Infected
    Based on our estimates parameter values, what would be the critical vaccination threshold required to prevent this epidemic, assuming an all - or - nothing vaccine with 80 % efficacy?
    We can calculate the critical effective vaccination coverage as
    Since the all - or - nothing vaccine is only 80 % effective, we can calculate the critical vaccination threshold as :

    Let' s do the same with vaccine coverage 78%

    Parameters
    mortality rate
    Days
    Vaccination coverage
    Define the system of differential equations
    Solve the system

    CITE THIS NOTEBOOK

    Modeling and mitigating the spread of a new respiratory virus​
    by Athanasios Paraskevopoulos​
    Wolfram Community, STAFF PICKS, September 10, 2023
    ​https://community.wolfram.com/groups/-/m/t/3010106