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 "Impact of variants and NPI measures: ",
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 StyleBox[" Endemic does not mean constant as SARS-CoV-2 continues to evolve\n\
Sarah P. Otto, Ailene MacPherson, & Caroline Colijn\n",
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 StyleBox[" (2024)",
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Model includes n waning classes, corresponding to reducing antibody levels \
over time.\
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Assume a standard SIR model with waning at rate \[Delta] and transmission at \
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Measuring the variables as a fraction of the full population, we have:\
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This system has two equilibria (disease absent and disease endemic):\
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We consider two vaccination rates, 0.012% corresponding to the low summer \
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The population-level reduction in incidence for the nominal parameter rates:\
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This is consistent with Figure 3 of Hethcote et al. (1981) with n=3, where \
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Here we allow the new variant to change the transmission rate (\[Beta]) \
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as well as those in the susceptible class\
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Before the new variant is present (I2=0), the final equation is the same as \
the previous one. Setting these equations to zero (with I2=0), we have \
R[j-1]=R[j] at equilibrium for the recovered classes besides the first \
recovered class.  Furthermore, adding the first four equations (with I2=0) \
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Measuring the variables as a fraction of the full population (S+I1+I2+n R == \
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COVID-19 infections are characterized by a short latent period, during which \
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By considering only cases yielding a stable endemic equilibrium, stability to \
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Here we allow the new variant to change the transmission rate (\[Beta]) \
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as well as those in the susceptible class\
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We assume that the endemic equilibrium is stable and determine when it would \
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The bottom row consists of zeros except for the last element.\n\nBy \
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By the same logic as used with only one class of individuals, the last \
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equations in each set indicates that R[1]=R[j] for both R1 and R2 as well.  \
We can thus focus on the first two equations in the above set to analyse the \
equilibrium, using the fact that R1=(1-f-S1-I1)/n and R2=(f-S2-I2)/n.\
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This system has two equilibria (disease absent and disease endemic):\
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In the simulations, we\[CloseCurlyQuote]ll allow a fraction of \
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Cell["\<\
Hence, the endemic equilibrium is stable when it exists when there is one \
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This is mathematically equivalent to the risk of becoming infected at any \
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Alternatively, if individuals transition rapidly in whether or not they \
follow NPI measures, we have to allow movement between the first and second \
class.  Here, we explore the opposite extreme, where movement is so rapid \
that the susceptible and recovered classes quickly approach an steady state \
number of followers and all that matters is the state of an individual \
(following with probability f or not following with probability 1-f) at the \
time of infection. \
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Cell["\<\
By the same logic as used with only one class of individuals, the last \
equation in each of the above two sets indicates that R[j-1]=R[j] at \
equilibrium. Furthermore, summing the first four equations indicates that \
R[1]=R[j] as well.  We can thus focus on the first three equations to analyse \
the equilibrium, using the fact that R=(1-f-S-I1)/n.\
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The disease-present equilibrium is stable (the real part of all eigenvalues \
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which must also be positive when the disease can spread when rare.\
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which again must be positive as long as the disease can spread when rare.\
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Hence, the endemic equilibrium is stable when it exists when there is one \
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Increasing the long-term waning rate by \[CapitalDelta]\[Delta] changes the \
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\[Bullet]\tExpected total waning time is 1/\|01d6ff
\[Bullet]\tRecovery distribution is exponential\
\>", "Text",
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Cell["\<\
SIR model: Variants that are more immune evasive or transmissible\
\>", "Section",
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  3.877274931156378*^9, 3.877274931474628*^9}, {3.8831718494463*^9, 
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Cell["\<\
Here we allow the new variant to change the transmission rate (\[Beta]) \
and/or the extent of leakiness of immunity (\[Beta]L):\
\>", "Text",
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We next determine when the endemic equilibrium is stable would be invaded by \
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\[Bullet]\tConsiders only two compartments, RH with higher immunity (not \
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\[Bullet]\tExpected total waning time is again 1/\|01d6ff
\[Bullet]\tBiologically reasonable to consider the R compartments as antibody \
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\[Bullet]\tCauses a slightly humped distribution, but not very noticeable\
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x determines the fraction of the waning period in which more immune evasive \
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or unstable) of the endemic equilibrium may change as we increase the number \
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Here we allow the new variant to change the transmission rate (\[Beta]) \
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We next determine when the endemic equilibrium is stable would be invaded by \
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COVID-19 infections are characterized by a short latent period, during which \
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with the latent and infectious period short relative to the spread of the \
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For the parameters considered typical of Omicon (next section), selection \
would only be ~70 % as strong with a latent period. We ignore this correction \
to simplify the model presentation.\
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Model includes two waning classes, where both wane to susceptibility at the \
same rate (\[Delta]), but the second class can be infected by the variant:\
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\[Bullet]\tConsiders only two compartments, RH (not susceptible) and RL \
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\[Bullet]\tNot biologically clear what these compartments are, given that all \
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\[Bullet]\tWaning time distribution is always exponential\
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x determines the fraction of the waning period in which more immune evasive \
variants can infect.\
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The disease model with transmission at rate \[Beta] is then described by the \
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Here we allow the new variant to change the transmission rate (\[Beta]) \
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Before the new variant is present (I2=0), the endemic equilibrium was found \
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We next determine when the endemic equilibrium is stable would be invaded by \
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