The Model

Model Description

The model presented here is a death-based SIR model [1] where S represents the number of people who are susceptible to the virus, I represents the number of people who are currently infected with the virus, and R represents the number of people who are resistant to the virus.  Prior to the introduction of the virus into a population the number of susceptible people in the population is equal to the number of people in the population, N, while I and R are equal to zero.  Once the virus is introduced into the population, S, I and R are governed by the following three differential equations:

(1)   \begin{equation*}\frac{dS}{dt} = \frac{-\beta IS}{N}\end{equation*}

(2)   \begin{equation*}\frac{dI}{dt} = \frac{\beta IS}{N} - \gamma I\end{equation*}

(3)   \begin{equation*}\frac{dR}{dt} = -\gamma I\end{equation*}

Here, \gamma is the inverse of the infectious period (assumption: 7 days), and

\beta = \gamma R_t, where R_t [2, 3] is the number of people an infectious person infects during the infectious period.

In this death-based SIR model the number of covid-19 deaths on any given day [4, 5] is shifted backward in time by the estimated duration from infection to death (assumption: 17 days), and divided by the effective mortality rate (link) to get new infections per day. The current infections, I, in the population is the sum of the new infections per day for the number of days corresponding to the infectious period.  Therefore, deaths per day gives us I(t), which along with N, S_0, R_0 and \gamma, gives us \beta(t) and hence R_t(t), completing the SIR model through the present day (minus the estimated duration from infection to death). 

Note that R_t(t) represents the effective transmission ratio, the number of people infected by each infectious person as a function of time.  The change in the number of deaths per day, dI/dt is driven by R_t(t), which indicates how efficiently the population is transmitting the virus at a any given time.

The model data and projections presented here are calculated for individual countries and in the case of the US, for states and counties.  Only a subset of possible locations are presented based on arbitrary thresholds of population.

Graphs vs. time are shown for the following for each location listed in Data / Projections:

  • Cumulative Confirmed Cases [6, 7]
  • Daily Confirmed Cases
  • Cumulative Deaths [4, 5]
  • Daily Deaths
  • Current Infections I(t)
  • Effective Transmission Ratio (R_t(t))
  • Fraction of Population Infected and Constant Infections Transmission Ratio (R_{tcl})
  • Probability of Covid-19 Contact
  • Mortality and Undercount
  • Fraction of Positive Tests (US only) [8]

The fraction of the population infected is the fraction of the population that has contracted the virus as function of time and is represented by the equation 1-S(t)/N.

From S(t) and N we also calculate the effective transmission ratio that results in a constant in the number of new daily infections. The ‘no change’ transmission ratio is the constant infections transmission ratio.  We get R_{tcI}, the constant infections transmission ratio, by setting dI/dt equal to zero:

(4)   \begin{equation*}R_tcI = \frac{N}{S}\end{equation*}

The constant infections transmission ratio is presented versus time and is based on how much of the population is currently susceptible.  In the model presented here, reinfection is not allowed.  Once a member of the population has the virus or dies from the virus that person is no longer counted as part of the susceptible population (S(t)). Given this assumption, R_{tcI} tells us how aggressively the population in a given location must project themselves (e.g. hygiene, physical distancing, masks) to control the spread of infection. Since the fraction of the population infected and the constant infections transmission ratio are both based on S(t) and N they are presented on a single graph.

Knowing I(t) and N we calculate the probability that any person who interacts with others in a given population will encounter 1 or more people infected with the virus. The probability of Covid-19 contact is given by the equation [9]:

(5)   \begin{equation*}P(t) = 1-(1-\frac{I(t)}{N})^k\end{equation*}

where k is the number of people that person interacts.

Above, mortality is shorthand for mortality rate. In the model presented here the mortality rate is 0.006 (assumption) outside of the US and variable with time and place inside of the US as described in detail in Local Mortality Rates.

Undercount is the ratio of new infections (daily deaths divided and mortality) to the number of daily confirmed cases.  For comparison with new infections, the daily confirmed cases are necessarily shifted back in time by the estimated time period between the onset of the infection and the reported date of the confirmed case (assumption:  10 days).  The undercount is a contrived measure of undertesting.

The fraction of positive tests is the ratio of positive live virus tests to total live virus tests [8].