A Solution to the Kermack and McKendrick Integro-Differential Equations
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Abstract
In this manuscript, we derive a closed form solution to the full Kermack and McKendrick integro-differential equations (Kermack and McKendrick 1927) which we call the KMES. We demonstrate the veracity of the KMES using independent data from the Covid 19 pandemic and derive many previously unknown and useful analytical expressions for characterizing and managing an epidemic. These include expressions for the viral load, the final size, the effective reproduction number, and the time to the peak in infections. The KMES can also be cast in the form of a step function response to the input of new infections; and that response is the time series of total infections.
Since the publication of Kermack and McKendrick’s seminal paper (1927), thousands of authors have utilized the Susceptible, Infected, and Recovered (SIR) approximations; expressions putatively derived from the integro-differential equations to model epidemic dynamics. Implicit in the use of the SIR approximation are the beliefs that there is no closed form solution to the more complex integro-differential equations, that the approximation adequately reproduces the dynamics of the integro-differential equations, and that herd immunity always exists. However, the KMES demonstrates that the SIR models are not adequate representations of the integro-differential equations, and herd immunity is not guaranteed. We suggest that the KMES obsoletes the need for the SIR approximations; and provides a new level of understanding of epidemic dynamics.
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SciScore for 10.1101/2022.04.28.22274442: (What is this?)
Please note, not all rigor criteria are appropriate for all manuscripts.
Table 1: Rigor
Ethics not detected. Sex as a biological variable not detected. Randomization not detected. Blinding not detected. Power Analysis not detected. Table 2: Resources
No key resources detected.
Results from OddPub: We did not detect open data. We also did not detect open code. Researchers are encouraged to share open data when possible (see Nature blog).
Results from LimitationRecognizer: We detected the following sentences addressing limitations in the study:With these caveats in mind, the curve in Figure 6, wholly derived from the Covid-19 data from several countries, certainly has the characteristics many authors have expected a viral load to have (Challenger et al 2022, Jones et al 2021). While these authors …
SciScore for 10.1101/2022.04.28.22274442: (What is this?)
Please note, not all rigor criteria are appropriate for all manuscripts.
Table 1: Rigor
Ethics not detected. Sex as a biological variable not detected. Randomization not detected. Blinding not detected. Power Analysis not detected. Table 2: Resources
No key resources detected.
Results from OddPub: We did not detect open data. We also did not detect open code. Researchers are encouraged to share open data when possible (see Nature blog).
Results from LimitationRecognizer: We detected the following sentences addressing limitations in the study:With these caveats in mind, the curve in Figure 6, wholly derived from the Covid-19 data from several countries, certainly has the characteristics many authors have expected a viral load to have (Challenger et al 2022, Jones et al 2021). While these authors reached their conclusions through direct measurement of the viral load of thousands of patients, we have derived the same form using only the country case data. In retrospect, this is remarkable. Since Equation 32 clearly shows that B(t) is dependent on KT(t), a property of the disease and PC(t), a function of the population behavior, this, in turn, means that the time variations of I(t) and R(t) also depend on these two parameters. It is natural to assume that both I(t) and R(t) will depend on properties of the disease, but it may be somewhat surprising to see that their values also depend on the behavior of the population. In supplement 1.1 we explain this dependency by showing that I(t) is best interpreted as the total infectiousness within the infected population N(t). As a complementary interpretation, R(t) is best thought of as the degree of recovery from infectiousness within N(t). Therefore, a previously infected individual is simultaneously a part of both the infected and recovered populations with the degree of membership determined by the parameter Ψ(t). As time goes on, the degree of membership inevitably moves the infected individuals towards membership in the recovered community, but during this time, the inf...
Results from TrialIdentifier: No clinical trial numbers were referenced.
Results from Barzooka: We did not find any issues relating to the usage of bar graphs.
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