Abstract
Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe. In this brief note, we endeavour to assess the efficiency of quarantine measures by means of mathematical modelling. Instead of the classical SIR model, we introduce a new model of infection progression under the assumption that all infected individual are isolated after the incubation period in such a way that they cannot infect other people. Disease progression in this model is determined by the basic reproduction numberR0(the number of newly infected individuals during the incubation period), which is different compared to that for the standard SIR model. IfR0> 1, then the number of latently infected individuals exponentially grows. However, ifR0< 1 (e.g.due to quarantine measures and contact restrictions imposed by public authorities), then the number of infected decays exponentially. We then consider the available data on the disease development in different countries to show that there are three possible patterns: growth dynamics, growth-decays dynamics, and patchy dynamics (growth-decay-growth). Analysis of the data in China and Korea shows that the peak of infection (maximum of daily cases) is reached about 10 days after the restricting measures are introduced. During this period of time, the growth rate of the total number of infected was gradually decreasing. However, the growth rate remains exponential in Italy. Arguably, it suggests that the introduced quarantine is not sufficient and stricter measures are needed.
Highlights
Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe
In the absence of vaccination and lack of effective treatment, the only way to influence the disease development is to act on the basic reproduction number R0 = kS0τ, that is to decrease the value of the parameter k characterizing the disease transmission rate by the infected individuals
If there are two different patches of the disease development with different basic reproduction numbers, the disease can be eradicated in the first patch due to the imposed restrictions but it can give a new outbreak in another patch if the restrictions are not adopted there or they are not sufficient
Summary
The classical susceptible-infected-recovered (SIR) model in epidemiology (see, e.g., [2]) allows the determination of critical condition of disease development in the population irrespective of total population size. Let us only determine the condition of the disease progression in the case where the number of infected/recovered/dead is much less than the number of susceptible, and S in the model above can be approximate by a constant, S ≈ S0. We obtain an ordinary differential equation with constant coefficients whose solution can be readily found, I(t) = I0eμt, where I0 is the number of infected at the initial moment of detection of the infection/disease. Substituting this expression into equation (1.4), we get μ = (kS0 − β − σ) = (β + σ)(R0 − 1), where the new parameter R0 = kS0/(β + σ) is called the basic reproduction number. A model that allows for the incubation period is considered
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