Mathematical Epidemiology, SIR Models and COVID-19
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Submission date: 25. May. 2020
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This paper is not strictly speaking a mathematical paper. It is meant for a more general audience. As was pointed out to me by my brother David Luckhaus, who is a theoretical chemist, sections 3 and 4, which are more mathematical, are not so easy to read. If you want to follow the arguments, in principle the mathematics you need is taught to students of science, economics, and engineering in the calculus classes, but it helps to have experience in many particle systems, ordinary and partial differential equations. In section 3 I start from the process of infection from the point of view of an individual, who gets infected at a random time and removed at a random time. Removed means dead, quarantined or immune. On the time scale of the epidemy the individual then stays immune. In the large number limit - the proof uses only the law of large numbers but is not given here - one arrives at a partial differential equation, the socalled age structured model. Here age is the time since infection. This model is analyzed in section 4. Roughly speaking, because age plays a role only for the infected before removal, it can be eliminated in the discussion of an epidemy, which starts with a verylow number of infected and ends also that way. At least that is true if you want to calculate the difference of removed before and after the epidemy. An important point though is the difference beetween the model for a homogeneous population and the one for several distinguishable subpopulations. For the homogeneous population the outcome is that of an equivalent classical SIR model: Below a certain percentage of immunized, the socalled herd immunization, states with zero infection are still unstable, eventually the epidemy will start, and finally it will end in a stable state. The same is true for the model with k subpopulations. But immunization has now k parameters und the region of stability is a region in k-parameter space. For the generalized SIR model that also means that instead an infection rate one has a matrix of (cross) infection rates. I do not advocate here to estimate the parameters in this matrix. That is because the age dependence makes observation of the trajectory of the epidemy difficult. But what we can do for Corona is, try to identify stable states which mean minimal loss of lives. Since mortality depends dramatically on age (here really the age of an individual) one should aim for a stable state where the immunization occurs mainly in the low age groups, and get there quickly. What would such a strategy be (for Corona) one might aim for 90% immunization among the under forties by encouraging say soccer matches, classes for students maybe political demonstrations and try to keep everyone above forty away from the under forties temporarily. When the epidemy has run its course, one first tells the under fifties they can behave normally. If your state was already stable everything is ok. Otherwise another epidemy will start and run its course. It is very likely that at the end of the second epidemy a stable state for the whole population has been reached.