ABSTRACT
In this work we use mathematical modeling to describe the potential phenomena which may occur if immunity to COVID-19 lasts for a finite time instead of being permanent, i.e. if a recovered COVID-19 patient may again become susceptible to the virus after a given time interval following his/her recovery. Whether this really happens or not is unknown at the current time. If it does happen, then we find that for certain combinations of parameter values (social mobility, contact tracing, immunity threshold duration etc), the disease can keep recurring in wave after wave of outbreaks, with a periodicity approximately equal to twice the immunity threshold. Such cyclical attacks can be prevented trivially if public health interventions are strong enough to contain the disease outright. Of greater interest is the finding that should such effective interventions not prove possible, then also the second and subsequent waves can be forestalled by a consciously relaxed intervention level which finishes off the first wave before the immunity threshold is breached. Such an approach leads to higher case counts in the immediate term but significantly lower counts in the long term as well as a drastically shortened overall course of the epidemic. As we write this, there are more than 1,00,00,000 cases (at least, detected cases) and more than 5,00,000 deaths due to COVID-19 all over the globe. The unknowns surrounding this disease outnumber the knowns by orders of magnitude. One of these unknowns is how long does immunity last i.e., once a person recovers from COVID-19 infection, how long does s/he remain insusceptible to a fresh infection. Most modeling studies assume lifetime immunity, or at least sufficiently prolonged immunity as to last until the outbreak is completely over. Among the exceptions are Giordano et. al. [1] and Bjornstad et. al. [2] who account for the possibility of re-infection - while the former find no special behaviour on account of this, the latter find an oscillatory approach towards the eventual equilibrium. In an article which appeared today, Kosinski [3] has found multiple waves of COVID-19 if the immunity threshold is finite. The question of whether COVID-19 re-infection can occur is completely open as of now. A study [4] has found that for benign coronaviruses (NOT the COVID-19 pathogen!), antibodies become significantly weaker six months after the original infection, and re-infection is common from one year onwards. Although it is currently unknown whether COVID-19 re-infections can occur, the mere possibility is sufficiently frightening as to warrant a discussion of what might happen if it is true. In this Article, we use mathematical modeling to present such a discussion. Before starting off, let us declare in the clearest possible terms that this entire Article is a what-if analysis, predicated on an assumption whose veracity is not known at the current time. The contents of this Article are therefore hypothetical - as of now they are neither factual nor counter-factual.
ABSTRACT
In this work we construct a mathematical model for the transmission and spread of coronavirus disease 2019 or COVID-19. Our model features delay terms to account for (a) the time lapse or latency period between contracting the disease and displaying symptoms, and (b) the time lag in testing patients for the virus due to the limited numbers of testing facilities currently available. We find that the delay introduces a significant disparity between the actual and reported time-trajectories of cases in a particular region. Specifically, the reported case histories lag the actual histories by a few days. Hence, to minimize the spread of the disease, lockdowns and similarly drastic social isolation measures need to be imposed some time before the reported figures are approaching their peak values. We then account for the social reality that lockdowns can only be of a limited duration in view of practical considerations. We find that the most effective interval for imposing such a limited-time lockdown is one where the midpoint of the lockdown period coincides with the actual peak of the spread of the disease in the absence of the lockdown. We further show that the true effectivity of imposing a lockdown may be misrepresented and grossly underestimated by the reported case trajectories in the days following the action.
