Counter-intuitive COVID-19 Trajectories - Explanations, Early Warning Indicator and Mitigation Strategies

The COVID-19 trajectories worldwide have shown several surprising features which are outside the purview of classical epidemiological models. These include (a) almost constant and low daily case rates over extended periods of time, (b) sudden waves emerging from the above solution despite no or minimal change in the level of non-pharmaceutical interventions (NPI), and (c) reduction or flattening of case counts even after relaxation of NPI. To explain these phenomena, we add contact tracing to our recently developed cluster seeding and transmission (CST) model, which is predicated on heterogeneous rather than homogeneous mixing of people in society. With this addition, we find no fewer than four effects which make prediction of epidemic trajectories uncertain. These are (a) cryptogenic instability, where a small increase in population-averaged contact rate causes a large increase in cases, (b) critical mass effect, where a wave can manifest after weeks of quiescence with no change in parameter values, (c) knife-edge effect, where a small change in parameter across a critical value can cause a huge change in the response of the system, and (d) hysteresis effect, where the timing and not just the strength of a particular NPI determines the subsequent evolution of the epidemic. Despite these effects however, it is a robust conclusion that a good contact tracing program can effectively substitute for much more invasive measures. We further find that the contact tracing capacity ratio - a metric of the stress to which the tracers are subject - can act as a reliable early warning indicator of an imminent epidemic wave. Extensive simulations demonstrate that whenever there is a drop in capacity ratio during a period of low daily infections, there is a very high probability of the case counts rising significantly in the immediate future. Author summaryClose to two years into the pandemic, the trajectories of COVID-19 in different places and at different times have shown wild variations and confounded modeling and forecasting efforts. Our new mathematical model can help to explain these variations. Some solutions of our model are non-standard but realistic. For example, we find an epidemic curve where daily cases remain on a plateau for a long time before suddenly exploding into a wave, despite interventions remaining constant throughout. We also find solutions showing that a specific intervention, for example capacity reduction at public gatherings, is very effective if implemented early on in a wave but useless if implemented a little later. Our proposed early warning indicator can be a game-changer for epidemic forecasting and model-based intervention strategies. Current forecasting algorithms have the weakest performance at the inflection points where there is an abrupt change in trend in the daily infection rates. The early warning indicator can give us advance notice of an approaching inflection point, and enable the authorities to take preventive measures before a wave actually arrives. Our results indicate that close communication between contact tracing personnel and public health authorities can achieve synergistic mitigation of the pandemic.

A Cluster-based Model of COVID-19 Transmission Dynamics

Many countries have manifested COVID-19 trajectories where extended periods of constant and low daily case rate suddenly transition to epidemic waves of considerable severity with no correspondingly drastic relaxation in preventive measures. Such solutions are outside the scope of classical epidemiological models. Here we construct a deterministic, discrete-time, discrete-population mathematical model which can explain these non-classical phenomena. Our key hypothesis is that with partial preventive measures in place, viral transmission occurs primarily within small, closed groups of family members and friends, which we call clusters. Inter-cluster transmission is infrequent compared to intra-cluster transmission but it is the key to determining the course of the epidemic. If inter-cluster transmission is low enough, we see stable plateau solutions. Above a cutoff level however, such transmission can destabilize a plateau into a huge wave even though its contribution to the population-averaged spreading rate still remains small. We call this the cryptogenic instability. We also find that stochastic effects when case counts are very low may result in a temporary and artificial suppression of an instability; we call this the critical mass effect. Both these phenomena are absent from conventional infectious disease models and militate against the successful management of the epidemic.

COVID-19 Spreading dynamics in an age-structured population with selective relaxation of restrictions for vaccinated individuals : a mathematical modeling study

BackgroundCOVID-19 vaccination of healthcare and other essential workers is underway in many countries while immunization of the general public is expected to begin in the next several weeks. We consider the question of whether people who have received the vaccine can be selectively and immediately permitted to return to normal activities. MethodsWe use a delay differential equation model to calculate the effects of vaccinee "immunity passports" on the epidemic spreading trajectories. The model incorporates age-structuring to account for children who are ineligible for vaccination, and senior citizens who are especially vulnerable to the disease. We consider consensus strains of virus as well as high-transmissibility variants such as B1.1.7 and B1.351 in our analysis. ResultsWe find that with high vaccine efficacy of 80 percent or greater, unrestricted vaccinee--vaccinee interactions do not derail the epidemic from a path towards elimination. Vaccinee--non-vaccinee interactions should however be treated with far more caution. At current vaccine administration rates, it may be the better part of a year before COVID-19 transmission is significantly reduced or ceased. With lower vaccine efficacy of approximately 60 percent, restrictions for vaccinees may need to remain in place until the elimination of the disease is achieved. In all cases, the death tolls can be reduced by vaccinating the vulnerable population first. ConclusionsDesigning high-efficacy vaccines with easily scalable manufacturing and distribution capacity should remain on the priority list in academic as well as industrial circles. Performance of all vaccines should continue to be monitored in real time during vaccination drives with a view to analysing socio-demographic determinants of efficacy, if any, and optimizing distribution accordingly. A speedy and efficacious vaccination drive augmented with selective relaxations for vaccinees will provide the smoothest path out of the pandemic with the least additional caseloads, death tolls and socio-economic cost.

Impact of immediate and preferential relaxation of social and travel restrictions for vaccinated people on the spreading dynamics of COVID-19 : a model-based analysis

BackgroundFour COVID-19 vaccine candidates developed by Pfizer, Moderna, University of Oxford/ Astra Zeneca (also Oxford/ Serum Institute of India) and ICMR/ Bharat Biotech have been granted emergency use authorization in the democratic world following established clinical trial procedures in their respective countries. Vaccination of the general public is expected to begin in several weeks. We consider the question of whether people who have received the vaccine can be selectively and immediately cleared to return to normal activities, including hassle-free travel. MethodsWe use a delay differential equation model developed previously by our group to calculate the effects of vaccinee "immunity passports" on the spreading trajectories of the disease. We consider default virus strains as well as high-transmissibility variants such as B1.1.7 in our analysis. ResultsWe find that with high vaccine efficacy of 80 percent or greater, vaccinees may be immediately cleared for normal life with no significant increase in case counts. Free travel of such vaccinees between two regions should not jeopardize the infection control performance of either. At current vaccine administration rates, it may be eight months or more before COVID-19 transmission is significantly reduced or eliminated. With lower vaccine efficacy of approximately 60 percent however, social as well as travel restrictions for vaccinees may need to remain in place until transmission of the disease is eliminated. ConclusionsDesigning high-efficacy vaccines with easily scalable manufacturing and distribution capacity should remain on the priority list in academic as well as industrial circles. Performance of all vaccines should continue to be monitored in real time during vaccination drive with a view to analysing socio-demographic determinants if any of efficacy, and optimizing distribution accordingly. A speedy and efficacious vaccination drive will provide the smoothest path out of the pandemic with the least additional caseloads, death toll and socioeconomic cost.

COVID-19 Spreading Dynamics with Vaccination - Allocation Strategy, Return to Normalcy and Vaccine Hesitancy

In this work we use mathematical modeling to analyse the dynamics of COVID-19 spread after a vaccination program is initiated. The model used is a delay differential equation developed earlier by our group. Basis of currently available data, our principal findings are as follows. (a) For fastest deceleration of the pandemic, people with high interaction rate such as grocers and airline cabin crew should be given priority in vaccine access. (b) Individuals who have been vaccinated may be selectively cleared to return to normal activities without significant risk of a resurgence in cases. (c) If an infection as well as a vaccine confers immunity for a duration{tau} 0, then the pandemic can be eliminated by vaccinating people at a sufficiently high rate. Unless{tau} 0 is very small, the cutoff rate required appears feasible to achieve in practice. (d) The presence of a substantial minority of vaccine-hesitant population might not amount to a significant threat or even an inconvenience to a vaccine-compliant majority population.

A New Approach to the Dynamic Modeling of an Infectious Disease

In this work we propose a delay differential equation as a lumped parameter or compartmental infectious disease model featuring high descriptive and predictive capability, extremely high adaptability and low computational requirement. Whereas the model has been developed in the context of COVID-19, it is general enough to be applicable mutatis mutandis to other diseases as well. Our fundamental modeling philosophy consists of a decoupling of public health intervention effects, immune response effects and intrinsic infection properties into separate terms. All parameters in the model are directly related to the disease and its management; we can measure or calculate their values a priori basis our knowledge of the phenomena involved, instead of having to extrapolate them from solution curves. Our model can accurately predict the effects of applying or withdrawing interventions, individually or in combination, and can quickly accommodate any newly released information regarding, for example, the infection properties and the immune response to an emerging infectious disease. After demonstrating that the baseline model can successfully explain the COVID-19 case trajectories observed all over the world, we systematically show how the model can be expanded to account for heterogeneous transmissibility, detailed contact tracing drives, mass testing endeavours and immune responses featuring different combinations of limited-time sterilizing immunity, severity-reducing immunity and antibody dependent enhancement.

Potential spreading dynamics of COVID-19 with temporary immunity - a mathematical modeling study

COVID-19 is caused by a hitherto nonexistent pathogen, hence the immune response to the disease is currently unknown. Studies conducted over the past few weeks have found that the antibody titre levels in the blood plasma of infected patients decrease over time, as is common for acute viral infections. Fully documented reinfection cases from Hong Kong, India, Belgium and USA, as well as credible to anecdotal evidence of second-time cases from other countries, bring into sharp focus the question of what profile the epidemic trajectories may take if immunity were really to be temporary in a significant fraction of the population. Here we use mathematical modeling to answer this question, constructing a novel delay differential equation model which is tailored to accommodate different kinds of immune response. We consider two immune responses here : (a) where a recovered case becomes completely susceptible after a given time interval following infection and (b) where a first-time recovered case becomes susceptible to a lower virulence infection after a given time interval following recovery, and becomes permanently immunized by a second infection. We find possible solutions exhibiting large number of waves of disease in the first situation and two to three waves in the second situation. Interestingly however, these multiple wave solutions are manifest only for some intermediate values of the reproduction number R, which is governed by public health intervention measures. For sufficiently low as well as sufficiently high R, we find conventional single-wave solutions despite the short-lived immunity. Our results cast insight into the potential spreading dynamics of the disease and might also be useful for analysing the spread after a vaccine is invented, and mass vaccination programs initiated.

Temporary Immunity and Multiple Waves of COVID-19

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.

Transmission Dynamics of COVID-19 and Impact on Public Health Policy

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.