‘Scientific Paradoxes’ in the SARS-COV2[1] Pandemic
Seemingly disparate fields have contributed to the concept of R0 using population, susceptibility, virulence, spread, diffusion, immunity, quarantine etc. Attempts has been made to understand R0 in mathematical epidemiology, non-linear dynamics, statistical & percolation physics, medical geography, spatial statistics, location science, social science, health economics and infection biology. However, there is no comprehensive scientific apparatus to take advantage of the insights from these disparage fields. We coherently combine the insights from above by incorporating a heterodox spatial framework. A spatial framework that is a macro-meso-micro (local-regional-global) space-time framework for R0. The most glaring lacuna is in micro-spatial (street and hospital floor level) R0 and the mechanics of quarantine. Ironically, micro-space-time R0 estimates can be the most effective in managing Covid19, reversing a scientific paradox.
Temporal quarantines, implemented through a macro/meso spatial scale (country and regional levels that is in vogue currently), creates a paradox. The more time is increased — the longer the pandemic persists, with albeit lower peaks so that the supply of Covid-‘care’ matches demand.
We propose a hierarchical spatial (macro-meso- micro) and temporal interaction model to understand COVID19 pandemic and create more efficient solutions to contain it, that also eliminates the paradox. But first, we need to understand the following R0 ‘flavors’ coming from different fields and then synthesize these into a micro-space-time quantity.
But before that we need to see some historical evidence of spatial heterodoxy as in the AIDS pandemic.
Spatial Diffusion
A 1994 paper by the geographer-physicist duo (Gould and Wallace 1994) showed the oversights of traditional epidemiologists in understanding the spread of AIDS due to their focus on the continuous time series differential equations and not on the spatial heterodoxy of quantized spatial jumps in AIDS diffusion. A quarter century later, with enormous progress in network epidemiology, spatial epidemiology, spatial statistics and the new field of network science we are still held back by temporal epidemiology, or at the most macro and meso scale space-time epidemiology (global and country/regional based units). Micro-level spatial models are absent. Moreover, Gould & Wallace, through a series of discoveries, showed the complex ‘backcloth’ of transportation networks (streets, highways, air routes etc.) that pushed AIDS into pandemicity. The difference between AIDS and COVID2 is that the latter is way faster. Here, time is squished, reducing further the influence of temporal differential models in understanding the spread of COVID19. Consequently the only way to manage this is controlling micro-scale spatial diffusion.
Before that let us connect the mathematics of R0 from unrelated sciences.
Mathematics of Epidemiological R0 (reproductive number)
R0 is an important derivation in Mathematical Epidemiology (Kermack, McKendrick, and Walker 1927). It measures quantities like susceptible, infected, recovered that conserve both endemicity and population. Because of conserving, equilibrium conditions arise so that measures like R0 acquire meaningfulness that can be manipulated to control an epidemic. Hysteresis and backward bifurcations can corrupt the R0 values below 1 and still ensure pandemicity.
Mathematics of Quarantine R0
Endemic Equilibria is introduced when vaccines are successful even partially. This kind of modeling extends the SIR model to SI’Q’R modeling.
Mathematics of Percolation Physics and Network Science derived R0
R0 can be viewed in network science & percolation physics as phase change.
Percolation physics, one of the least understood areas of science (Achlioptas, D’Souza, and Spencer 2009; Pan et al. 2011) can be continuous and discrete. However, much recent progress has been made in percolation science using network science – a recent development in discrete mathematics, statistical mechanics and computer science. Discrete percolation results in phase change. We see discrete percolation happening in R0 hence pandemicity crossing a threshold (R0 = 1) implies a phase change. Hysteresis (when contact chains have memory as well as loss of immunity like Covid-19) and backward bifurcations (happens spatially when infected move) can concur phase change even when R0 is way below one or approaching 0!
Mathematics of Micro-space-time R0
| Time | Space | Space*Time (interaction) | |||
| Macro | Meso | Micro | |||
| R0 Pandemicity | ≥1 | ≥1 | ≥1 | <1 | <<1 |
| Hysteresis | — | Yes | Yes | Yes | Yes |
| Backward bifurcation | — | Yes | Yes | Yes | Yes |
| Quarantine (SI’Q’R) | Does not work | Does not work | Does not work | Works partially | Works best |
| herd immunity threshold (Meyerowitz-Katz n.d.) | Does not work | Does not work | Does not work | Works partially | Works best |
TABLE1: showing the efficacy of different epidemiologically influenced controls
A simple model could explain R0 in a spatial network setting. Suppose you live in a street network and can reach anyone connected to the network. Fig 1 shows that each intersection (node) has 3 or 4 segments (and one when you live on a cul-de-sac). Now start deleting street segments from the nodes so that on an average each node has less than 1 link. What happens is that the network is not connected universally anymore (you may have islands of connectivity but no universal connectivity). The network R0 is below 1. Something magical happens when R0 reaches 1. You have universal connectivity. [Figure 1]. This is equivalent to the concept of a discrete phase change in statistical physics.
| R0 = 10 links/10 nodes = 1.0 [every node is connected to every other] | R0 = 9 links/10 nodes = 0.9 [every node is connected to every other] | R0 = 9 links/9 nodes = 1.0 [every node is connected to every other because of 2 nodes fusing into one] |
| Figure 1: R0 in a network setting. On the left you have full connectivity (R0 is >1); islands of connectivity in the center (R0 is <1); due to hysteresis, backward bifurcation, reinfections etc. R0 turns back to >= 1 latently. Red boundaries enclose a fully connected network | ||
Revisiting Spatial R0
What we see in the above R0 models is that space creates problems because people move, destabilizing the R0 mathematics. Due to population mixing, simple SI(x)R models fail to maintain the phase change at R0 = 1, invoking backward bifurcations, hysteresis etc. As illustrated below (Fig. 2) (Brauer and Castillo-Chavez 2013) different geographic locations form what we call ‘Core Groups’ and the SIR models succeeds the R0 test of phase change.
| · A susceptible (S) population of Covid-19
· are converted to infected (I) at rate b (contact process #1). · Active users “recover” (R) from use at rate g · and relapse back to active status at rate r. o (m “mortality” rate; steady state) |
|
| Fig 2: ‘Core Groups’ geographic distribution: Increasing infections in certain neighborhoods or hospital floors (local level) produces rapid systemic growth due to local/regional transportation routes | |
Behavioral R0
With economists developing SIR models using behavioral economics (“Covid Economics: Vetted and Real-Time Papers | Centre for Economic Policy Research” n.d.) where people reduce their contacts proportional to getting infected. R0 is modified by changing rate of infection, rate b, using ‘deaths’ as a way to inform people when no testing is available (Cochrane 2020).
Conclusion
Managing R0 at the micro-spatial scale, like street level and floor level can create an effective quarantine strategy that does not create the paradox of low peak but extended pandemicity.
References
Achlioptas, Dimitris, Raissa M. D’Souza, and Joel Spencer. 2009. “Explosive Percolation in Random Networks.” Science 323 (5920): 1453–55. https://doi.org/10.1126/science.1167782.
Brauer, Fred, and Carlos Castillo-Chavez. 2013. Mathematical Models in Population Biology and Epidemiology. Springer Science & Business Media.
Cochrane, John H. 2020. “The Grumpy Economist: An SIR Model with Behavior.” The Grumpy Economist (blog). May 4, 2020. https://johnhcochrane.blogspot.com/2020/05/an-sir-model-with-behavior.html.
“Covid Economics: Vetted and Real-Time Papers | Centre for Economic Policy Research.” n.d. Accessed May 13, 2020. https://cepr.org/content/covid-economics-vetted-and-real-time-papers-0.
Gould, Peter, and Rodrick Wallace. 1994. “Spatial Structures and Scientific Paradoxes in the AIDS Pandemic.” Geografiska Annaler. Series B, Human Geography 76 (2): 105–16. https://doi.org/10.2307/490593.
Kermack, William Ogilvy, A. G. McKendrick, and Gilbert Thomas Walker. 1927. “A Contribution to the Mathematical Theory of Epidemics.” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 115 (772): 700–721. https://doi.org/10.1098/rspa.1927.0118.
Klompas, Michael. 2020. “Coronavirus Disease 2019 (COVID-19): Protecting Hospitals From the Invisible.” Annals of Internal Medicine, March. https://doi.org/10.7326/M20-0751.
Meyerowitz-Katz, Gideon. n.d. “Here’s Why Herd Immunity Won’t Save Us From The COVID-19 Pandemic.” ScienceAlert. Accessed May 10, 2020. https://www.sciencealert.com/why-herd-immunity-will-not-save-us-from-the-covid-19-pandemic.
Pan, Raj Kumar, Mikko Kivelä, Jari Saramäki, Kimmo Kaski, and János Kertész. 2011. “Using Explosive Percolation in Analysis of Real-World Networks.” Physical Review E 83 (April): 046112. https://doi.org/10.1103/PhysRevE.83.046112.
[1] “It causes mild but prolonged disease, infected persons are contagious even when minimally symptomatic or asymptomatic, the incubation period can extend beyond 14 days, and some patients seem susceptible to reinfection … These factors make it inevitable that patients with respiratory viral syndromes that are mild or nonspecific will introduce the virus into hospitals, leading to clusters of nosocomial infections” (Klompas 2020).