Course Outline: My plan for the structure of this course is to take guidance from those in attendance. I have collected some background references and a couple dozen papers with substantial mathematical content involving COVID. Many also have code which is freely available. Students are encouraged to look at a few of these and/or look around for others and choose two topics to learn something about and present for the class. In between I will plan to lecture on topics informed by student interest. Some possible topics include: Basic SARS-CoV-2 biology, phylogenetics including maximum liklihood and moduli spaces of trees, evolutionary models including quasispecies and Wright-Fischer, Itô integrals for stochastic odes (used to model human interactions), quantum computing (used for protein interactions), fluid dynamics (used for aerosols), zero knowlege proofs (as related to privacy). I am open to other ideas and outside speakers.
Here is a short summary of how the listed papers relate to math and to the virus:
SARS... by the numbers gives a quick overview of the virus. It consists of a single strand of 30000 bases of ribonucleaic acid decorated with around 1000 beadlike proteins (each bead is four copies of a protein called N for nucleocapsid) and surrounded by a bilipid membrane which has about 100 spikes (each spike is three copies of a protein called S for spike each of which is about half embedded in the membrane). To get a feel for the sizes of things the RNA molucule forms a spiral strand with radius 1/10^9 m and length 1/10^5 m and the N-beads make it about 3 times as thick. The entire virus is roughly round, 1/10^7 m across and 5/10^9 m thick with spikes of height similar to their spacing of 1/10^8 m while the cells they infect are about 1/10^5 m and a person has 10^9 to 10^11 such target cells. This means that the beaded RNA would just fit neatly tethered to the inner surface but could also be balled up in the interior where it fills under 1/100 of the volume. The N proteins have regions (domains) which bind the RNA and others which bind each other and the paper The SARS-CoV-2 nucleocapsid... gives a simple mathematical model to predict a phase transition it shows which also informs how the RNA is organized within the virus.
The virus enter cells by binding between the spikes and the protein ACE2 in the host cell membrane. This makes the spikes a tempting drug target as well as a worry for mutations evading vaccines. The papers Energetics Based... and Drug Repositioning... discuss attempts to discover such binding computationally either using a quantum computer or taking inspiration from such attempts to improve the speed as reviewed in Prospects ... for problems detailed in Mixed.... Something similar is done experimentally in Deep Mutational.... Entry takes around 10 minutes.
Once inside some proteins can be made immediately from the RNA template and many more copies are made with a mix of different new smaller subtemplates copied from the original and degradation of long proteins made from them into several smaller functional ones resulting in around 24 different proteins. The Virus information repository has details of what is known. In the process there is a frame shift expedited by a pseudoknot structure in the RNA which is explored in To Knot or not... using some graphs and A Mathematical.... The replicating virus wind up in specialized compartments of the Golgi apparatus of the cells where they bud off and migrate to the surface to exit without killing the infected cell. This process takes about 8 hours and a typical infected cell sheds several hundred virus.
Studying both spread and evolution of the virus involves taking RNA sequence data and building a phylogenetic tree that is likely to have produced them which usually means a maximum liklihood structure. Computationally this can be slow as in A Short Proof... and there are provably fast approximations. One idea here is the observation that the spaces of metric trees (see Moduli Spaces... and Tropical Geometry...) are also the real positive versions of moduli spaces of marked rational curves and a Gromov nonpositive curvature result there suggests a way to average trees. Once obtained these phylogenies can be used to figure out how the virus is evolving.
If the sequences are all from one patient this evolution can often be modeled as a quasispecies (see Quasispecies theory..., Viral Quasispecies... or Rigorous...) in which the distribution of sequences seen is modeled as an eigenvector of a replication/mutation matrix. One interesting result here is the frequent observed/conjectural appearance of a sharp threshold for the mutation rate beyond which the virus would lose its information and disappear (the mutational catastrophy idea of Eigen) and is related to the eigenvector structure of mutation operators. This has been proposed as a way to attack diseases which are close to the threshold. Covid has among the largest genomes (30000 bases) of any RNA virus which means it requires a lower mutation rate for this catastrophy and relatedly one of its proteins (called ExoN) helps it correct errors and yields an error rate of 1/10^6 per site per cycle and without it the virus fares poorly as in High Mutation.... One researcher (Prof Inoue) gave several interviews claiming this effect could explain the retreat of the delta variant from Japan. The stability of multiple strains is also considered via bifurcation analysis in Some Stability... and Theoretical Conditions.... A second relevant model is the Wright-Fisher model of genetic drift which models loss of variation due to population size. The combination of these suggests trying to understand which nearby mutations are more or less viable. The paper Simple Quasi... claims that genomes with more irrelevant mutations (called a flat mutation landscape- hence: survival of the flattest) are favored and this could have an eigenvector explanation and help explain the emergence of omicron which is considered in SARS-CoV-2 infection... and Competition for Dominance... where the possible role of either mutations in the correcting protein ExoN or a flattened mutation landscape from the patient's weakened immune system (due to HIV) could have driven the observed rapid mutation. The framework has also been used to try to predict future mutations as in Patterns of Volatility... and Deep Mutational.... Another possible origin of omicron is considered in Evidence for....
Maximum liklihood phylogenies also help understand the past spread as with Viral genome... which traces an outbreak to an ill-advised rose garden event and The Molecular... which tracks the first appearance of the virus and posits that there were likely around 10 separate infections from animals (zoonotic) in the market before two took hold. Between species evolution is also different due to a bigger role for exchange between different virus as in Predicting Mammalian ....
Studying the spread between humans involves the basic model assuming people are identical and perfectly mixing as in Mathematical... yielding the famous effective reproduction number and debates about herd immunity. Improvements to this model can include more realistic inhomogeneities in interaction yielding the idea of superspreading and an improved contact tracing approach in Implications of..., The Effectiveness... and Dynamic Graph.... There can also be feedback with people modulating social activity based on percieved threat levels but the paper Stochastic Social Behavior... posits that much of the peak and plateau structure seen in rates of infection can be explained more simply using the stochastic ode model of human behavior familiar from market analysis.
Attempts to trace in real time using cell phone data (which also help studies of interaction behavior) by warning people of possible exposure have been embraced in some societies and not others with provable data security a major mathematical issue as discussed in Privacy-Preserving... and Mind the GAP....
The issue of how the virus is usually spread has also been interesting with surfaces, large droplets and aerosols all contending for importance. The papers Fluid Dynamics... and The Dispersion ... find that a fairly simple analysis of particle flow with gravity suggest three behavior regimes depending on particle size. This in turn informs mitigation measures from appropriate distances, sanitization and Mask-Ematics... to HVAC upgrades and moving events outdoors.