MIT OpenCourseWareSession 2: SIRSb with Vaccination
MIT OpenCourseWare
Opening logistics 0:00
The session begins with routine housekeeping. People are asked to close files when they finish with them because a shared Dropbox folder can create conflicts and multiple versions. The work for the day is to review submitted models, discuss them, and then prepare for the next modeling task.
Risk model review 5:34
The model under review uses a basic SIR structure with deaths kept separate. It adds a perceived risk pathway built from indicated infection and death risk, then smooths that signal with two time constants before letting it affect contact rate. The discussion focuses on where the risk signal should come from and whether using cumulative deaths or net infection flow creates unit problems, negative values, and too many free parameters.
Risk responds unevenly 20:01
Risk perception rises faster than it falls. When real risk is higher than what people think, anxiety grows and people notice more warning signs. That is why COVID responses can tighten in days or weeks, yet take months to ease. The same asymmetry shows up in customer service, trust, and quality. A bad experience hurts more than a good one helps.
Exponential response limits 22:33
The contact-rate formula uses one sensitivity parameter and an exponential term. If the sensitivity is zero, the effect disappears and contact stays at its base value. If perceived risk is zero, the multiplier is one. If perceived risk is high, the multiplier drops quickly and cuts contact. The same form can misbehave when perceived risk goes negative, so the exponential can be too strong for some ranges.
Modeling illness duration 39:30
Rahmandad explains that the infectious stock should drain by a total outflow based on its average illness duration, not by a separate recovery stock. That total exit is then split into death and recovery using infection fatality rate: the fraction who die goes to death, and the rest goes to recovery. He says this keeps infection fatality rate and illness duration from getting mixed together. If death and recovery take different times, the model needs separate disease stages or separate stocks. He closes this critique by stressing unit consistency before you build the model.
Vaccination and risk 48:00
He then turns to Russ’s vaccination model, where vaccinated and unvaccinated people each move through susceptible, infected, and recovered states. The main concern is the risk-perception term, since if perceived risk is not kept between 0 and 1, the factor one minus perceived risk can make infection negative. He checks the infection flow and confirms that the model counts infections from both vaccinated and unvaccinated people. He also accepts the weighted infectivity setup, where vaccinated cases contribute less to transmission, as a consistent way to handle the two groups.
Trend and delay 58:00
The delayed COVID statistics are used to track a slope, but fixed delays are discouraged. They hide many stock variables inside Vensim and can create sharp, hard-to-read behavior. The same trend can be built more cleanly with the built-in trend function or with a guarded division such as XIDZ, which returns zero when the denominator is zero.
Risk and vaccination 1:02:02
Perceived risk should follow the absolute level of danger, not the sign of the trend. A sigmoid centered at zero can still push interaction above the baseline or stop infection only partway, so the form is too loose. The vaccination flow also misses a time-to-vaccinate term and uses a fraction in a way that would vaccinate everyone unless the flow shuts off as the target level is reached.
Vacination gaps and stock flow 1:18:01
The discussion works through how to move only the people who are still willing to be vaccinated, not the whole susceptible group. The key step is to compare the total number vaccinated with the total population, then turn that difference into a fraction of the population that is still open to vaccination. That fraction should drive the flow from susceptible to vaccinated. Several versions fail because they tie the rate to the susceptible fraction alone or let the rate go negative when the two fractions do not match. Hazhir also rejects formulas that treat infected vaccinated people as if they were simply part of the general vaccinated pool. They still matter for the count of people who have already taken the vaccine, even if they later become infectious.
Risk perception setup 1:37:43
Gerardo’s model is opened and the risk perception sector is checked first. It takes infection rate and death rate, turns them into reference values, and then builds an attended risk perception from them. The point is that people judge whether current infection or death is high or low by comparing it with a baseline. That baseline can change over time, so the model lets people adjust their norms instead of keeping them fixed.
Vaccination structure 1:49:04
The vaccination section assumes everyone is willing to be vaccinated, and it treats vaccinated and unvaccinated people as equally infectious unless another factor is added. The more careful issue is waning immunity. If vaccinated recovered people are not separated from the general recovered group, the model can hide differences in how long vaccine immunity and natural immunity last. That matters when vaccine protection is longer, shorter, or very different from natural protection.
Waning immunity choices 1:58:31
Hazhir says the model can treat vaccine and natural immunity in different ways, depending on the story you believe. If vaccination and recovery both matter, you can move people from vaccinated recovered to susceptible unvaccinated, or keep a separate waning time for vaccinated recovered people. He says the biology is not settled, so the simpler stochastic story is usually the one to use.
Simple model design 2:04:01
He then lists the core parameters: risk perception, infection fatality, vaccine effect on fatality, contact and infectivity, recovery and waning times, and willingness to vaccinate. He says about a dozen parameters is enough, and some can be fixed from outside data. Vaccine efficacy comes through reduced infection, reduced death, and lower transmission to others.
Model structure checks 2:17:00
Rahmandad steers the group toward a simple retrofit model with three building cohorts, such as 1950s, 1960s, and 1970s stock, even though more detailed data could support finer splits. He warns that the model should sit in equilibrium only when there is no retrofit investment, because once retrofits are changing efficiency over time, a true equilibrium is hard to define. He then lays out the key design choice: either let one aging chain carry buildings through time, or use separate cohort stocks and flows and keep aging as a separate coflow. He says either choice is acceptable here, but the cohort route helps you see what memory the system needs and which changing quantities must be stored in stocks.
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