# Infectious Disease Outbreak-Immunity durability and booster vaccination

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## WHAT IS IT?

This model allows students to examine the effect of immunity durability on the max daily cases and epidemic duration in a population. It helps students understand why booster vaccination is needed.

## HOW IT WORKS

Susceptible-infectious-recovered (SIR) model is used to build the present model.

The model starts with a population of susceptible people (green) and one infectious person (orange). The infectious person carries the virus strain with the defined initial transmission rate.

In each tick, each infectious person produces a new virus. The virus transmission rate either stays the same or mutates to become more or less transmissible at the defined mutation rate. The transmission rates are color codes.

The new virus may float 0.5 steps in a random direction and infect one of the susceptible people within a radius of 1.5 at its transmission rate. The virus loses its ability to infect people if no susceptible people are within the infection radius.

Infectious people remain contagious for six days. They either recover (become blue) or die at a mortality rate of 10%. No new people join this model to maintain the population density.

Users may set up the percentage of vaccinated people in the population. The present model assumes the vaccinated people will not be infected even if viruses mutate.

Vaccinated and recovered people lose immunity after the time defined by the "immunity durability." This model assumes that the durabilities of natural infection and vaccine-induced immunity are the same.

The booster vaccination extends the immunity durability of a defined percentage of vaccinated people. For example, when immunity durability last 30 days, a booster rate of 50% will extend 50% of the vaccinated people's immunity durability for another 30 days. The unboosted people will not be boosted anymore. Therefore, vaccinated people will decrease over time if the booster rate is less than 100%.

## HOW TO USE IT

First, choose the factors, such as population size, transmission rate, etc.

Click on

**Set up/Reset**, then**Run/Pause**. The model is set to stop when there are no infectious people.Observe the infection changes in the population in the plot and monitor.

Use

**Run one day**to run the model in a controlled way and collect day-by-day data.The people in the model are color-coded in two ways: SIR coloring and transmission rate coloring. The mode of SIR coloring displays the population based on people's health status (i.e., susceptible, infectious, or recovered); the mode of transmission rate coloring displays the population based on the virus transmissibility when the people get infected (i.e., cyan and brown indicate the people who are infected by the less transmissible strains while red and purple indicate the people who are infected by the highly transmissible strains). Use the button

**Switch color-coding mode**or the switch**Color-coding-modes**to switch between the two modes.Use the slider "Immunity-durability" to test how the durability of immunity affects the severity and duration of an outbreak.

Use

**"booster-rate"**to test how booster rate affect the severity and duration of an outbreak.

## THINGS TO TRY

Observe the prevalence of color-coded strains in the simulation window and plots to examine the prevalence of different strains over time.

Collect the data of the max daily cases and epidemic duration to examine the effect of durabilities of immunity in populations with different densities.

Collect the data of the max daily cases and epidemic duration to examine the effects of booster rates in a population.

## RELATED MODELS

Find this model series at http://3dsciencemodeling.com

- Infectious Disease Outbreak-Basic Phenomenon
- Infectious Disease Outbreak-Transmission and mortality
- Infectious Disease Outbreak-Population Comparison
- Infectious Disease Outbreak-HealthCare, Isolation and Quarantine
- Infectious Disease Outbreak-Social distancing
- Infectious Disease Outbreak-Vaccination
- Infectious Disease Outbreak-SEIR model
- Infectious Disease Outbreak-variants

## CREDITS AND REFERENCES

Dr. Lin Xiang (lin.xiang@uky.edu) created this model at the University of Kentucky in 2023. If you mention this model in a publication, we ask that you include the citations below.

Xiang, L. (2023). Infectious Disease Outbreak-Immunity durability and booster vaccination. Department of STEM Education, University of Kentucky, Lexington, KY.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 License. To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/.

## Comments and Questions

breed [persons person] breed [viruses virus] persons-own [ infectious-period susceptible infectious recovered vaccination p-contagious immunity immunity-duration ] viruses-own [ infection ;viral tranmission rate infection-period ;viral viability ] Patches-own [ ] Globals [max-daily-cases infection-number] ;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; ; Setup procedures ; ;;;;;;;;;;;;;;;;;;;;;;;;;;;; to setup clear-all set-default-shape viruses "c-virus-0" set-default-shape persons "person-0" set-patches setup-people add-an-infectious color-code reset-ticks end to set-patches ask patches [set pcolor 9 + random-float 1] end to setup-people ask n-of population patches [sprout-persons 1 [set size 0.9 set-susceptible set immunity 0 set p-contagious 0 set infectious-period 0 set immunity-duration 0] ] let vaccinated (population * Vaccination-rate / 100) ask n-of vaccinated persons with [susceptible = true and infectious = false] [set-vaccinated set immunity Initial-Transmission-Rate set p-contagious Initial-Transmission-Rate ] set infection-number 0 end to add-an-infectious ask one-of persons [set-infectious set p-contagious Initial-Transmission-Rate hatch-viruses 1 [ set size 0.5 set infection Initial-Transmission-Rate set-v-color ]] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; ; Go procedures ; ;;;;;;;;;;;;;;;;;;;;;;;;;;;; to go if ticks > 0 and not any? persons with [susceptible = false and infectious = true] [stop] move produce-viruses infect virus-ability recover immunization-vanishing color-code find-max-daily-cases plot-levels tick end to move ask viruses [right random 360 forward 0.5] end to produce-viruses ask persons with [susceptible = false and infectious = true] ;ask infectious people [hatch-viruses 1 ;produce a new virus [set size 0.5 ;set the size of a virus ifelse random 100 < Virus-mutation-rate ;set virus mutation rate [set infection 1 + random 100] ;infection mutates to a rate between 1 to 100% [set infection [p-contagious] of myself] ;not mutate set-v-color]] ;set virus color based on infection rate end to infect ask viruses [if any? persons with [susceptible = true] in-radius 1.5 ;if any susceptible people nearby [let infected-person one-of persons with [susceptible = true] in-radius 1.5 ;pick one of the susceptible people if random 100 < infection [move-to infected-person ask infected-person [set-infectious set p-contagious [infection] of myself set immunity [infection] of myself ;update immunity to the infected strain ] set infection-number infection-number + 1 die]] ;ask virus to die after infecting a person, update total of infected people ] end to virus-ability ask viruses [ifelse infection-period >= 2 [die] [set infection-period infection-period + 1] ] end to recover ask persons with [susceptible = false and infectious = true] [ifelse infectious-period >= 7 [ifelse random 100 < 10 [die] [set-recovered set infectious-period 0]] ;mortality equals 10% [set infectious-period infectious-period + 1] ] end ;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; ; supporting procedures ; ;;;;;;;;;;;;;;;;;;;;;;;;;;;; to set-v-color ; set colors of viruses (ifelse infection >= 80 [set color 15] infection < 80 and infection >= 60 [set color 127] infection < 60 and infection >= 40 [set color 117] infection < 40 and infection >= 20 [set color 35] infection < 20 [set color 75]) end to plot-levels ;; this creates creates the bar graph set-current-plot "People infected by variants with different transmissibilities" clear-plot plot-pen-down set-current-plot-pen "< 20%" plotxy 0 count persons with [p-contagious < 20 and p-contagious > 0] set-current-plot-pen "20%~39%" plotxy 1 count persons with [p-contagious < 40 and p-contagious >= 20] set-current-plot-pen "40%~59%" plotxy 2 count persons with [p-contagious < 60 and p-contagious >= 40] set-current-plot-pen "60%~79%" plotxy 3 count persons with [p-contagious < 80 and p-contagious >= 60] set-current-plot-pen ">= 80%" plotxy 4 count persons with [p-contagious >= 80] end to set-p-color ; set colors of people ask persons [ (ifelse p-contagious >= 80 and vaccination = false [set color 15] p-contagious < 80 and p-contagious >= 60 and vaccination = false [set color 127.5] p-contagious < 60 and p-contagious >= 40 and vaccination = false [set color 117.5] p-contagious < 40 and p-contagious >= 20 and vaccination = false [set color 36] p-contagious < 20 and p-contagious > 0 and vaccination = false [set color 75] p-contagious = 0 [set color 7] )] end to set-SIR-color ask persons [ (ifelse susceptible = true and infectious = false and recovered = false and vaccination = false [set color 56] susceptible = false and infectious = true and recovered = false and vaccination = false [set color 27] susceptible = false and infectious = false and recovered = true and vaccination = false [set color 97] susceptible = false and infectious = false and recovered = false and vaccination = true [set color 97])] end to color-code (ifelse Color-coding-modes = "Susceptible-infectious-recovered" [set-SIR-color] Color-coding-modes = "Variant transmission rates" [set-p-color]) end to find-max-daily-cases if count persons with [susceptible = false and infectious = true] > max-daily-cases ;Count the infectious.If it is greater than the current record of max daily cases [set max-daily-cases count persons with [susceptible = false and infectious = true]] ;update the max daily case end to immunization-vanishing ask persons with [susceptible = false and infectious = false] ;people are vaccinated or recovered [ifelse immunity-duration < Immunity-Durability ;if they have not been vaccinated/recovered for the immunity duration [set immunity-duration immunity-duration + 1] ; update the immunity duration [ifelse vaccination = true [ifelse random 100 < Booster-rate ;otherwise, at the chance of booster rate [set immunity-duration 0] ;get a booster--reset the duration [set-susceptible ; or become susceptiable set immunity 0 set p-contagious 0 set infectious-period 0 set vaccination false set immunity-duration 0]] [set-susceptible ; or become susceptiable set immunity 0 set p-contagious 0 set infectious-period 0 set vaccination false set immunity-duration 0]] ] end to set-susceptible set susceptible true set infectious false set recovered false set vaccination false end to set-infectious set susceptible false set infectious true set recovered false set vaccination false end to set-recovered set susceptible false set infectious false set recovered true set vaccination false end to set-vaccinated set susceptible false set infectious false set recovered false set vaccination true set color 97 end

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