Epidemiology modeling with Stella CSCI 1210

Stochastic vs. deterministic  Suppose there are 1000 individuals and each one has a 30% chance of being infected:  Stochastic method: run the model on the right 1000 times  Deterministic method: 1000 * 30% = 300 get infected (Law of Mass Action)

Stella Stocks and Flows  A flow takes “stuff” out from a stock or puts stuff into a stock

Result of simple flow model

Simple Epidemic Flow models  A short-term illness with recovery and permanent immunity

Simple Epidemic Flow Models  Short-term lethal illness with no recovery or immunity  Examples: “Martian flu”, measles in Incas  Note the flow into a sink outside the model

Simple Epidemic Flow Models  Short-term illness with recovery and temporary immunity  Example: malaria

Filling out the model  These are dynamic models  The value of each stock depends only on the initial value and the flows over time  The flows depend on the assumptions and state of the model – this is what determines how the model works

The Infection process  Simplest model: small population in which everyone is in contact  Each sick person has a certain constant probability of infecting each susceptible person in one time unit  Size of infection flow depends on the number of sick people and the number of susceptibles.

Modeling infection in Stella  The thin arrows represent influences. Note that all the influences affect the rate of infection.  We leave out incubation for simplicity: everyone is either susceptible or ill.

Qualitative analysis of infection  When there are few sick people, there can be little infection  When nearly everyone is sick, there can be little infection  Maximum infection will occur when the population is between these cases  Eventually, everyone will get sick.

Results of simple SI model

A model with recovery and immunity  After recovery, people are neither susceptible nor ill  A certain fraction of ill people will recover each time period.  The rate of recoveries depends on the number of ill people.

Results of the SIS model

Infection and recovery rates

Effect of immunization  Reduces the initial number of susceptibles  This reduces the infection rate, but does not alter the recovery rate  If the infection rate is small enough, the disease will die out without becoming an epidemic (herd immunity).

Infection and recovery, with herd immunity

Results of immunization campaign

Notes on Herd immunity  Not necessary to vaccinate the entire population.  Even individuals who were not vaccinated share the benefits.

HIV  Human Immunodiciency Virus (HIV)  A retrovirus  Originated in Africa, probably in 20 th century  Descended from simian virus (SIV) which “jumped hosts”  Long, contagious incubation period

From HIV to AIDS  Virus attacks human immune system  Death is from opportunistic secondary infections, not HIV itself  Anti-retroviral drugs can slow the virus and prolong life.

AIDS and Africa  42 million HIV/AIDS cases worldwide  29 million cases in Africa  Origin of the virus  Anarchy in central Africa (Uganda, Rwanda, Congo) helps spread the disease

AIDS: the “Gay Plague”?  Initially, US AIDS cases were almost all in gay men  However, African AIDS cases are mostly heterosexual  More US heterosexual AIDS cases as time has passed  What gives?

A two-tier model  High-risk group initially contracts the disease  Low-risk group does not have the disease  Slight interaction between groups  Two submodels proceed separately but have a weak coupling

Two-tier model

Results of the two-tier model

AIDS and the “Martian Flu”  HIV/AIDS is incurable, fatal, and has no known immunity  However, US AIDS epidemic may have peaked  So, “Martian Flu” model needs elaboration

Elaborated AIDS model  Add birth and death flows for susceptibles who do not get infected  Either die naturally, retire from sex, or enter monogamous relationships  Creates a situation similar to “herd immunity” model

Elaborated single-pool model

AIDS model with high riskiness

AIDS model with low riskiness