CS300b - Special Topics in Computer Science (Agent Modeling)

Diffusion of Innovations

Spontaneous Adoption Model:
- N is the number of agents.
- This model assumes a fixed probability ( $\alpha$ ) of individuals adopting an innovation per unit time.
- The change in the number of adopters ( $\Delta I$ ) is proportional to the product of $\alpha$ and the number of potential adopters:
  - $\Delta I = I_{t+1} - I_t = \alpha(N - I_t)$ .
- This model can be simulated in NetLogo or solved analytically.

Social Influence: The SI Model

The SI model incorporates social influence alongside spontaneous adoption.
- Individuals can become infected (adopt) through contact with infected individuals.
- The rate of infection is determined by the transmissibility rate ( $\tau$ ).
- The discrete-time SI model can be simulated in NetLogo.

Social Influence: The SI Model

The SI Model - Analytical Model

Because individuals meet at random, the probability of a susceptible person meeting an infected person is given by:

$Pr(S,I)=\frac{I}{N}\big(\frac{N-I}{N}\big)$

Therefore, the chance of infection is given by:

$Pr(infection)=\tau Pr(S,I)$

$Pr(infection)=\tau \frac{I}{N}\big(\frac{N-I}{N}\big)$

The dynamics of infections are given as follows:

$I_{t+1} = I_t + N \tau \frac{I_t}{N}\big(\frac{N-I_t}{N}\big)$

The SI Model - Analytical Model (simplified)

$I_{t+1} = I_t + \tau I_t\big(1-\frac{I_t}{n}\big)$

The SI Model - ABM

Transmissibility is caused by multiple neighbors who are infected at a rate of infection $\tau$ .

$Pr(infection)=1-(1-\tau)^n$

The SIS Model

The SIS (susceptible-infected-susceptible) model adds recovery to the SI model.
- Infected individuals recover with a fixed probability ( $\gamma$ ) per unit time.
- The SIS model explores the balance between infection and recovery.

The SIS Model

$I_{t+1}=I_t+\tau I_t\big(1-\frac{I_t}{N}\big)-\gamma I_t$

The SIS Model - Equilibrium

Dynamic equilibrium happens when the number of people getting infected equals the number of people getting recovered – $I_t = I_{t+1} = I$ .

$I = I + \tau I\big(1-\frac{I}{N}\big)-\gamma I$

$\boxed{\frac{I}{N} = 1 - \frac{\gamma}{\tau}}$

$R_0$ and Herd Immunity

$R_0 = \frac{\tau}{\gamma} > 1$

R-naught or basic reproduction number
Social and behavioral measures can change $R_0$ .

$R_0$ and Herd Immunity

$\frac{\tau}{\gamma}(1-V) > 1$

V is the proportion of vaccinated people.
This is the threshold vaccination rate for herd immunity:

$V^* = 1 - \frac{1}{R_0}$

The SIR Model

The SIR (susceptible-infected-recovered) model incorporates permanent immunity or removal.
- Recovered individuals are no longer susceptible to infection.
- The SIR model is often used to study epidemics with lasting immunity.

The SIR Model

$\Delta S = -\tau S\frac{I}{N}$
$\Delta I = \tau S\frac{I}{N} - \gamma I$
$\Delta R = \gamma I$

Flattening the Curve

The SIR model illustrates the concept of “flattening the curve”.
- Reducing transmissibility ( $\tau$ ) through interventions like social distancing can lower the peak infection rate.
- This prevents overwhelming healthcare systems and provides time for treatment development.

Flattening the Curve

Limitations of Simple Contagion Models

We must ackwnoledge the limitations of the presented models.
- Individual variation in transmission, adoption, and recovery is not considered.
- Models assume random interactions, ignoring population structure and mobility patterns.
- Cultural norms, socioeconomic factors, and identity are not accounted for.

Simple vs. Complex Contagion

Simple contagions, like diseases, spread through single contact.
Complex contagions require multiple contacts for adoption, often involving social reinforcement.
- This concept was introduced by Centola and Macy (2007).
- Examples include the spread of certain behaviors, beliefs, and innovations.

Compartment Models

The SI, SIS, and SIR models belong to a family of models known as compartment models
- The groups of individuals are compartmentalized in the different categories each model provides
- There are limitations!

W₂: Contagion

Week 2 – Contagion

Contagion

Contagion

Diffusion of Innovations

Diffusion of Innovations

The SI Model - Analytical Model

The SI Model - Analytical Model (simplified)

The SI Model - ABM

The SIS Model

The SIS Model

The SIS Model - Equilibrium

$R_0$ and Herd Immunity

$R_0$ and Herd Immunity

The SIR Model

The SIR Model

Flattening the Curve

Flattening the Curve

Limitations of Simple Contagion Models

Simple vs. Complex Contagion

Compartment Models

W2: Contagion

W₂: Contagion