ECC3841 Chap.4 Contagion and Diffusion
Contagion and Diffusion
Contagion and diffusion connect network reachability to dynamic thresholds. Epidemic models compare transmission with recovery, while protection reduces the effective reproduction number. SIR removes recovered individuals from immediate susceptibility; SIS returns them to the susceptible state. Bass adoption combines an external innovation channel with internal imitation.
Simple contagion can spread after one exposure, whereas complex contagion requires reinforcement. The central skill is selecting the state process and interpreting a threshold without treating it as a full forecast.
What this chapter covers
- 01
Giant-component thresholds
- 02
Basic and effective reproduction numbers
- 03
Protection and herd thresholds
- 04
SIR compartment flows
- 05
SIS reinfection dynamics
- 06
Bass innovation and imitation channels
- 07
Simple versus complex contagion
- 08
Network seeding and intervention
Worked example · free
Compute a protection threshold
- +1Write the condition four times one minus v is below one.
- +1Divide to obtain one minus v below one quarter.
- +1Rearrange to v above three quarters.
- +1Interpret equality as the boundary, not strict decline.
Key terms
- Reproduction number
- The expected secondary cases generated under specified transmission and susceptibility conditions.
- SIR model
- A compartment model moving people from susceptible to infected to recovered.
- SIS model
- A compartment model returning recovered individuals to susceptibility.
- Bass model
- An adoption model combining external innovation and internal imitation.
- Complex contagion
- Spread that requires multiple exposures or threshold reinforcement.
Contagion and Diffusion FAQ
What does a reproduction threshold tell me?
It indicates whether a small infection tends to grow or contract near the relevant state. It does not by itself determine peak size, timing or final reach. Name the state transition, threshold comparison and model assumption before recommending a diffusion intervention.
When should I use SIS?
Use SIS when recovery does not confer lasting protection and individuals can become susceptible again. Name the state transition, threshold comparison and model assumption before recommending a diffusion intervention.
Why can Bass adoption be S-shaped?
Early adoption is driven by the external channel, imitation strengthens as adopters accumulate, and growth later slows as the remaining market shrinks. Name the state transition, threshold comparison and model assumption before recommending a diffusion intervention.
Is the highest-degree seed always best?
No. Clustering can make contacts redundant, and complex contagion may reward several nearby seeds. Match targeting to the transmission rule. Name the state transition, threshold comparison and model assumption before recommending a diffusion intervention.
Exam move
For every diffusion problem, write states and flows before updating stocks. Draw boxes for susceptible, infected and recovered people, then label the flow between boxes with the beginning-of-period stocks used in the formula. Calculate new infections and recoveries separately. Update all compartments only after both flows are known.
Check that shares remain nonnegative and sum to one; failure of either check signals an arithmetic error or an unsuitable time step. Distinguish SIR from SIS by asking what happens after recovery. For thresholds, identify the force creating new cases or adopters and the force removing or blocking them. Write the effective reproduction inequality and preserve the strict sign when converting it to a protection share.
Equality is the boundary. Separate reach from realized transmission: a giant component creates a possible route but does not guarantee that every edge transmits. Compare simple contagion, where one exposure can suffice, with complex contagion, where reinforcement is required. This difference changes whether a bridge or a cluster of seeds is useful. For Bass adoption, label external innovation and internal imitation separately.
Predict how increasing each parameter changes takeoff and curve shape before graphing. Do not call every S-shaped curve an epidemic result. Build a policy table that maps vaccination, distancing, faster recovery, advertising and seeding to the parameter or state each changes. Then state the objective: reduce early growth, lower the peak, shrink final reach or accelerate beneficial adoption.
On the final practice round, recompute one SIR step, one protection threshold and one early Bass interpretation without notes. Finish each with a regime statement and a limitation about heterogeneity, clustering or behavioral response.
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