7 summer solstice2012-a cognitive heuristic model of epidemics
1. A Cognitive Heuristic model for
Epidemics Modelling
A. Guazzini*
Department of Psychology, University of Florence
*: CSDC, Centre for the study of Complex Dynamics,
University of Florence, Italy
Contacts: andrea.guazzini@complexworld.net Webpage: http://www.complexworld.net/
2. A Cognitive Heuristics model for Epidemiology
Compelling features in modeling epidemics
• Social structure.
• Viral dynamics.
• Psychological and Cognitive effects.
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
3. A Cognitive Heuristics model for Epidemiology
The Classical Modelling of Epidemics
• The simplest models of epidemics correspond to percolation problems on a social network.
• The two key ingredients are the probability of infections and the viral dynamics.
• The simplest viral dynamics are SIS and SIR.
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
4. A Cognitive Heuristics model for Epidemiology
Are we still alive?
• In spite of the scale-free social structure, and long-range connections, we are still alive.
• Prophylaxis, fast intervention and education are valid in preventing epidemics.
• How can we include these elements in a simple model?
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
5. A Cognitive Heuristics model for Epidemiology
Role of Perception, Alarmism and Prejudice
(i.e.The cognitive Strategy)
• We are able to modify our behavior, either lowering the probability of infection or reducing contacts.
• These modifications are triggered by the alarm level and perception of a danger.
• Both local and global information became in such scenarios fundamental.
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
6. A Cognitive Heuristic model for Epidemiology
Standard modeling of Epidemics
Epidemic diffusion is usually modeled by means of spreading processes acting
within networks with a given (frequently complex) topology.
Such approaches have proven to be quite effective for the forecasting of
“simple/typical” diseases, such as the seasonal flu.
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
7. A Cognitive Heuristic model for Epidemiology
Cognitive Epidemics Modeling
fundamental hypothesis
A- Homogeneous Vs Multilayer/Nested/Multi-scale representation of the Network.
Rigid and Fixed Unweighted Dynamical and Rewiring Weighted
Symmetrical Lattice Like Networks and Asymmetrical Networks
Topology affects:
- Spreading of Viruses, Information, Money and Strategies
- Economical aspects such as the “Value of an Encounter”
- The selection and reproduction of the agents/strategies
Time evolution of number of
infected agents of an classical
“SIR” model on different
networks topologies
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
8. A Cognitive Heuristic model for Epidemiology
Cognitive Epidemics Modeling
fundamental hypothesis
B- “Rigid” and “Passive” nodes Vs “Smart” and “Adapting” agents
Encoding
A coherent and ecological approach to make an
agent cognitive should consider:
Decision
Making
- A bounded memory/knowledge
- An economic principle driving the learning
Environment Action - An evolution/diffusion of the (best) strategies
Learning
Knowledge A Cognitive Agent should provide:
Exp. Gain - Sensitivity to the environmental conditions
Decision Making
- Spontaneous evolution of new strategies
Exp. Risk - Adaptive and coherent behaviors
Encoding
Cognitive
Heuristic
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
9. A Cognitive Heuristic model for Epidemiology
Cognitive Epidemics Modeling
fundamental hypothesis
C- Multiple Time Scaling of the Epidemics Phenomena
- The typical Timescale of the Virus depends on:
- Infectious rate
(v) - Death rate
⌧i - Mutation rate
- Spontaneous infectious rate, etc..
- The Timescale of the Agents
- Learning dynamics,
(a) - Strategies evolution,
⌧i - Reproduction,
- Lifetime, etc ...
- The Timescale of the Network
- Information spreading,
(n)
⌧i - Diffusion rate of the epidemic
- Economical cycles, etc....
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
10. A Cognitive Heuristic model for Epidemiology
A new operative framework for the modeling of Human Cognitive Heuristics:
The tri-partite model
Reaction time
Module I Flexibility
Unconscious knowledge
perceptive and attentive processes
Cognitive costs
Relevance Heuristic
Module II
Reasoning
Goal Heuristic
External Recognition Heuristic
Solve Heuristic
Data
Module III
Learning
Behavior
Evaluation Heuristic
The minimal structure of a Self Awareness
cognitive agent
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
11. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe for the
epidemics modeling
The Environment
- Topology of the network (i.e. Weighted directed Random network)
- Viruses’ Features (e.g. Infectious Rate, Death Rate, Spontaneous Infectious Rate)
- Economical Features (e.g.Value Function, Gain Function)
- Informational Features (e.g. Media!!)
The Agent
- Bounded Knowledge/Memory
- A function of fitness
- Adaptive Cognitive Strategy of decision making
The Timescaling
- Encounters/Infection Phase (i.e Decision Phase)
- Economical Phase (i.e Fitness Estimation Phase)
- Learning/Genetic Phase (i.e Reproduction phase)
Time
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
12. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe for the The Environment
epidemics modeling
Topology of the network Viruses’ Features
%% PHASE 0: Network Structure
Topology=rand(N,N); % Virus
Mean_connectivity=30; %N
Topology=Topology<Mean_connectivity/N; SIr=Prob(1); % Spontaneous infectious rate
Ir=Prob(2); % Infectious rate
for i=1:N,
for j=i:N, Dr=Prob(3); % Death rate
Topology(i,j)=Topology(j,i); Itime=#Steps; % Incubation time
end
end Etime=#Steps; % Expression time
Rtime=#Steps; % Resilience time
Weighted undirected Random network with k=30
Economical Features Informational Features
P ⇤ X
i Ci H1 = fA (
t t
Ii )
t
Encounter Value
Function Vet = e P ⇤ i
i ⇥ Ki
Where:
t The state of the subject i at time t
Where:
I i (1 if infected and 0 if sane)
⇤
Ci t t Functions that describe the
e Set the maximum possible gain (here 2) Total number of encounters made by i
fA , gA Media Behavior (Trustability)
Ki Degree of the node (connectivity) t
X X
⇤ ⇤
t⇤ ⇤
⌧ Ci =
Typical economical period (days)
⇤
=t t0 t⇤ =t0 j
Cij t
H2 = gA (Vet
t
)
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
13. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe for the The Agent
epidemics modeling
Fitness Function Bounded Knowledge/Memory
⇤
⇤ Ci t
Mij = t 1
Mij m1 + Ij (1
t
m1 )
Gain Function Gi = Vet
⇤K
i ˜t ˜t
H2 = H2 1
m2 + gA (Vet
t
)(1
⇤
m2 )
Where: Encounter X
⇤
˜t ˜t
H 1 = H1 1
m2 + fA (
t
Ii )(1
t
m2 )
Ki Degree of the node (connectivity) Ci Total number of encounters made by i
i
t
X X Iit
⌧ ⇤ Typical economical period (days) Ci =
⇤
t⇤
Cij
The state of the subject i at time t (1 if infected and 0 if sane)
Mij 2 (0, 1)
t
Memory Matrix of past encounters: 0-Safe 1-Dangerous
⇤
= t t0 t⇤ =t0 j m1 , m2 2 (0, 1) Agent Memory Factors (Past Encounters and MEDIA)
Adaptive Cognitive Strategy of decision making
Cognitive
CDNAt
˜t ˜t
i
The agent strategy is represented by a
vector (e.g. Cognitive DNA) where the
Pi|j = exp(Mij
t t
1 (i)
t
+ H1 2 (i)
t
+ H2 3 (i))
t
three evolving components weight the
three informational sources.
!
c
DN At = [ 1;
t
2;
t
3]
t
1 (i), 2 (i), 3 (i) are dynamically evolved by a Montecarlo Method:
i t t t
Where:
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
14. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe for the The Timescaling:
epidemics modeling Ht 1 - H t2 Encounters/Infection Phase
Pi|j = exp(Mij
t t
1 (i)
t ˜t
+ H1 2 (i)
t ˜t
+ H2 3 (i))
t
Pj|i = exp(Mji
t t
1 (j)
t ˜t
+ H1 2 (j)
t ˜t
+ H2 3 (j))
t
IF
t t t
Pi|j Pj|i <
i j
Encounter
t
2 (0, 1)
Possible Cases
(SIR Models)
Uniformly distributed random variable
A- Both the agents are expressing the disease
- The encounter is forbidden (e.g. the Gain is not increased)
- Memory Updating: The trustability factors (Mtij e Mtji) are increased (Trustable=0, Untrastable=1)
B- Both the agents are sane
- The encounter is possible (e.g. the Gain is always increased if the encounter happens)
- Memory Updating: The trustability factors (Mtij e Mtji) are decreased (Trustable=0, Untrastable=1)
C- Only one agent is Infective but not Expressing the disease
- The encounter is possible (e.g. the Gain is always increased if the encounter happens)
- Memory Updating: The trustability factor Mtij is decreased if i get no the infection, and is
increased alternatively (Trustable=0, Untrastable=1)
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
15. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe for the The Timescaling:
epidemics modeling Economical Phase
Sane
Infected Every Economical Temporal Step the following recipe is
applied to compute the agents’ “gain”
$
Expressing $
X P ⇤
i Ci
$
Encounter Value
Function Vet = e P ⇤
Resilient i ⇥ Ki
⇤
⇤ Ci
Ki Degree of the node (connectivity)
⇤
Gain Function Gi = Vet
⇤K
⌧ Typical economical period (days)
i
⇤
=t t0
⇤
Ci Total number of encounters made by i
t
X X
Ci =
⇤
t⇤
Cij Finally the agents are sorted with respect to their
t⇤ =t0 j “richness” (i.e. fitness)
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
16. A Cognitive Heuristic model for Epidemiology
A Social Cognition inspired recipe Timescales
The Timescaling:
(A) (SE) (R) (I)
for the epidemics modeling > > > ReproductionEvolution Phase
Reproduction Control Parameter: Birthrate R(B) Strategies Evol. Control Parameter: Crossing Over C (O)
(R) (R) (SE) An Uniformly distributed
8(i, j) : G(i,j) > M e(G ) Where Me is the Median 8 #s (i, j)
t
variable C(O) is generated
#s (i, j) = |( (R) ⇥(R(B) ) ) + R | IF (O) 1
t t (B)
C < c
DN A 3
=c DN A S(i,j) i
1 2
(R) Gaussian Noise with Mean=0 and SD=1
3
< C (O) <
3
c
DN AS(i,j) =c DN Aj
Births Standard Deviation
R(B) 2
#t (i, j) Number of sons of the couple (i,j) at time t
s
C (O)
>
3
c
DN AS(i,j) = Random
Death (Infection) Control Parameter: Deathrate R(D) Death (Aging) Control Parameter: Critical Age A(C)
t
(I)
8 i Given Ai Age of the agent i
(I) Average time duration
8 i : Ii =1 ⌧ of infection (A) Gaussian Noise with Mean A
(C)
and SD (A(C) )
t t
With probability P1 = R (D) The Agent Dies
IF Ai > (A)
Agent Dies
Where
(A)
= A(C)
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
17. A Cognitive Heuristic model for Epidemiology
Preliminary Results 5
Final Number of Infected (Seed=10)
Final Number of Infected (Seed=4)
Final Number of Infected
5
P=1
P=f(M)
100 P=f(H1)
P=f(H2)
P=f(M,H1,H2)
50
Final Number of Infected
25
10
5
0.1 0.2 0.4 0.8 1
P=1 Infectious Rate
P=f(M)
100 P=f(H1) Final Number of Infected (Seed=20)
5
P=f(H2)
P=f(M,H1,H2)
Final Number of Infected
50
P=1
P=f(M)
100 P=f(H1)
P=f(H2)
25 P=f(M,H1,H2)
10
5 50
0.1 0.2 0.4 0.8 1
Infectious Rate 25
10
5
Conditions: N=225, K=30, Death Rate=0.1, Tmax=1000 0.1 0.2 0.4
Infectious Rate
0.8 1
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
18. A Cognitive Heuristic model for Epidemiology
Preliminary Results 1000
Relaxing Time (Seed=10)
Relaxing Time (Seed=4)
1000
Time
500
P=1
200 P=f(M)
P=f(H1)
P=f(H2)
100
P=f(M,H1,H2)
0.1 0.2 0.4 0.8 1
Infectious Rate
Time
500
Relaxing Time (Seed=20)
1000
P=1
200 P=f(M)
Time
P=f(H1) 500
P=f(H2)
100
P=f(M,H1,H2)
P=1
0.1 0.2 0.4 0.8 1 200 P=f(M)
P=f(H1)
Infectious Rate 100
P=f(H2)
P=f(M,H1,H2)
Conditions: N=225, K=30, Death Rate=0.1, Tmax=1000 0.1 0.2 0.4
Infectious Rate
0.8 1
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
19. A Cognitive Heuristic model for Epidemiology
Average Gain (Seed=10)
Preliminary Results 20
P=1
P=f(M)
P=f(H1)
P=f(H2)
Average Gain (Seed=4) P=f(M,H1,H2)
20
P=1
Gain
10
P=f(M)
P=f(H1)
P=f(H2)
P=f(M,H1,H2)
4
2
1
0.1 0.2 0.4 0.8 1
Infectious Rate
Gain
10 Average Gain (Seed=20)
20
P=1
P=f(M)
P=f(H1)
P=f(H2)
P=f(M,H1,H2)
4
Gain
10
2
1
0.1 0.2 0.4 0.8 1 4
Infectious Rate 2
1
Conditions: N=225, K=30, Death Rate=0.1, Tmax=1000 0.1 0.2 0.4
Infectious Rate
0.8 1
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
22. A Cognitive Heuristic model for Epidemiology
A step forward: Some open problems
- Role of the network topology on the evolution of the system.
- Description of the Strategies evolution dynamics, with particular
attention toward the social segregation and the equilibrium “Mixtures”.
- Role of the Virus parameters on the equilibrium state of the system
- Role of the Media Trustability Functions (f() and g()) on the system
dynamics
- Real Vs Simulated scenarios.
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
23. Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
... and thank you for the attention!
24. A Cognitive Heuristic model for Epidemiology
Preliminary Results
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
25. A Cognitive Heuristic model for Epidemiology
Preliminary Results
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June
26. A Cognitive Heuristic model for Epidemiology
Preliminary Results
Summer Solstice 2012 & Biophys 2012
Arcidosso, 26-29 June