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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/
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
A Cognitive Heuristic model for Epidemiology



                                                                                                                                           Encounters (Seed=10)
       Preliminary Results                                                                                     100


                                                                                                                                                                               P=1
                                                                                                                                                                               P=f(M)
                                                                                                                                                                               P=f(H1)
                                                                                                                                                                               P=f(H2)
                                                                                                                                                                               P=f(M,H1,H2)

                                 Encounters (Seed=4)
             100




                                                                                                  Encounters
                                                                    P=1                                         50

                                                                    P=f(M)
                                                                    P=f(H1)
                                                                    P=f(H2)
                                                                    P=f(M,H1,H2)
                                                                                                                20


                                                                                                                10
Encounters




                                                                                                                 0.1                 0.2                 0.4            0.8                        1
                                                                                                                                                   Infectious Rate
              50                                                                                                                                 Encounters (Seed=20)
                                                                                                                             100


                                                                                                                                                                                    P=1
                                                                                                                                                                                    P=f(M)
                                                                                                                                                                                    P=f(H1)
                                                                                                                                                                                    P=f(H2)
                                                                                                                                                                                    P=f(M,H1,H2)




                                                                                                                Encounters
              20
                                                                                                                              50


              10



               0.1         0.2             0.4                0.8                   1                                         20

                                     Infectious Rate
                                                                                                                              10



             Conditions: N=225, K=30, Death Rate=0.1, Tmax=1000                                                                0.1         0.2                 0.4            0.8                      1
                                                                                                                                                      Infectious Rate



                                   Summer Solstice 2012 & Biophys 2012
                                                       Arcidosso, 26-29 June
A Cognitive Heuristic model for Epidemiology



                                                                                                                                     Victims (Seed=10)
     Preliminary Results
                                                                                                                20


                                                                                                                                                                      P=1
                                                                                                                                                                      P=f(M)
                                                                                                                                                                      P=f(H1)
                                                                                                                                                                      P=f(H2)
                              Victims (Seed=4)                                                                                                                        P=f(M,H1,H2)

          20




                                                                                                      Victims
                                                                P=1
                                                                                                                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
Victims




                                                                                                                                        Infectious Rate
          10                                                                                                                       Victims (Seed=20)
                                                                                                        20


                                                                                                                                                                 P=1
                                                                                                                                                                 P=f(M)
                                                                                                                                                                 P=f(H1)
                                                                                                                                                                 P=f(H2)
                                                                                                                                                                 P=f(M,H1,H2)




                                                                                            Victims
           4
                                                                                                        10



           2
           1
                                                                                                          4
           0.1          0.2           0.4                 0.8                    1
                                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
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
Summer Solstice 2012 & Biophys 2012
            Arcidosso, 26-29 June




... and thank you for the attention!
A Cognitive Heuristic model for Epidemiology




Preliminary Results




                      Summer Solstice 2012 & Biophys 2012
                                 Arcidosso, 26-29 June
A Cognitive Heuristic model for Epidemiology




Preliminary Results




                      Summer Solstice 2012 & Biophys 2012
                                 Arcidosso, 26-29 June
A Cognitive Heuristic model for Epidemiology




Preliminary Results




                      Summer Solstice 2012 & Biophys 2012
                                 Arcidosso, 26-29 June

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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
  • 20. A Cognitive Heuristic model for Epidemiology Encounters (Seed=10) Preliminary Results 100 P=1 P=f(M) P=f(H1) P=f(H2) P=f(M,H1,H2) Encounters (Seed=4) 100 Encounters P=1 50 P=f(M) P=f(H1) P=f(H2) P=f(M,H1,H2) 20 10 Encounters 0.1 0.2 0.4 0.8 1 Infectious Rate 50 Encounters (Seed=20) 100 P=1 P=f(M) P=f(H1) P=f(H2) P=f(M,H1,H2) Encounters 20 50 10 0.1 0.2 0.4 0.8 1 20 Infectious Rate 10 Conditions: N=225, K=30, Death Rate=0.1, Tmax=1000 0.1 0.2 0.4 0.8 1 Infectious Rate Summer Solstice 2012 & Biophys 2012 Arcidosso, 26-29 June
  • 21. A Cognitive Heuristic model for Epidemiology Victims (Seed=10) Preliminary Results 20 P=1 P=f(M) P=f(H1) P=f(H2) Victims (Seed=4) P=f(M,H1,H2) 20 Victims P=1 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 Victims Infectious Rate 10 Victims (Seed=20) 20 P=1 P=f(M) P=f(H1) P=f(H2) P=f(M,H1,H2) Victims 4 10 2 1 4 0.1 0.2 0.4 0.8 1 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

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