Hello Products • macpan2

Read Models

Here we read the target product model, prod, and its two factor models, si and age, into R.

prod = Compartmental(prod_dir)
si = Compartmental(si_dir)
age = Compartmental(age_dir)

The goal of this document is to multiply si and age to see if we can recover our target, prod.

Types of Variables

Each Model has the following set of variables.

si$variables$all()
#>            Epi
#> 1            S
#> 2            I
#> 3    infection
#> 4     recovery
#> 5 transmission
age$variables$all()
#>     Age
#> 1 young
#> 2   old
#> 3 aging
prod$variables$all()
#>             Epi   Age
#> 1             S young
#> 2             S   old
#> 3             I young
#> 4             I   old
#> 5     infection young
#> 6     infection   old
#> 7      recovery young
#> 8      recovery   old
#> 9             S aging
#> 10            I aging
#> 11 transmission

These variables are categorized into various types that distinguish their roles in the dynamics.

The state variables represent the nodes in the compartmental diagram.

si$variables$state()
#>   Epi
#> 1   S
#> 2   I
age$variables$state()
#>     Age
#> 1 young
#> 2   old
prod$variables$state()
#>   Epi   Age
#> 1   S young
#> 2   I young
#> 3   S   old
#> 4   I   old

The model definition also recognizes two subsets of these states – infectious and infected – which in these models are identical because there is no latency period.

si$variables$infectious_state()
#>   Epi
#> 1   I
age$variables$infectious_state()
#> [1] Age
#> <0 rows> (or 0-length row.names)
prod$variables$infectious_state()
#>   Epi   Age
#> 1   I young
#> 2   I   old

si$variables$infected_state()
#>   Epi
#> 1   I
age$variables$infected_state()
#> [1] Age
#> <0 rows> (or 0-length row.names)
prod$variables$infected_state()
#>   Epi   Age
#> 1   I young
#> 2   I   old

Note that the age model has no infection, and so these types of states are not present.

The flow variables represent the flows between states in the compartmental diagram.

si$variables$flow()
#>         Epi
#> 1 infection
#> 2  recovery
age$variables$flow()
#>     Age
#> 1 aging
prod$variables$flow()
#>         Epi   Age
#> 1 infection young
#> 2 infection   old
#> 3  recovery young
#> 4  recovery   old
#> 5         S aging
#> 6         I aging

And a subset of flows are those involved in infection.

si$variables$infection_flow()
#>         Epi
#> 1 infection
age$variables$infection_flow()
#> [1] Age
#> <0 rows> (or 0-length row.names)
prod$variables$infection_flow()
#>         Epi   Age
#> 1 infection young
#> 2 infection   old

But what about other variables that are neither states nor flows? We can list these other variables.

si$variables$other()
#>            Epi
#> 1            S
#> 2            I
#> 3    infection
#> 4     recovery
#> 5 transmission
age$variables$other()
#> Warning in age$variables$other(): 
#> There are no other variables in this model
#> except for state and flow variables.
#> NULL
prod$variables$other()
#>             Epi   Age
#> 1             S young
#> 2             S   old
#> 3             I young
#> 4             I   old
#> 5     infection young
#> 6     infection   old
#> 7      recovery young
#> 8      recovery   old
#> 9             S aging
#> 10            I aging
#> 11 transmission

We see that the only variables in the age model are states and flows. But we also see that the si and prod models have a transmission variable.

Multiplying Variables

We can recover each kind of variable in the prod model by operating on the factor models, si and age. The state variables in prod are a Cartesian product of the state variables in si and age.

cartesian(si$variables$state(), age$variables$state())
#>   Epi   Age
#> 1   S young
#> 2   I young
#> 3   S   old
#> 4   I   old
prod$variables$state()
#>   Epi   Age
#> 1   S young
#> 2   I young
#> 3   S   old
#> 4   I   old

The flow variables in prod are a union of two Cartesian products.

union_vars(
  cartesian(si$variables$flow(), age$variables$state()),
  cartesian(si$variables$state(), age$variables$flow())
)
#>         Epi   Age
#> 1 infection young
#> 2  recovery young
#> 3 infection   old
#> 4  recovery   old
#> 5         S aging
#> 6         I aging
prod$variables$flow()
#>         Epi   Age
#> 1 infection young
#> 2 infection   old
#> 3  recovery young
#> 4  recovery   old
#> 5         S aging
#> 6         I aging

The other variable in prod, transmission., can be obtained by taking the union of the other variables in si and age.

union_vars(si$variables$other(), age$variables$other())
#> Warning in age$variables$other(): 
#> There are no other variables in this model
#> except for state and flow variables.
#>            Epi
#> 1            S
#> 2            I
#> 3    infection
#> 4     recovery
#> 5 transmission
prod$variables$other()
#>             Epi   Age
#> 1             S young
#> 2             S   old
#> 3             I young
#> 4             I   old
#> 5     infection young
#> 6     infection   old
#> 7      recovery young
#> 8      recovery   old
#> 9             S aging
#> 10            I aging
#> 11 transmission

Putting this all together we have.

union_vars(
  cartesian(si$variables$state(), age$variables$state()),
  cartesian(si$variables$flow(), age$variables$state()),
  cartesian(si$variables$state(), age$variables$flow()),
  si$variables$other(),
  age$variables$other()
)
#> Warning in age$variables$other(): 
#> There are no other variables in this model
#> except for state and flow variables.
#>             Epi   Age
#> 1             S young
#> 2             I young
#> 3             S   old
#> 4             I   old
#> 5     infection young
#> 6      recovery young
#> 7     infection   old
#> 8      recovery   old
#> 9             S aging
#> 10            I aging
#> 11            S      
#> 12            I      
#> 13    infection      
#> 14     recovery      
#> 15 transmission
prod$variables$all()
#>             Epi   Age
#> 1             S young
#> 2             S   old
#> 3             I young
#> 4             I   old
#> 5     infection young
#> 6     infection   old
#> 7      recovery young
#> 8      recovery   old
#> 9             S aging
#> 10            I aging
#> 11 transmission

Multiplying Flows

si_flows = si$flows()
age_flows = age$flows()

si_flows$from_to_partition = "Age"
si_flows$from_flow_partition = "Age"
age_flows$from_to_partition = "Epi"
age_flows$from_flow_partition = "Epi"
prod_flows = rbind(si_flows, age_flows)
prod_flows
#>    from  to      flow       type from_partition to_partition flow_partition
#> 1     S   I infection per_capita            Epi          Epi            Epi
#> 2     I   S  recovery per_capita            Epi          Epi            Epi
#> 3 young old     aging per_capita            Age          Age            Age
#>   from_to_partition from_flow_partition to_flow_partition
#> 1               Age                 Age              Null
#> 2               Age                 Age              Null
#> 3               Epi                 Epi              Null
prod$flows()
#>    from  to      flow       type from_partition to_partition flow_partition
#> 1     S   I infection per_capita            Epi          Epi            Epi
#> 2     I   S  recovery per_capita            Epi          Epi            Epi
#> 3 young old     aging per_capita            Age          Age            Age
#>   from_to_partition from_flow_partition to_flow_partition
#> 1               Age                 Age              Null
#> 2               Age                 Age              Null
#> 3               Epi                 Epi              Null

Combining the Transmission Matrices

Combining the Derivations

Simulating the Resulting Product Model

Roadmap

Here we outline what development needs to take place in order to complete this document.

Add transmission matrix step to StandardExprs
Align transmission matrix specifications
Pass required indices for lining up the transmission matrix to the flow vector and state vector.

Scratch

prod$labels$state()
#> [1] "S.young" "I.young" "S.old"   "I.old"
prod$labels$flow()
#> [1] "infection.young" "infection.old"   "recovery.young"  "recovery.old"   
#> [5] "S.aging"         "I.aging"

si$variables$all()
#>            Epi
#> 1            S
#> 2            I
#> 3    infection
#> 4     recovery
#> 5 transmission
si$flows_expanded()
#>   from to      flow       type
#> 1    S  I infection per_capita
#> 2    I  S  recovery per_capita

age$variables$all()
#>     Age
#> 1 young
#> 2   old
#> 3 aging
age$flows_expanded()
#>    from  to  flow       type
#> 1 young old aging per_capita

prod$variables()
prod$flows_expanded()
UserExpr(prod)$expand_vector_expressions()
prod$expr_list()$print_exprs()

Simulating from the Product Model

Do the model definition by hand so that it works
Specify numerical inputs
Run the the simulation
Verify the results

Steps in Taking Model Products

We have a nested set of operations.

Cartesian product of state variables
Cartesian product of state and flow variables (once in each direction)
Matrix product of transmission matrix (lots of variations available here, for example Kronecker product)

A modeller may choose to do just 1, just 1-2, or 1-3.