RobertNachbar/CompartmentalModeling

RobertNachbar`EpidemiologyModeling`

NextGenerationMatrix

NextGenerationMatrix [eqns,vars,t,forceOfInfection,infectedCompartments]
	returns the next generation matrix for the system of differential equations eqns with independent variables vars in terms of t with specified forceOfInfection and infectedCompartments .

NextGenerationMatrix [modelDataAssociation,infectedCompartments]
	takes the equations, variables, time variable, and force of infection from the Association modelData .

NextGenerationMatrix […, All ]
	returns an association with the next generation matrix and intermediate vectors and matrices.

Details and Options



Examples

(10)

Basic Examples

(2)

Compute the next generation matrix for an SIR with vital demographics model:

In[1]:=

NextGenerationMatrix

[{

′



[t]Λ-μ[t]-β[t]λ[t],

′

ℰ

[t]β[t]λ[t]-ζℰ[t],

′

ℐ

[t]-γℐ[t]-μℐ[t]+ζℰ[t],

′

ℛ

[t]γℐ[t]-μℛ[t]},{,ℰ,ℐ,ℛ},t,{λ[t]ℐ[t]},{ℰ,ℐ}]

Out[1]=



βζΛ

μ(γζ+ζμ)

βζΛ

μ(γζ+ζμ)

,{0,0}

Use

All

to obtain the next generation matrix and the intermediate vectors and matrices:

In[1]:=

result=

NextGenerationMatrix

[{

′



[t]Λ-μ[t]-β[t]λ[t],

′

ℰ

[t]β[t]λ[t]-ζℰ[t],

′

ℐ

[t]-γℐ[t]-μℐ[t]+ζℰ[t],

′

ℛ

[t]γℐ[t]-μℛ[t]},{,ℰ,ℐ,ℛ},t,{λ[t]ℐ[t]},{ℰ,ℐ},All];

The next generation matrix is:

In[2]:=

MatrixForm@result["NextGenerationMatrix"]

Out[2]//MatrixForm=

βζΛ μ(γζ+ζμ)	βζΛ μ(γζ+ζμ)
0	0

The intermediate matrices are:

In[3]:=

MatrixForm/@Most@result//Normal//Column

Out[3]=

InfectedVariables

ℰ[t]

ℐ[t]

TransmissionTerms

βℐ[t][t]

TransferTerms

ζℰ[t]

-ζℰ[t]+(γ+μ)ℐ[t]

DiseaseFreeEquilibrium

[t]

ℰ[t]0

ℐ[t]0

ℛ[t]0

TransmissionJacobianMatrix

0	βΛ μ
0	0

TransferJacobianMatrix

ζ	0
-ζ	γ+μ

Scope

(1)



Options

(4)



Applications

(2)



Neat Examples

(1)



SeeAlso

Solve

▪

DynamicTransmissionModel

▪

ForceOfInfection

RelatedGuides

▪

Compartmental Modeling