Signature
lambda_1 = beta_11 * I_1 / N_1 + beta_12 * I_2 / N_2
| Inputs | Definition | Unit |
|---|---|---|
beta_11 | Transmission rate from infectious people in group 1 to a susceptible person in group 1, applied to the infectious proportion of group 1 | per day |
I_1 | Number of infectious people in group 1 | people |
N_1 | Total number of people in group 1, above zero | people |
beta_12 | Transmission rate from infectious people in group 2 to a susceptible person in group 1, applied to the infectious proportion of group 2 | per day |
I_2 | Number of infectious people in group 2 | people |
N_2 | Total number of people in group 2, above zero | people |
lambda_1 | Rate at which each susceptible person in group 1 is infected | per day |
|---|
Function
Compartment flow balance function of a compartmental model
Maps the numbers of people in the compartments of a population model, the flows between compartments per unit time and any inflow from or outflow to outside the model to the rate of change of each compartment, or, after a time step of length dt, to the next set of compartment counts. The flows can be fixed rates times compartment sizes, as in a static model, or can depend on the number infectious through the force of infection, which makes the model dynamic and nonlinear. A Markov cohort update (HE-FN-MM-001, HE-FM-MM-001) applies one fixed transition matrix to a closed cohort, the agent-based update HE-FN-ABM-001 with the per-contact infection probability HE-FM-ABM-001 follows individuals instead of pools, and a per-cycle probability from a constant rate follows HE-FM-TP-001. Reproduction numbers of the SIR model and of stratified models are in HE-FN-BRN-001 (HE-FM-BRN-001 to HE-FM-BRN-005 and HE-CF-BRN-001), and the final size of an epidemic and R_0 estimated from it are in HE-FN-ATR-001, HE-CF-ATR-001 and HE-FM-ATR-005, so none of them is restated here. Notation follows the Compartmental Model article.
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Implementations
Excel
Two-group force of infection on group 1 from named cells
With the transmission rates in TransWithin1 and TransFrom2To1, the infectious counts in Infectious1 and Infectious2 and the group sizes in PopGroup1 and PopGroup2, the cell returns the force of infection on group 1, held in ForceGroup1. Swapping the group indices in a second cell gives the force on group 2.
=TransWithin1*Infectious1/PopGroup1+TransFrom2To1*Infectious2/PopGroup2
Assumptions
Random mixing within each group and a transmission matrix fixed over the period
People within a group are alike and mix at random, so only the infectious proportion of each group matters, and the matrix of beta_ab stays fixed over the period modelled. Pitman and colleagues ask published models to tabulate the mixing matrix and give the type of mixing assumed. Rows index the group being infected and columns the group of the infectious person, so beta_12 and beta_21 are different entries.
Worked examples
Illustrative children and adults with mostly within-group transmission
Illustrative inputs. In a group of 2,000 children with 40 infectious and a group of 8,000 adults with 60 infectious, a within-child transmission rate of 0.5 and an adult-to-child rate of 0.2 per day give a force of infection on children of 0.01 plus 0.0015, or 0.0115 per day.
beta_11 = 0.5; beta_12 = 0.2; I_1 = 40; N_1 = 2000; I_2 = 60; N_2 = 8000; lambda_1 = 0.0115
Random mixing reduces the two-group force of infection to the single-group form
With beta of 0.3 and beta_1b set to 0.3 times N_b/N, that is 0.06 and 0.24, the same groups give lambda_1 of 0.003, the single-group value 0.3 times 100/10,000 of the article's day 1, a limiting case (derived).
beta_11 = 0.06; beta_12 = 0.24; I_1 = 40; N_1 = 2000; I_2 = 60; N_2 = 8000; lambda_1 = 0.003
Common errors
Group infectious counts divided by the whole population instead of the group size
Dividing I_b by N rather than N_b understates the force on the group whose own prevalence is high. In the illustrative children and adults example it gives 0.0032 instead of 0.0115 per day, about 72% too low, and in the random-mixing example 0.00168 instead of 0.003 (computed here for illustration).
Sources
Hethcote on the force of infection in an SEIR model with age groups
Hethcote HW. The mathematics of infectious diseases. SIAM Review. 2000;42(4):599-653. Section 6.1, Formulation of the SEIR Model with Age Groups (pages 634 and 635), which assumes contact rates that are constant on each age interval and writes, in equation 6.2, the force of infection on age group i as lambda_i = b_i times the sum over groups j of b~_j i_j, with i_j the infectious people of group j as a fraction of the whole population. The section does not print the full-matrix form lambda_a = sum over b of beta_ab I_b/N_b; the row derives Hethcote's separable form as a special case of it.
Pitman and colleagues on reporting the mixing matrix
Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. Dynamic transmission modeling. A report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health. 2012;15(6):828-834. Best practice V-6, which asks models that use differential equations to provide them, to tabulate all initial values and parameters, including the mixing matrix, and to supply details of the type of mixing considered.
Canonical Identity
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