BioModels Database logo

BioModels Database

spacer

BIOMD0000000294 - Restif2007_Vaccination_Invasion

 

 |   |   |  Send feedback
Reference Publication
Publication ID: 17210532
Restif O, Grenfell BT.
Vaccination and the dynamics of immune evasion.
J R Soc Interface 2007 Feb; 4(12): 143-153
Department of Veterinary Medicine, University of Cambridge, Cambridge Infectious Diseases Consortium, Madingley Road, Cambridge CB3 0ES, UK. or226@cam.ac.uk  [more]
Model
Original Model: BIOMD0000000294.xml.origin
Submitter: Lukas Endler
Submission ID: MODEL1012210000
Submission Date: 21 Dec 2010 04:00:08 UTC
Last Modification Date: 28 Apr 2014 15:31:18 UTC
Creation Date: 20 Apr 2010 23:54:23 UTC
Encoders:  Lukas Endler
set #1
bqbiol:isVersionOf Gene Ontology defense response, incompatible interaction
Gene Ontology entry of bacterium into host cell
set #2
bqmodel:isDerivedFrom BioModels Database Restif2006_Whooping_Cough
PubMed 460424
PubMed 460412
set #3
bqbiol:hasProperty Mathematical Modelling Ontology MAMO_0000046
set #4
bqbiol:hasTaxon Taxonomy Bordetella pertussis
Taxonomy Homo sapiens
set #5
bqbiol:hasVersion ICD A37
Notes

This is the model described in the article:
Vaccination and the dynamics of immune evasion.
Restif O, Grenfell BT. J R Soc Interface. 2007 Feb 22;4(12):143-53. PMID:17210532, doi:10.1098/rsif.2006.0167;
Abstract:
Vaccines exert strong selective pressures on pathogens, favouring the spread of antigenic variants. We propose a simple mathematical model to investigate the dynamics of a novel pathogenic strain that emerges in a population where a previous strain is maintained at low endemic level by a vaccine. We compare three methods to assess the ability of the novel strain to invade and persist: algebraic rate of invasion; deterministic dynamics; and stochastic dynamics. These three techniques provide complementary predictions on the fate of the system. In particular, we emphasize the importance of stochastic simulations, which account for the possibility of extinctions of either strain. More specifically, our model suggests that the probability of persistence of an invasive strain (i) can be minimized for intermediate levels of vaccine cross-protection (i.e. immune protection against the novel strain) and (ii) is lower if cross-immunity acts through a reduced infectious period rather than through reduced susceptibility.

This version of the model can be used for both the stochastic and the deterministic simulations described in the article. For deterministic interpretations with infinite population sizes, set the population size N = 1. The model does reproduces the deterministic time course. The initial values are set to the steady state values for a latent infection with strain 1 with an invading infection of strain 2 (I2=1e-06), 100 percent vaccination with a susceptibility reduction τ=0.7 at birth (p=1), and all other parameters as in figure 3 of the publication.

To be compatible with older software tools, the english letter names instead of the greek symbols were used for parameter names:

parameter symbol name
transmission rate β beta
recovery rate γ gamma
birth/death rate μ mu
rate of loss of natural immunity σ sigma
rate of loss of vaccine immunity σv sigmaV
reduction of susceptibility by primary infection θ theta
reduction of infection period by primary infection ν nu
reduction of susceptibility by vaccination τ tau
reduction of infection period by vaccination η eta

Originally created by libAntimony v1.4 (using libSBML 3.4.1)

Model
Publication ID: 17210532 Submission Date: 21 Dec 2010 04:00:08 UTC Last Modification Date: 28 Apr 2014 15:31:18 UTC Creation Date: 20 Apr 2010 23:54:23 UTC
Mathematical expressions
Reactions
Birth S (unvaccinated) Birth V (vaccinated) Death in S Death in V
Death in I1 Death in I2 Death in Iv2 Death in R1
Death in R2 Death in J1 Death in J2 Death in Rp
Primary Infection with strain 1 Primary Infection with strain 2 Primary Infection of V with strain 2 Recovery (I1)
Recovery (I2) Secondary Infection with strain 1 Secondary Infection with strain 2 Recovery (J1)
Recovery (J2) Recovery (Iv2) Loss of Immunity (R1) Loss of Immunity (R2)
Loss of Immunity (Rp) Loss of Immunity (V)    
Rules
Assignment Rule (variable: mu) Assignment Rule (variable: beta) Assignment Rule (variable: gamma) Assignment Rule (variable: sigma)
Assignment Rule (variable: sigmaV) Assignment Rule (variable: strain1_frac) Assignment Rule (variable: strain2_frac) Assignment Rule (variable: S_frac)
Assignment Rule (variable: V_frac) Assignment Rule (variable: R_1_frac) Assignment Rule (variable: R_2_frac) Assignment Rule (variable: R_frac)
Physical entities
Compartments Species
environment N S I1
I2 R1 R2
V Iv2 J2
J1 R  
Global parameters
mu life expectancy beta R0
gamma p tau theta
nu eta sigma sigmaV
infectious period (d) immune period (yr) vaccine immune period (yr) strain1_frac
strain2_frac S_frac V_frac R_1_frac
R_2_frac R_frac    
Reactions (26)
 
 Birth S (unvaccinated)  → [S];   {N}
 
 Birth V (vaccinated)  → [V];   {N}
 
 Death in S [S] → ;  
 
 Death in V [V] → ;  
 
 Death in I1 [I1] → ;  
 
 Death in I2 [I2] → ;  
 
 Death in Iv2 [Iv2] → ;  
 
 Death in R1 [R1] → ;  
 
 Death in R2 [R2] → ;  
 
 Death in J1 [J1] → ;  
 
 Death in J2 [J2] → ;  
 
 Death in Rp [R] → ;  
 
 Primary Infection with strain 1 [S] → [I1];   {J1} , {N}
 
 Primary Infection with strain 2 [S] → [I2];   {J2} , {N} , {Iv2}
 
 Primary Infection of V with strain 2 [V] → [Iv2];   {J2} , {N} , {I2}
 
 Recovery (I1) [I1] → [R1];  
 
 Recovery (I2) [I2] → [R2];  
 
 Secondary Infection with strain 1 [R2] → [J1];   {N} , {I1}
 
 Secondary Infection with strain 2 [R1] → [J2];   {N} , {I2} , {Iv2}
 
 Recovery (J1) [J1] → [R];  
 
 Recovery (J2) [J2] → [R];  
 
 Recovery (Iv2) [Iv2] → [R];  
 
 Loss of Immunity (R1) [R1] → [S];  
 
 Loss of Immunity (R2) [R2] → [S];  
 
 Loss of Immunity (Rp) [R] → [S];  
 
 Loss of Immunity (V) [V] → [S];  
 
Rules (12)
 
 Assignment Rule (name: mu) mu = 1/l_e
 
 Assignment Rule (name: beta) beta = R0*(gamma+mu)
 
 Assignment Rule (name: gamma) gamma = 365/tInf
 
 Assignment Rule (name: sigma) sigma = 1/tImm
 
 Assignment Rule (name: sigmaV) sigmaV = 1/tImm_V
 
 Assignment Rule (name: strain1_frac) strain1_frac = (I1+J1)/N
 
 Assignment Rule (name: strain2_frac) strain2_frac = (I2+J2+Iv2)/N
 
 Assignment Rule (name: S_frac) S_frac = S/N
 
 Assignment Rule (name: V_frac) V_frac = V/N
 
 Assignment Rule (name: R_1_frac) R_1_frac = (R1+R)/N
 
 Assignment Rule (name: R_2_frac) R_2_frac = (R2+R)/N
 
 Assignment Rule (name: R_frac) R_frac = R/N
 
 environment Spatial dimensions: 3.0  Compartment size: 1.0
 
 N
Compartment: environment
Initial amount: 1.0
 
 S
Compartment: environment
Initial amount: 0.0588235
 
 I1
Compartment: environment
Initial amount: 0.00176967
 
 I2
Compartment: environment
Initial amount: 1.0E-6
 
 R1
Compartment: environment
Initial amount: 0.439407
 
 R2
Compartment: environment
Initial amount: 0.0
 
 V
Compartment: environment
Initial amount: 0.9
 
 Iv2
Compartment: environment
Initial amount: 0.5
 
 J2
Compartment: environment
Initial amount: 0.0
 
 J1
Compartment: environment
Initial amount: 0.0
 
 R
Compartment: environment
Initial amount: 0.0
 
Global Parameters (22)
 
  mu
Value: NaN   (Units: per_year)
 
 life expectancy
Value: 50.0   (Units: years)
Constant
 
  beta
Value: NaN   (Units: per_year)
 
 R0
Value: 17.0   (Units: dimensionless)
Constant
 
  gamma
Value: NaN   (Units: per_year)
 
 p
Value: 1.0   (Units: dimensionless)
Constant
 
 tau
Value: 0.7   (Units: dimensionless)
Constant
 
 theta
Value: 0.5   (Units: dimensionless)
Constant
 
 nu
Value: 0.5   (Units: dimensionless)
Constant
 
 eta
Value: 0.5   (Units: dimensionless)
Constant
 
  sigma
Value: NaN   (Units: per_year)
 
  sigmaV
Value: NaN   (Units: per_year)
 
 infectious period (d)
Value: 21.0   (Units: days)
Constant
 
 immune period (yr)
Value: 20.0   (Units: years)
Constant
 
 vaccine immune period (yr)
Value: 50.0   (Units: years)
Constant
 
  strain1_frac
Value: NaN   (Units: dimensionless)
 
  strain2_frac
Value: NaN   (Units: dimensionless)
 
  S_frac
Value: NaN   (Units: dimensionless)
 
  V_frac
Value: NaN   (Units: dimensionless)
 
  R_1_frac
Value: NaN   (Units: dimensionless)
 
  R_2_frac
Value: NaN   (Units: dimensionless)
 
  R_frac
Value: NaN   (Units: dimensionless)
 
Representative curation result(s)
Representative curation result(s) of BIOMD0000000294

Curator's comment: (updated: 10 Jan 2011 23:17:27 GMT)

Reproduction of figure 3 from the original publication using Copasi 4.6 for deterministic simulation.

For each value of tau, the model was started with the steady state values for a latent infection with strain 1. As described in the article, invasion of strain 2 was simulated by using an initial value of 1e-06/N for strain 2.

The parameters plotted are:

V : V_frac

strain 1 : strain1_frac

strain 2 : strain2_frac

spacer
spacer