public model
Model Identifier
BIOMD0000000294
Short description
Restif2007 - Vaccination invasion

This model is described in the article:

Restif O, Grenfell BT.
J R Soc Interface 2007 Feb; 4(12): 143-153

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 σ v 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)

To the extent possible under law, all copyright and related or neighbouring rights to this encoded model have been dedicated to the public domain worldwide. Please refer to CC0 Public Domain Dedication for more information.

Format
SBML (L2V4)
Related Publication
  • Vaccination and the dynamics of immune evasion.
  • Restif O, Grenfell BT
  • Journal of the Royal Society, Interface , 2/ 2007 , Volume 4 , pages: 143-153 , PubMed ID: 17210532
  • Department of Veterinary Medicine, University of Cambridge, Cambridge Infectious Diseases Consortium, Madingley Road, Cambridge CB3 0ES, UK. or226@cam.ac.uk
  • 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.
Contributors
Lukas Endler

Metadata information

is
BioModels Database MODEL1012210000
BioModels Database BIOMD0000000294
isDescribedBy
PubMed 17210532
isDerivedFrom
PubMed 460412
BioModels Database BIOMD0000000249
PubMed 460424
hasTaxon
hasProperty
Mathematical Modelling Ontology Ordinary differential equation model
Human Disease Ontology pertussis

Curation status
Curated


Tags
Name Description Size Actions

Model files

BIOMD0000000294_url.xml SBML L2V4 representation of Restif2007 - Vaccination invasion 49.77 KB Preview | Download

Additional files

BIOMD0000000294.m Auto-generated Octave file 8.55 KB Preview | Download
BIOMD0000000294.xpp Auto-generated XPP file 6.11 KB Preview | Download
BIOMD0000000294.svg Auto-generated Reaction graph (SVG) 46.08 KB Preview | Download
BIOMD0000000294-biopax2.owl Auto-generated BioPAX (Level 2) 27.14 KB Preview | Download
BIOMD0000000294-biopax3.owl Auto-generated BioPAX (Level 3) 43.27 KB Preview | Download
BIOMD0000000294.sci Auto-generated Scilab file 162.00 bytes Preview | Download
BIOMD0000000294.png Auto-generated Reaction graph (PNG) 203.24 KB Preview | Download
BIOMD0000000294_urn.xml Auto-generated SBML file with URNs 49.51 KB Preview | Download
BIOMD0000000294.pdf Auto-generated PDF file 239.85 KB Preview | Download
BIOMD0000000294.vcml Auto-generated VCML file 65.61 KB Preview | Download

  • Model originally submitted by : Lukas Endler
  • Submitted: 21-Dec-2010 04:00:08
  • Last Modified: 18-May-2017 12:16:20
Revisions
  • Version: 2 public model Download this version
    • Submitted on: 18-May-2017 12:16:20
    • Submitted by: Lukas Endler
    • With comment: Current version of Restif2007 - Vaccination invasion
  • Version: 1 public model Download this version
    • Submitted on: 21-Dec-2010 04:00:08
    • Submitted by: Lukas Endler
    • With comment: Original import of restif07
Legends
: Variable used inside SBML models


Species
Species Initial Concentration/Amount
R

Homo sapiens
0.0 item
S

Homo sapiens
0.0588235 item
I1

Homo sapiens ; Bordetella pertussis
0.00176967 item
I2

Homo sapiens ; Bordetella pertussis
1.0E-6 item
Reactions
Reactions Rate Parameters
gamma/(1-eta)*Iv2

gamma/(1-eta)*Iv2
gamma = NaN per_year; eta = 0.5 dimensionless
mu*(1-p)*N

mu*(1-p)*N
p = 1.0 dimensionless; mu = NaN per_year
mu*S

mu*S
mu = NaN per_year
beta*S*(I1+J1)/N

beta*S*(I1+J1)/N
beta = NaN per_year
beta*S*(I2+J2+Iv2)/N

beta*S*(I2+J2+Iv2)/N
beta = NaN per_year
sigma*R1

sigma*R1
sigma = NaN per_year
sigma*R2

sigma*R2
sigma = NaN per_year
sigma*R

sigma*R
sigma = NaN per_year
sigmaV*V

sigmaV*V
sigmaV = NaN per_year
mu*I1

mu*I1
mu = NaN per_year
gamma*I1

gamma*I1
gamma = NaN per_year
mu*I2

mu*I2
mu = NaN per_year
gamma*I2

gamma*I2
gamma = NaN per_year
Curator's comment:
(added: 10 Jan 2011, 23:17:27, updated: 10 Jan 2011, 23:17:27)
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