Modelling small population outbreaks
Modelling small population outbreaks <latexit - - PowerPoint PPT Presentation
Modelling small population outbreaks <latexit - - PowerPoint PPT Presentation
Modelling small population outbreaks <latexit
Deterministic models
S I R dS dt = − β N SI + γ(N − S − I) dI dt = β N SI − νI
ν
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Continuous trajectory
Life is discrete & stochastic
Event Transition Jump intensity Infection (s, i) → (s − 1, i + 1) βsi/N Recovery (s, i) → (s, i − 1) νi Loss of immunity (s, i) → (s + 1, i) γ(N − s − i)
S I R
ν
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One 𝚰 = Many trajectories
time prevalence 100 200 300 400 20 40 60 80 100Variability
⎨
⎨
Extinction
Exponential distribution
S I R
ν
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<latexit sha1_base64="9c1F3TvkaNKYDBtjwjXDjADXxLw=">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</latexit><latexit sha1_base64="9c1F3TvkaNKYDBtjwjXDjADXxLw=">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</latexit><latexit sha1_base64="9c1F3TvkaNKYDBtjwjXDjADXxLw=">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</latexit><latexit sha1_base64="9c1F3TvkaNKYDBtjwjXDjADXxLw=">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</latexit>γ
<latexit sha1_base64="A9UzQ6vRa9KjBcsXUPEkYBsX+EQ=">ACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipYB/QFplMp3VsXiYTsRZX/oBb/THxD/QvDOmoAuxNyS5c+45Z+6d8WJfpspx3gvW3PzC4lJxubSyura+Ud7caqZRlnDR4JEfJW2PpcKXoWgoqXzRjhPBAs8XLW90qutO5GkMgov1TgWvYANQzmQnCmCmt0hCwJ2Va64VceE/XdSQR71qPyGLvqIwJEhgEAIRbkPhpSeDlw4iAnrYUJYQpk0dYFHlEibEUsQgxE6ou+QVp0cDWmtPVOj5rSLT29CSht7pImIl1Cud7NPTPOGv3Le2I8dW9j+nu5V0CowjWh/+mzFl1ehaFAY7NDJmig2ip+O5S2ZORXdu/5hKkUNMmM7VE8o50Y5PWfbaFIzuz5bZuofhqlRveY5N8On7nK2C24eVF2n6l4cVmon+VUXsYNd7N9HqGM9TRIO8bPOMFr9a5dWvdWw/fVKuQa7bxK6ynL7mkZk=</latexit><latexit sha1_base64="A9UzQ6vRa9KjBcsXUPEkYBsX+EQ=">ACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipYB/QFplMp3VsXiYTsRZX/oBb/THxD/QvDOmoAuxNyS5c+45Z+6d8WJfpspx3gvW3PzC4lJxubSyura+Ud7caqZRlnDR4JEfJW2PpcKXoWgoqXzRjhPBAs8XLW90qutO5GkMgov1TgWvYANQzmQnCmCmt0hCwJ2Va64VceE/XdSQR71qPyGLvqIwJEhgEAIRbkPhpSeDlw4iAnrYUJYQpk0dYFHlEibEUsQgxE6ou+QVp0cDWmtPVOj5rSLT29CSht7pImIl1Cud7NPTPOGv3Le2I8dW9j+nu5V0CowjWh/+mzFl1ehaFAY7NDJmig2ip+O5S2ZORXdu/5hKkUNMmM7VE8o50Y5PWfbaFIzuz5bZuofhqlRveY5N8On7nK2C24eVF2n6l4cVmon+VUXsYNd7N9HqGM9TRIO8bPOMFr9a5dWvdWw/fVKuQa7bxK6ynL7mkZk=</latexit><latexit sha1_base64="A9UzQ6vRa9KjBcsXUPEkYBsX+EQ=">ACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipYB/QFplMp3VsXiYTsRZX/oBb/THxD/QvDOmoAuxNyS5c+45Z+6d8WJfpspx3gvW3PzC4lJxubSyura+Ud7caqZRlnDR4JEfJW2PpcKXoWgoqXzRjhPBAs8XLW90qutO5GkMgov1TgWvYANQzmQnCmCmt0hCwJ2Va64VceE/XdSQR71qPyGLvqIwJEhgEAIRbkPhpSeDlw4iAnrYUJYQpk0dYFHlEibEUsQgxE6ou+QVp0cDWmtPVOj5rSLT29CSht7pImIl1Cud7NPTPOGv3Le2I8dW9j+nu5V0CowjWh/+mzFl1ehaFAY7NDJmig2ip+O5S2ZORXdu/5hKkUNMmM7VE8o50Y5PWfbaFIzuz5bZuofhqlRveY5N8On7nK2C24eVF2n6l4cVmon+VUXsYNd7N9HqGM9TRIO8bPOMFr9a5dWvdWw/fVKuQa7bxK6ynL7mkZk=</latexit><latexit sha1_base64="A9UzQ6vRa9KjBcsXUPEkYBsX+EQ=">ACyXicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0I7ipYB/QFplMp3VsXiYTsRZX/oBb/THxD/QvDOmoAuxNyS5c+45Z+6d8WJfpspx3gvW3PzC4lJxubSyura+Ud7caqZRlnDR4JEfJW2PpcKXoWgoqXzRjhPBAs8XLW90qutO5GkMgov1TgWvYANQzmQnCmCmt0hCwJ2Va64VceE/XdSQR71qPyGLvqIwJEhgEAIRbkPhpSeDlw4iAnrYUJYQpk0dYFHlEibEUsQgxE6ou+QVp0cDWmtPVOj5rSLT29CSht7pImIl1Cud7NPTPOGv3Le2I8dW9j+nu5V0CowjWh/+mzFl1ehaFAY7NDJmig2ip+O5S2ZORXdu/5hKkUNMmM7VE8o50Y5PWfbaFIzuz5bZuofhqlRveY5N8On7nK2C24eVF2n6l4cVmon+VUXsYNd7N9HqGM9TRIO8bPOMFr9a5dWvdWw/fVKuQa7bxK6ynL7mkZk=</latexit>Mean = 1/𝝃 Var = Mean Memory less
Erlang distribution
S I1 R
βI/N
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kν
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10 20 30 40 16Aug 23Aug 30Aug 06Sep 13Sep 20Sep 27Sep 04Oct 11Oct
date incidence
Tristan da Cunha
284 ind - 32% reinfected
One possible model…
S E I L T
Temporary protection Long-term protection
One possible model…
S E I L T
Temporary protection Long-term protection
Already implemented as a fitmodel!