# Epidemics

Virtual society Virus spread Literature Version
Polish virtual society epidemics model
Guinea pigs epidemics model
Virus genetic evolution epidemics model

## Review of epidemics models

### Ferguson's, in France[1][2]

Quite large amount of data on epidemics in France from Sentinel and Centre National de Reference de la Grippe, region Sud, Laboratoire de Virology.

• workplaces not explicitly included
• transmission model: probability of the infection of the individual i, $p_i=1-\exp^{-\lambda_{i,y}(t) \Delta T}$, where $\lambda_i$ is the instantaneous infection risk of the individual i.
• transmission rate in a household $= \beta^{a}_{hous} f(t)/n$, where a - age (Adult(>=18) or Child), f(t) - relative infectiousness according to time t since infection, n - the size of the household.
• transmission rates for: within household infection (A,C), within school, within community (A->A,C->C,A<->C). Hence, the overall risk of infection includes: the household risk, the within school risk, the community risk
• time step = 6h
• annual variations of influenza infections modeled via: the strength of transmission during the year and the relative contribution of children to transmission during the year (included in $\lambda_{i,y}(t)$), where y - year

### Ferguson's, in UK and USA[3]

• households, schools, workplaces included in the model
• commuting data also included (the average distances traveled from hh to wp); the UK distance distribution more-less like power law, that of USA - surprisingly a bit different
• air travel data included
• time step = 6h
• transmission model: probability of the infection of the individual i, $p_i=1-\exp^{-\lambda_i(t) \Delta T}$, where $\lambda_i$ is the instantaneous infection risk of the individual i.

### Ferguson's, in Thailand[4]

• households, schools, workplaces included in the model
• travel data - not enormous, but some, generalized into probabilities of commuting within district
• seasonality not included (available data suggested that climate changes do not influence the infectivity rate)

### Germann's, in USA[7]

• no seasonal or environmental factors included, no virus evolution effects included
• basic geographic unit corresponds to census tract, it is a community of 2000 individuals: age, household size, and employment status distributions match US census data; agents divided into 5 age groups (0-4, 5-18, 9-29, 30-64, 64+ years) or community with no residents, only a workplace of about 100 individuals; in total, about 180000 communities are included; also travel data for the members of community included (in census tracts info);
• playgroups or daycare centers, schools, workplaces included; also contacts with people from neighborhood, occasional contacts in churches, supermarket, etc, included
• time step = 12h
• transmission model: probability of infection from a single contact, $p = P_{trans}\cdot c$, where $c$ is the contact probability (dependent on the age of both contacting people), $P_{trans}$ is the probability of transmission; antiviral treatment of the infectious contact person or the vaccination of the given individual diminish the $P_{trans}$;

Total probability of the susceptible person to get infected, $P\;=\;1-\prod_{i}^{N_{ic}}(1-p_{i})$, where the product goes over all $N_{ic}$ contacts with infectious individuals, with infection probability at ith contact given by $p_i$.

• disease history includes latent, incubation, and contagious periods of average durations: 1.2, 1.9, and 4.1 days, respectively.
• model implemented in the molecular dynamics SPaSM code, substituting atoms by agents
• basic reproductive number $R_0$ in the range [1.6;2.4] - as parameter of simulations
• initial conditions: 1-10 per 10000 arriving flight passengers randomly infected, only once for the whole simulation

### Stroud's, in southern California, USA[8]

• model implemented in EpiSimS platform

## References

1. CAUCHEMEZ S, Valleron A J, Boelle P Y, Flahault A, and Ferguson N M, Estimating the impact of school closure on influenza transmission from Sentinel data, Nature, 452 (2008), pp.750-754
2. CAUCHEMEZ S, Carrat F, Viboud C, Valleron A J, Boelle P Y, A Bayesian MCMC approach to study transmission of influenza: application to household longitudinal data, Stat. Med., 23, (2004), p3469
3. FERGUSON N M, Cummings D A T, Fraser C, Cajka J C, Cooley P C, and Burke D S, Strategies for mitigating an influenza pandemic, Nature, 442 (2006), pp. 448-452
4. FERGUSON N M, Cummings D, Cauchemez S, Fraser C, Riley S, Meeyai A, Iamsirithaworn S, and Burke D, Strategies for containing an emerging influenza pandemic in Southeast Asia, Nature, 437 (2005), pp. 209-214
5. FERGUSON N M, Donnelly C, and Anderson R A, Transmission intensity and impact of control policies on the foot and mouth epidemic in Great Britain, Nature, 413 (2001), pp. 542-548
6. DUNHAM J B, An Agent-Based Spatially Explicit Epidemiological Model in MASON Journal of Artificial Societies and Social Simulation,2005, 9(1)3
7. GERMANN T C, Kadau K, Longini I, and Macken C, Mitigation strategies for pandemic influenza in the United States, Proc. Nat. Acad. Sci., 103 (2006), pp.5935-5940
8. STROUD P, Del Valle S, Sydoriak S, Riese J and Mniszewski S (2007). Spatial Dynamics of Pandemic Influenza in a Massive Artificial Society,Journal of Artificial Societies and Social Simulation 10(4)9