Data
We have collected and processed provincial-level geo-spatial data for Italy (ISTAT 2020a) in order to construct the mobility graph, whose nodes represent provinces which are populated according to age-stratified census data (ISTAT 2020b) and whose normalized weighted directed edges encode inter-provincial traffic flows (Pepe et al. 2020).
The heterogeneous contact patterns among provincial sub-populations are governed by synthetic contact matrices (Prem, Cook, and Jit 2017) whose components \(C_{ij}^{L}\) represent the daily contacts that a given agent belonging to age group \(i\) has with any agent of age group \(j\) in a given location \(L \in\) {Home, School, Work, Other}.
The number of agents in each provincial node and each age group is determined by the actual provincial population adjusted by a scale factor. The internal structure of the agent is completely specified by the following attributes:
- identifier code \(\text{id} \in [1,N] \subset \mathbb{N}\)
- current location \(p \in [1,M] \subset \mathbb{N}\)
- age group \(a \in [1,A] \subset \mathbb{N}\)
- residence (origin) location \(r \in [1,M] \subset \mathbb{N}\)
- workplace location \(\text{wp} \in [1,M] \subset \mathbb{N}\)
- household community \(\vec{h}, h_j \in \{j \in [1,N] : p_j=p_i\}\)
- workplace community \(\vec{w}, w_j \in \{j \in [1,N] : \text{wp}_j=\text{wp}_i\}\)
- infection state \(x \in \{S,E,I_a,I_p,I_s,R,D\}\)
- diagnostic state \(d \in \{O,N,P,W,R\}\)
In order to initialize the workplace locations we let each agent migrate from its residence node according to the probability vector defined by the mobility weights \(M_{ij}\). The workplace community is populated by the agents who have been contacted according to \(C_{ij}^{\text{Work}}\) sharing the workplace location, while the household community is populated by the agents who have been contacted according to \(C_{ij}^{\text{Home}}\).
Model
The model consists of two main (extendable) components: the infectious disease transmission module governing the micro, individual-level dynamics and the public health surveillance module driving the macro, system-level dynamics.
Transmission dynamics
The state space of the transmission compartmental module developed to describe the natural history of the disease is defined by the following states: susceptible (\(S\)), exposed or latent (\(E\)), asymptomatic infected (\(I_a\)), pre-symptomatic infected (\(I_p\)), symptomatic infected (\(I_s\)), recovered (\(R\)), died (\(D\)). All the relevant epidemiological parameters and timing distributions are reported in the Appendix.
The scheduled behaviors of the patient is structured as follows:
- contact household members (if present at home);
- move to workplace with probability \(p_w\) or, with probability \(1-p_w\) to another location driven by the mobility weights;
- contact workplace members;
- contact a sample of the population in the current nodal location determined by \(C_{ij}^{\text{School}}\);
- contact a sample of the population in the current nodal location determined by \(C_{ij}^{\text{Other}}\);
- move back home with probability \(p_h\);
Surveillance dynamics
The state space of the surveillance compartmental module developed to describe the diagnostic protocol is defined by the following states: untested (\(O\)), tested negative (\(N\)), tested positive and not yet negative (\(P\)), tested negative after being tested positive at least once waiting for second negative test confirming recovery (\(W\)), tested negative twice consecutively after being positive (\(R\)). All the relevant diagnostic performance parameters are taken into account and reported in the Appendix. All patients diagnosed as \(P\) or \(W\) are quarantined at home so that all the contacts in other locations are prevented.
The surveillance module controls all the policy interventions such as inter- and intra-provincial mobility restrictions: we have aggregated the mobility dataset into four distinct phases by integrating information about the timing and stringency of national and sub-national policy interventions (Thomas Hale and Kira 2020; Desvars-Larrive et al. 2020). In particular in phase 3 we modeled the national lockdown enforcement by modulating the contact matrices by an amplification factor (Prem et al. 2020).
Objectives
- Model calibration on national and sub-national epidemiological data for Italy
- Scenario analysis to explore the impact of exogenous/policy-driven and endogenous/behavior-driven social distancing
- Sensitivity analysis to test the robustness of the model to changes in timing distributions, behavioral and mobility patterns, household and workplace structures, geo-spatial and temporal resolutions, digital contact-tracing coverage
- Multi-objective assessment (e.g. public health vs. epistemic, clinical vs. epidemiological value) of the portfolio of diagnostic strategies implemented by the surveillance module
Current limitations
- All typical limitations of agent-based and metapopulation models are inherited (Ajelli et al. 2010)
- Geospatial and temporal resolutions
- Conservation of global population size
- Large uncertainty in the initial condition (date, location and number of infected patients)
- Lack of realistic regional-level heterogeneity in diagnostic protocols
- Absence of severity levels (mild, severe, critical) sub-compartments for symptomatic patients in transmission module
- Absence of hospitalization (\(H\), \(ICU\)) compartments in transmission module
- Absence of behavioral module
- Absence of diagnostic performance of medical doctors (in absence of laboratory resources)
Future developments
- Implementation of severity levels (mild, severe, critical) sub-compartments for symptomatic patients in transmission module
- Implementation of hospitalization (\(H\), \(ICU\)) compartments in transmission module
- Implementation of global and local, prevalence-based and opinion-based behavioral module
- Violation of conservation of global population size with airport mobility
- Fine-graining of household and workplace structures
- Model calibration on a given region with municipal-level mobility and epidemiological data
Appendix
Computational framework
| Language | Activity |
|---|---|
| Python | Data collection |
| Data wrangling | |
| Data visualization | |
| Julia | Modelling |
| Scenario analysis | |
| Multi-objective assessment |
Diagnostic strategies
| Role | Scale | Priority | Distribution | Digital Contact-Tracing |
|---|---|---|---|---|
| Passive | National | Random | Uniform | No |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes | ||||
| Regional | Random | Uniform | No | |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes | ||||
| Provincial | Random | Uniform | No | |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes | ||||
| Active | National | Random | Uniform | No |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes | ||||
| Regional | Random | Uniform | No | |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes | ||||
| Provincial | Random | Uniform | No | |
| Yes | ||||
| Targeted | Centrality-based | Yes | ||
| Targeted | Age-based / Ex-Ante IFR | No | ||
| Yes | ||||
| Symptom-based / Ex-Post IFR | No | |||
| Yes |
Parameters & distributions
| Name | Value | Description | References |
|---|---|---|---|
| \(y\) | \(0-29\) \((1-6)\) | Range of young age groups | Davies et al. (2020) |
| \(m\) | \(30-59\) \((7-12)\) | Range of middle age groups | Davies et al. (2020) |
| \(o\) | \(60-80\) \((13-16)\) | Range of old age groups | Davies et al. (2020) |
| \(\sigma_1\) | \(\mathcal{N}(\mu=0.5,\sigma=0.1;[0,0.5])\) | Symptomatic fraction on infection for young age groups | Davies et al. (2020) |
| \(\sigma_2\) | 0.5 | Symptomatic fraction on infection for middle age groups | Davies et al. (2020) |
| \(\sigma_3\) | \(\mathcal{N}(\mu=0.1,\sigma=0.1;[0.5,1])\) | Symptomatic fraction on infection for old age groups | Davies et al. (2020) |
| \(\beta_S\) | \(\mathcal{N}(\mu=0.5,\sigma=0.023;[0,+∞])\) | Transmissibility of symptomatic infectious person | Davies et al. (2020) |
| \(\beta_P\) | \(0.15 \cdot \beta_S\) | Transmissibility of pre-symptomatic infectious person | Aleta et al. (2020) |
| \(\beta_A\) | \(0.5 \cdot \beta_S\) | Transmissibility of a-symptomatic infectious person | Davies et al. (2020) |
| \(d_E\) | \(\mathcal{\Gamma}(\mu=3,k=4)\) | Incubation period | Davies et al. (2020) |
| \(d_P\) | \(\mathcal{\Gamma}(\mu=1.5,k=4)\) | Duration of infectiousness in days during the pre-symptomatic phase | Davies et al. (2020) |
| \(d_A\) | \(\mathcal{\Gamma}(\mu=3.5,k=4)\) | Duration of infectiousness in days during the a-symptomatic phase | Davies et al. (2020) |
| \(d_S\) | \(\mathcal{\Gamma}(\mu=5,k=4)\) | Duration of infectiousness in days during the symptomatic phase | Davies et al. (2020) |
| \(\delta_1\) | \(0\) | Infection fatality ratio for the 0-50 age group | Poletti et al. (2020) |
| \(\delta_2\) | \(0.46\) | Infection fatality ratio for the 50-60 age group | Poletti et al. (2020) |
| \(\delta_3\) | \(1.42\) | Infection fatality ratio for the 60-70 age group | Poletti et al. (2020) |
| \(\delta_4\) | \(6.87\) | Infection fatality ratio for the 70-80 age group | Poletti et al. (2020) |
| \(\nu_S\) | \(mean(0.20,0.38)\) | False negative rate in symptomatic phase | Kucirka et al. (2020) |
| \(\nu_P\) | \(mean(0.38,0.67)\) | False negative rate in pre-symptomatic phase | Kucirka et al. (2020) |
| \(\nu_E\) | \(mean(0.67,1)\) | False negative rate in incubation phase | Kucirka et al. (2020) |
References
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Aleta, Alberto, David Martı'n-Corral, Ana Pastore y Piontti, Marco Ajelli, Maria Litvinova, Matteo Chinazzi, Natalie E. Dean, et al. 2020. “Modelling the Impact of Testing, Contact Tracing and Household Quarantine on Second Waves of COVID-19.” Nature Human Behaviour, August. Springer Science and Business Media LLC. https://doi.org/10.1038/s41562-020-0931-9.
Davies, Nicholas G., and Petra Klepac, Yang Liu, Kiesha Prem, Mark Jit, and Rosalind M. Eggo. 2020. “Age-Dependent Effects in the Transmission and Control of COVID-19 Epidemics.” Nature Medicine 26 (8). Springer Science and Business Media LLC: 1205–11. https://doi.org/10.1038/s41591-020-0962-9.
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———. 2020b. “Resident Population by Sex, Age and Marital Status.” http://demo.istat.it/pop2020/index_e.html.
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