Mobility restrictions for the control of epidemics: When do they work?

Background Mobility restrictions—trade and travel bans, border closures and, in extreme cases, area quarantines or cordons sanitaires—are among the most widely used measures to control infectious diseases. Restrictions of this kind were important in the response to epidemics of SARS (2003), H1N1 influenza (2009), Ebola (2014) and, currently in the containment of the ongoing COVID-19 pandemic. However, they do not always work as expected. Methods To determine when mobility restrictions reduce the size of an epidemic, we use a model of disease transmission within and between economically heterogeneous locally connected communities. One community comprises a low-risk, low-density population with access to effective medical resources. The other comprises a high-risk, high-density population without access to effective medical resources. Findings Unrestricted mobility between the two risk communities increases the number of secondary cases in the low-risk community but reduces the overall epidemic size. By contrast, the imposition of a cordon sanitaire around the high-risk community reduces the number of secondary infections in the low-risk community but increases the overall epidemic size. Interpretation Mobility restrictions may not be an effective policy for controlling the spread of an infectious disease if it is assessed by the overall final epidemic size. Patterns of mobility established through the independent mobility and trade decisions of people in both communities may be sufficient to contain epidemics.

The basic reproductive number of model (1) gives the average number of secondary infections produced by a typical infectious individual during its infectious period β α+γ1+δ , the secondary cases generated by asymptomatic exposed individuals qβ κ and, the secondary cases produced by the fraction of diagnosed individuals α (α+γ1+δ)(γ2+δ) at the reduced infectiousness lβ.

COVID-19 Dynamics in Heterogeneous Risk Communities
In heterogeneous communities, new cases of infection per unit time among the Community j resident population are modeled by incorporating the effective density or effective population size -the expected amount of residents and visitors sojourning in each community at time t. Community j residents can get infected at their community June 25, 2020 1/5 of residency (with risk of infection β j ) or while spending time in Community i (with risk of infection β i ), modeled as follows Since we are budgeting times of residency proportions, then p ii + p ij = 1. In the two communities system, we use t i to denote the Community i residents' average proportion of time in Community j while 1 − t i denotes the average proportion of time that Community i residents spend in their place of residency. Hence, the expected proportion of infected population sojourning in Community i at time t is expressed as follows: ( The proposed model assumes that COVID-19-diagnosed cases do not travel across communities and these individuals are exclusively in contact with the local population. By using the single community model (1) as baseline model, the dynamics of COVID-19 on a two communities landscape, with distinct risk of infection, can be described by the following system of differential equations where i, j ∈ {1, 2} and i = j.

COVID-19 Basic Reproductive Number in Heterogeneous Risk
Communities System's (2) basic reproductive number is computed by following the next generation approach [36,37]. Consider the infectious compartments E 1 , I 1 , J 1 , E 2 , I 2 June 25, 2020 2/5 and J 2 , evaluating at the DFE, leads to S 1 (0) = N 1 and S 2 (0) = N 2 . Then then, the basic reproductive number of model (2), in the presence of mobility, is given by the spectral radius of the next generation matrix −F V −1 . Note that the basic reproductive number is a function of the community-specific mobility, risk levels and community density, (P, β i and N i , respectively).

COVID-19 Final Epidemic Size in Heterogeneous Risk Communities
We compute the community-specific final epidemic size, as function of residency times. By assuming S i (0) = N i , E i (0) = I i (0) = J i (0) = R i (0) = 0 and, by adding the first two equations in model (2), Model's (2) final size relation is denoted by the community-specific final proportion of infected individuals (or attack rate).
The eigenvalues of the matrix B, on the final epidemic size expression (3), are the same as those of the next generation matrix. Therefore, the global -communities integrated -basic reproductive number is also the spectral radius of B, [26]. In addition, under this Lagrangian framework the global basic reproductive number is a function of the mobility matrix (P) and the community-specific basic reproductive numbers (R 0i ), defined in the absence of mobility (t 1 = t 2 = 0), so that, R 0 = f (P, R 01 , R 02 ).
Let s ∞ i = lim t→∞ S i (t) N i , represent the proportion of the population remained susceptible