Table 1 Definitions and scenarios in the study

Risk

Interpreted based on levels of infection rate, prevalence,

or average contacts (via population size)

High-risk patch

Defined either by high direct first time infection rate (i.e., high β

which leads to high corresponding \(\mathcal {R}_{0}\)) or by high exogenous re-infection rate

(i.e., high δ)

Enhanced socio-economic conditions (reducing health disparity)

Defined by better healthcare infrastructure which is incorporated by high prevalence of a disease (i.e., high I(0)/N) in a large population (i.e., large N)

Mobility

Captured by average residence times of an individual in different patches (i.e., by using \(\mathbb {P}\) matrix)

Scenario 1

\(\underbrace {\beta _{1} > \beta _{2}, \ \delta _{1} = \delta _{2}}_{\text {high risk}}\); \(\underbrace {\frac {I_{1}(0)}{N_{1}} > \frac {I_{2}(0)}{N_{2}}, \ N_{1} > N_{2};}_{\text {enhanced socio-economic conditions}}\) \(\underbrace {\text {vary} \ p_{12} \ \text {when} \ p_{21}\approx 0}_{\text {mobility}}\)

Scenario 2

\(\underbrace {\beta _{1} = \beta _{2}, \ \delta _{1} > \delta _{2}}_{\text {high risk}}\); \(\underbrace {\frac {I_{1}(0)}{N_{1}} > \frac {I_{2}(0)}{N_{2}}, \ N_{1} > N_{2};}_{\text {enhanced socio-economic conditions}}\) \(\underbrace {\text {vary} \ p_{12}\ \text {when} \ p_{21}\approx 0}_{\text {mobility}}\)