Fig. 3

Control strategies and optimization. A given country (orange pentagon) is placed in the structural parameter space (K, Q) with epidemic threshold \(\beta ^{2D}_c\) obtained from Eq. (3). For an epidemic spread with infection probability \(\beta\) it is desired to position the country in such a way that \(\beta < \beta ^{2D}_c(K,Q)\), such that there will be no epidemic. This can be achieved by the following strategies: a Social strategy. Assume that the natural epidemic infection rate is \(\beta = 0.8 > \beta ^{2D}_c\) (thick black dashed line) above the epidemic threshold of the country. By using social distancing or mask-wearing the infection probability could be reduced to \(\beta ^\prime = 0.7 < \beta ^{2D}_c\) (grey dashed line) and thus becomes below the epidemic threshold and the disease will not spread. b Quarantine strategies. By reducing the infection channels in and between the cities (reducing K and Q respectively) the country’s position in the structural parameter space can be changed and the epidemic threshold will increase such that the infection probability will be below the epidemic threshold. Three ways are suggested: (1) local quarantine strategy within cities by reducing \(K \rightarrow K^{\prime }\). (2) Global quarantine strategy between cities by reducing \(Q \rightarrow Q^\prime\). (3) Mixed strategy by reducing both \(K \rightarrow K^{\prime \prime }\),\(Q \rightarrow Q^{\prime \prime }\). c Strategies optimization. A weight function, \(W(\beta )\), can be evaluated for optional locations for the parameters space of the country based on economical, health, and social arguments such that \(\beta _c(K,Q) \rightarrow \beta ^{+}\). Optimization of the weight function will yield the optimal location for the country \((K_{opt}, Q_{opt})\). Here we used the Euclidean distance in the parameters space as a weight function \(W(\beta ) = \sqrt{(K-K^{\prime \prime })^2 + (Q-Q^{\prime \prime })^2}\) and its optimization (minimizing) will yield the shortest Euclidean distance which represents minimal reduction of the inter and intra links