Populations evolve in spatially heterogeneous environments. While a certain trait might bring a fitness advantage in some patch of the environment, a different trait might be advantageous in another patch. Here, we study the Moran birth–death process with two types of individuals in a population stretched across two patches of size
N
, each patch favouring one of the two types. We show that the long-term fate of such populations crucially depends on the migration rate
μ
between the patches. To classify the possible fates, we use the distinction between polynomial (short) and exponential (long) timescales. We show that when
μ
is high then one of the two types fixates on the whole population after a number of steps that is only polynomial in
N
. By contrast, when
μ
is low then each type holds majority in the patch where it is favoured for a number of steps that is at least exponential in
N
. Moreover, we precisely identify the threshold migration rate
μ
⋆
that separates those two scenarios, thereby exactly delineating the situations that support long-term coexistence of the two types. We also discuss the case of various cycle graphs and we present computer simulations that perfectly match our analytical results.
Motivated by COVID-19, we develop and analyze a simple stochastic model for the spread of disease in human population. We track how the number of infected and critically ill people develops over time in order to estimate the demand that is imposed on the hospital system. To keep this demand under control, we consider a class of simple policies for slowing down and reopening society and we compare their efficiency in mitigating the spread of the virus from several different points of view. We find that in order to avoid overwhelming of the hospital system, a policy must impose a harsh lockdown or it must react swiftly (or both). While reacting swiftly is universally beneficial, being harsh pays off only when the country is patient about reopening and when the neighboring countries coordinate their mitigation efforts. Our work highlights the importance of acting decisively when closing down and the importance of patience and coordination between neighboring countries when reopening.
Structural balance theory is an established framework for studying social relationships of friendship and enmity. These relationships are modeled by a signed network whose energy potential measures the level of imbalance, while stochastic dynamics drives the network towards a state of minimum energy that captures social balance. It is known that this energy landscape has local minima that can trap socially-aware dynamics, preventing it from reaching balance. Here we first study the robustness and attractor properties of these local minima. We show that a stochastic process can reach them from an abundance of initial states, and that some local minima cannot be escaped by mild perturbations of the network. Motivated by these anomalies, we introduce Best Edge Dynamics (BED), a new plausible stochastic process. We prove that BED always reaches balance, and that it does so fast in various interesting settings.
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