Projet de thèse en Mathématiques aux interfaces
Sous la direction de Nicolas Vayatis.
Thèses en préparation à Paris Saclay , dans le cadre de Mathématiques Hadamard , en partenariat avec CMLA - Centre de Mathématiques et de Leurs Applications (laboratoire) , Apprentissage statistique et données massives (equipe de recherche) et de Ecole normale supérieure Paris-Saclay (établissement de préparation de la thèse) depuis le 01-10-2017 .
The question that we put in the center of the proposed research is how societies can dynamically distribute resources that could effectively suppress a diffusive phenomenon. Our aim is to design new models and methods for assessing and controlling diffusion processes, either by extending traditional epidemic models such as SIS and SIR to fit in more specific context, or by working on more modern processes such as information cascades or point processes (e.g. Hawkes processes [16, 17]). A natural way to devise effective control strategies is to rely on a criticality score that assesses how important a person is for the diffusion on a specific network structure. Thus, spending healing resources on those individuals following some priority indicator, can improve the efficiency of control policies. In fact, a number of state-of-the-art control strategies [2, 15] rely on such scores by giving the treatment resources to the top-k infected nodes. Furthermore, the Dynamic Resource Allocation [5, 14] (DRA) framework describes formally a family of such allocation problems stated around the continuous-time SIS epidemic model, after making some reasonable assumptions such as the limitation of the effect of accumulated resources on a node. The Largest Reduction of Infectious Edges (LRIE) [5, 15] is an optimal greedy strategy which lays into the aforementioned framework. The LRIE criterion (or score) is computed locally for a node and has an intuitive interpretation: it combines the notion of virality to its healthy neighbors, and the notion of vulnerability for future reinfections due to its infected neighbors. Despite the limitations of being local and greedy (i.e. lack of global and long-term healing planning, ignoring the macroscopic network structure), and its requirement to know the infection state of each single node at every time instance, we believe that the logic of LRIE score can be improved and generalized for more complex settings without losing much of its intuitiveness and interpretability. Moreover, the locality of LRIE score can become a major advantage as it could allow for distributed policies against a diffusion. The proposed research plan is a composition of two parts: Part A: theoretical and algorithmic extensions of the existing DRA framework and development of improved strategies such as LRIE, that can i) perform diffusion control under uncertainties (e.g. incomplete data - unknown graph connections). ii) use arbitrary infection functions that model the probability for a node to get infected. These functions are not necessarily linear with the number of infected neighbors and cannot be encoded by a constant rate per infectious edge as in traditional compartmental models (e.g. SIS, SIR). For instance, they could be functions that saturate, which has been shown to be the case in social influence scenarios such as smoking/drinking behavior [9]. iii) extend the analysis to competing processes [10] where both infected and healthy state are diffusive. This type of competition characterizes the environment in which many of the diffusions take place in real-life (e.g. limited attention for information, competing ideas, competing or substitute products, etc.). iv) introduce macroscopic features related to the network structure in a more sophisticated the LRIE-like score as a `structure-based regularization' of the otherwise very local focus of the LRIE criterion. Part B: modelling the diffusion control as a sequential decision process. There are three basic assumptions in the DRA framework [5, 14] that can be reconsidered: i) the administrator has full information about the network structure and the state of each node at any moment (this is related to Part A.i point), ii) she also has full access to infected nodes and hence is able to assign healing resources directly to any of them, iii) there is no cost of reassigning the resources whenever the network infection state changes and the new scores indicate different healing priority of nodes. The introduction of constraints to one or more of these points generates many variations of the diffusion control task, which are very challenging and in essence get closer to the reality of epidemic control. Consider for example the setting where the administrator has direct access for assigning resources to only a random sample of the infected individuals per time frame (imagine that only a subset of individuals make use of the healthcare system when getting infected e.g. visit a doctor), also there is a cost whenever a resource unit is reassigned (imagine that being healthcare professional that does not help in the recovery of any patient while moving from patient to patient). In such a setting, at the moment when an infected node is sampled (i.e. by asking for medical help), the administrator needs to decide if it's worth assigning unassigned resources to that patient, or reassign already assigned resources and account for the respective cost, or alternatively not treat that patient hoping that other more critical patients will soon appear on which the resources will have better effect. Therefore, in the second part of the work, we propose to model such control problems and investigate ways to distribute treatments incrementally based on the above random sampling process under a number of reasonable assumptions. Theoretical analysis and simulations of indicative scenarios on real and artificial data will be used to support the developed resource allocation strategies. Potential findings not only can improve our understanding on the related problems, but also to have an important impact in the debate on the corrective administration strategies for public affairs [11] that usually involve the spending of money (allocation of resources) in a social network.
Sequential epidemic control with dynamic resource allocation
The question that we put in the center of the proposed research is how societies can dynamically distribute resources that could effectively suppress a diffusive phenomenon. Our aim is to design new models and methods for assessing and controlling diffusion processes, either by extending traditional epidemic models such as SIS and SIR to fit in more specific context, or by working on more modern processes such as information cascades or point processes (e.g. Hawkes processes [16, 17]). A natural way to devise effective control strategies is to rely on a criticality score that assesses how important a person is for the diffusion on a specific network structure. Thus, spending healing resources on those individuals following some priority indicator, can improve the efficiency of control policies. In fact, a number of state-of-the-art control strategies [2, 15] rely on such scores by giving the treatment resources to the top-k infected nodes. Furthermore, the Dynamic Resource Allocation [5, 14] (DRA) framework describes formally a family of such allocation problems stated around the continuous-time SIS epidemic model, after making some reasonable assumptions such as the limitation of the effect of accumulated resources on a node. The Largest Reduction of Infectious Edges (LRIE) [5, 15] is an optimal greedy strategy which lays into the aforementioned framework. The LRIE criterion (or score) is computed locally for a node and has an intuitive interpretation: it combines the notion of virality to its healthy neighbors, and the notion of vulnerability for future reinfections due to its infected neighbors. Despite the limitations of being local and greedy (i.e. lack of global and long-term healing planning, ignoring the macroscopic network structure), and its requirement to know the infection state of each single node at every time instance, we believe that the logic of LRIE score can be improved and generalized for more complex settings without losing much of its intuitiveness and interpretability. Moreover, the locality of LRIE score can become a major advantage as it could allow for distributed policies against a diffusion. The proposed research plan is a composition of two parts: Part A: theoretical and algorithmic extensions of the existing DRA framework and development of improved strategies such as LRIE, that can i) perform diffusion control under uncertainties (e.g. incomplete data - unknown graph connections). ii) use arbitrary infection functions that model the probability for a node to get infected. These functions are not necessarily linear with the number of infected neighbors and cannot be encoded by a constant rate per infectious edge as in traditional compartmental models (e.g. SIS, SIR). For instance, they could be functions that saturate, which has been shown to be the case in social influence scenarios such as smoking/drinking behavior [9]. iii) extend the analysis to competing processes [10] where both infected and healthy state are diffusive. This type of competition characterizes the environment in which many of the diffusions take place in real-life (e.g. limited attention for information, competing ideas, competing or substitute products, etc.). iv) introduce macroscopic features related to the network structure in a more sophisticated the LRIE-like score as a `structure-based regularization' of the otherwise very local focus of the LRIE criterion. Part B: modelling the diffusion control as a sequential decision process. There are three basic assumptions in the DRA framework [5, 14] that can be reconsidered: i) the administrator has full information about the network structure and the state of each node at any moment (this is related to Part A.i point), ii) she also has full access to infected nodes and hence is able to assign healing resources directly to any of them, iii) there is no cost of reassigning the resources whenever the network infection state changes and the new scores indicate different healing priority of nodes. The introduction of constraints to one or more of these points generates many variations of the diffusion control task, which are very challenging and in essence get closer to the reality of epidemic control. Consider for example the setting where the administrator has direct access for assigning resources to only a random sample of the infected individuals per time frame (imagine that only a subset of individuals make use of the healthcare system when getting infected e.g. visit a doctor), also there is a cost whenever a resource unit is reassigned (imagine that being healthcare professional that does not help in the recovery of any patient while moving from patient to patient). In such a setting, at the moment when an infected node is sampled (i.e. by asking for medical help), the administrator needs to decide if it's worth assigning unassigned resources to that patient, or reassign already assigned resources and account for the respective cost, or alternatively not treat that patient hoping that other more critical patients will soon appear on which the resources will have better effect. Therefore, in the second part of the work, we propose to model such control problems and investigate ways to distribute treatments incrementally based on the above random sampling process under a number of reasonable assumptions. Theoretical analysis and simulations of indicative scenarios on real and artificial data will be used to support the developed resource allocation strategies. Potential findings not only can improve our understanding on the related problems, but also to have an important impact in the debate on the corrective administration strategies for public affairs [11] that usually involve the spending of money (allocation of resources) in a social network.