How to Run a Campaign: Optimal Control of SIS and SIR Information Epidemics

Information spreading in a population can be modeled as an epidemic. Campaigners (e.g. election campaign managers, companies marketing products or movies) are interested in spreading a message by a given deadline, using limited resources. In this paper, we formulate the above situation as an optimal control problem and the solution (using Pontryagin's Maximum Principle) prescribes an optimal resource allocation over the time of the campaign. We consider two different scenarios --- in the first, the campaigner can adjust a direct control (over time) which allows her to recruit individuals from the population (at some cost) to act as spreaders for the Susceptible-Infected-Susceptible (SIS) epidemic model. In the second case, we allow the campaigner to adjust the effective spreading rate by incentivizing the infected in the Susceptible-Infected-Recovered (SIR) model, in addition to the direct recruitment. We consider time varying information spreading rate in our formulation to model the changing interest level of individuals in the campaign, as the deadline is reached. In both the cases, we show the existence of a solution and its uniqueness for sufficiently small campaign deadlines. For the fixed spreading rate, we show the effectiveness of the optimal control strategy against the constant control strategy, a heuristic control strategy and no control. We show the sensitivity of the optimal control to the spreading rate profile when it is time varying.Comment: Proofs for Theorems 4.2 and 5.2 which do not appear in the published journal version are included in this version. Published version can be accessed here: http://dx.doi.org/10.1016/j.amc.2013.12.16

Percolation on Networks with Antagonistic and Dependent Interactions

Drawing inspiration from real world interacting systems we study a system consisting of two networks that exhibit antagonistic and dependent interactions. By antagonistic and dependent interactions, we mean, that a proportion of functional nodes in a network cause failure of nodes in the other, while failure of nodes in the other results in failure of links in the first. As opposed to interdependent networks, which can exhibit first order phase transitions, we find that the phase transitions in such networks are continuous. Our analysis shows that, compared to an isolated network, the system is more robust against random attacks. Surprisingly, we observe a region in the parameter space where the giant connected components of both networks start oscillating. Furthermore, we find that for Erdos-Renyi and scale free networks the system oscillates only when the dependency and antagonism between the two networks is very high. We believe that this study can further our understanding of real world interacting systems

Evaluating the Usefulness of Paratransgenesis for Malaria Control

Malaria is a serious global health problem which is especially devastating to the developing world. Mosquitoes are the carriers of the parasite responsible for the disease, and hence malaria control programs focus on controlling mosquito populations. This is done primarily through the spraying of insecticides, or through the use of insecticide treated bed nets. However, usage of these insecticides exerts massive selection pressure on mosquitoes, resulting in insecticide resistant mosquito breeds. Hence, developing alternative strategies is crucial for sustainable malaria control. Here we explore the usefulness of paratransgenesis, i.e., introducing genetically engineered bacteria which secrete anti-plasmodium molecules, inside the mosquito midgut. The bacteria enter a mosquito's midgut when it drinks from a sugar bait, i.e., a sugar solution containing the bacterium. We formulate a mathematical model for evaluating the number of such baits required for preventing an outbreak. We study scenarios where vectors and hosts mix homogeneously as well as heterogeneously. We perform a full stability analysis and calculate the basic reproductive number for both the cases. Additionally, for the heterogeneous mixing scenario, we propose a targeted bait distribution strategy. The optimal bait allocation is calculated and is found to be extremely efficient in terms of bait usage. Our analyses suggest that paratransgenesis can prevent an outbreak, and hence it offers a viable and sustainable path to malaria control

Optimal Resource Allocation Over Time and Degree Classes for Maximizing Information Dissemination in Social Networks

We study the optimal control problem of allocating campaigning resources over the campaign duration and degree classes in a social network. Information diffusion is modeled as a Susceptible-Infected epidemic and direct recruitment of susceptible nodes to the infected (informed) class is used as a strategy to accelerate the spread of information. We formulate an optimal control problem for optimizing a net reward function, a linear combination of the reward due to information spread and cost due to application of controls. The time varying resource allocation and seeds for the epidemic are jointly optimized. A problem variation includes a fixed budget constraint. We prove the existence of a solution for the optimal control problem, provide conditions for uniqueness of the solution, and prove some structural results for the controls (e.g. controls are non-increasing functions of time). The solution technique uses Pontryagin's Maximum Principle and the forward-backward sweep algorithm (and its modifications) for numerical computations. Our formulations lead to large optimality systems with up to about 200 differential equations and allow us to study the effect of network topology (Erdos-Renyi/scale-free) on the controls. Results reveal that the allocation of campaigning resources to various degree classes depends not only on the network topology but also on system parameters such as cost/abundance of resources. The optimal strategies lead to significant gains over heuristic strategies for various model parameters. Our modeling approach assumes uncorrelated network, however, we find the approach useful for real networks as well. This work is useful in product advertising, political and crowdfunding campaigns in social networks.Comment: 14 + 4 pages, 11 figures. Author's version of the article accepted for publication in IEEE/ACM Transactions on Networking. This version includes 4 pages of supplementary material containing proofs of theorems present in the article. Published version can be accessed at http://dx.doi.org/10.1109/TNET.2015.251254