Data-driven efficient network and surveillance-based immunization

Abstract

Given a contact network and coarse-grained diagnostic information such as electronic Healthcare Reimbursement Claims (eHRC) data, can we develop efficient intervention policies from data to control an epidemic? Immunization is an important problem in multiple areas, especially epidemiology and public health. However, most existing studies rely on assuming prior epidemiological models to develop pre-emptive strategies, which may fail to adapt to the change in new epidemiological patterns and the availability of rich data such as eHRC. In practice, disease spread is usually complicated, hence assuming an underlying model may deviate from true spreading patterns, leading to possibly inaccurate interventions. Additionally, the abundance of health care surveillance data (such as eHRC) makes it possible to study data-driven strategies without too many restrictive assumptions. Hence, such a data-driven intervention approach can help public-health experts take more practical decisions. In this paper, we take into account propagation log and contact networks for controlling propagation. Different from previous model-based approaches, our solutions are solely data driven in a sense that we develop immunization strategies directly from the network and eHRC without assuming classical epidemiological models. In particular, we formulate the novel and challenging data-driven immunization problem. To solve it, we first propose an efficient sampling approach to align surveillance data with contact networks, then develop an efficient algorithm with the provably approximate guarantee for immunization. Finally, we show the effectiveness and scalability of our methods via extensive experiments on multiple datasets, and conduct case studies on nation-wide real medical surveillance data.

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Notes

  1. 1.

    Code in Python: https://goo.gl/tsMueB.

  2. 2.

    http://www.sociopatterns.org.

  3. 3.

    We extract vaccine reports based on ICD-9 codes V04.81. These are actual vaccine allocations as given in the eHRC data.

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Acknowledgements

This paper is based on work partially supported by the NSF (IIS-1353346, CAREER IIS-1750407), the NEH (HG-229283-15), ORNL, the Maryland Procurement Office (H98230-14-C-0127), and a Facebook faculty gift to BAP. AV is partially supported by the following grants: DTRA CNIMS Contract HDTRA1- 11-D-0016-0010, NSF BIG DATA Grant IIS-1633028 and NSF DIBBS Grant ACI-1443054, NSF EAGER SSDIM-1745207. Publication of this article was also funded by ORNL LDRD funding to AR. Oak Ridge National Laboratory (ORNL) (Grant No. Order 4000143330) is operated by UT-Battelle, LLC, for the US Department of Energy under contract DE-AC05-00OR22725. The US Government retains and the publisher, by accepting the article for publication, acknowledges that the US Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US Government purposes.

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Appendix

Appendix

Proof of Lemma 4.4

When \(\alpha _{\mathbf {M}, \ell }\) is optimal, \(\alpha _{\mathbf {M}, \ell }=\alpha ^*_{\mathbf {M}, \ell }\).

Second, let \(\beta _{S_{\ell }}\) be the number of nodes without any parents. Maximizing \(\alpha _{\mathbf {M}, \ell }\) for Problem 3.1 is equivalent to minimizing \(\beta _{S_{\ell }}\) at location \(L_{\ell }\). Suppose \(\beta ^*_{S_{\ell }}\) is maximum number of nodes without any parents in the sample at location \(L_{\ell }\). It is obvious \(\beta ^*_{S_{\ell }} = CN(L_{\ell }, t_0)= |S_L|\). For each timestep \(t_i\), if \(CF_i (S_{\ell }) <CN(L_{\ell }, t_i)\), then \(CN(L_{\ell }, t_i)- CF_i(S_{\ell })\) is the number of nodes that cannot be mapped to the cascade generated by \(S_{\ell }\) at timestep \(t_i\). Hence, \(\theta (S_{\ell })\) is the number of nodes that cannot be mapped to the cascade generated by \(S_{\ell }\). If there exists any \(t_i\) that \(CF_i (S_{\ell }) <CN(L_{\ell }, t_i)\), we can always generate a cascade by mapping all \(CF_i (S_{\ell }) \) nodes into the cascade, then uniformly at random map other \(\theta (S_{\ell })\) nodes into cascade. This way, the number of nodes without any parents, \(\beta _{S_{\ell }} \le \beta ^*_{L_{\ell }}+ \theta (S_{\ell }) \) as \(\theta (S_{\ell })\) nodes can have connection within themselves. Since \(\beta _{S_{\ell }} + \alpha _{S_{\ell }} = \sum _{t_i} N(L_i, t_i)\), then \( \alpha _{\mathbf {M}, \ell } \ge \alpha ^*_{\mathbf {M}, \ell } - \theta (S_{\ell })\). Hence, \(\alpha ^*_{\mathbf {M}, \ell } - \theta (S_{\ell }) \le \alpha _{\mathbf {M}, \ell } \le \alpha ^*_{\mathbf {M}, \ell }\). When \(\alpha _{\mathbf {M}, \ell }=\alpha ^*_{\mathbf {M}, \ell }\), \(\theta (S_{\ell }) =0\). \(\square \)

Proof of Lemma 4.5

First, it is clear that \(g(\emptyset )=0\).

Second, to prove g(S) is monotonic increasing, we need to prove \(\theta (S)\) is a monotonic decreasing function. To do that, we first show that \(CF_{i}(S_{\ell })\) is monotone non-decreasing and submodular functions for any i and \(L_{\ell }\). First, let us define \(f_{i}(S_{\ell })\) as the number of nodes in \(L_{\ell }\) that \(S_{\ell }\) can reach in i-hops; hence, \(f_{i}(S_{\ell }) \le f_{i}(S_k)\) when \(S_{\ell }\subseteq S_k\). Second, given \(S_{\ell }\subseteq S_k\) and a node u, \(f_{i}(S_{\ell }\cup \{u\}) - f_{i}(S_{\ell })\) is marginal gain of a set union. Since the function in the set union problem is submodular [14], \(f_{i}(S_{\ell })\) is also submodular. Since \(f_{i}(S_{\ell })\) is monotone non-decreasing and submodular, the cumulative function \(CF_{i}(S_{\ell })\) is also non-decreasing and submodular.

Let \(X_i=[{\mathbb {1}}_{CF_i (A \cup B) <CN_i } (CN_i - CF_i(A \cup B))]\), \(Y_i=[{\mathbb {1}}_{CF_i (A) <CN_i } (CN_i - CF_i(A))] \). For any set A and B,

$$\begin{aligned} \theta (A \cup B) - \theta (A)= \sum _{i=1}^T (X_i - Y_i ) \end{aligned}$$
(6)

For any i, let us consider the following two cases:

(1) If \({\mathbb {1}}_{CF_i (A) <CN_i }=0\), it means \(CF_i (A) \ge CN_i\), then \(CF_i (A\cup B) \ge CN_i\); hence, \({\mathbb {1}}_{CF_i (A \cup B) <CN_i }=0\). We have \(X_i - Y_i=0\).

(2) If \({\mathbb {1}}_{CF_i (A) <CN_i }=1\), we have two cases:

(2a) \({\mathbb {1}}_{CF_i (A \cup B) <CN_i }=0\), then \(X_i -Y_i = -Y_i = - (CN_i - CF_i(A)) <0\);

(2b) \({\mathbb {1}}_{CF_i (A \cup B) <CN_i }=1\), then \( X_i - Y_i = (CN_i - CF_i(A \cup B))- (CN_i - CF_i(A)) = CF_i(A) - CF_i(A\cup B) \le 0\) (using Claim 2).

Putting together, we have \(\theta (A \cup B) \le \theta (A)\). Hence, \(\theta (S)\) is monotonic decreasing, and hence g(S) is monotonic increasing.

Third, to prove g(S) is submodular, For any location l, we need to prove that, given \(S \subseteq T\), \(g(S \cup \{a\}) - g(S) \ge g(T \cup \{a\}) - g(T)\), which is equivalent to \(\theta (S )-\theta (S\cup \{a\}) \le \theta (T)-\theta (T\cup \{a\}) \) (supermodularity). Let us write

\(\delta (S,a,i)= [{\mathbb {1}}_{CF_i (S \cup \{a\})<CN_i } (CN_i - CF_i(S \cup \{a\}))] -[{\mathbb {1}}_{CF_i (S) <CN_i } (CN_i - CF_i(S))]\), and

\(\delta (T,a,i) = [{\mathbb {1}}_{CF_i (T \cup \{a\})<CN_i } (CN_i - CF_i(T \cup \{a\}))] -[{\mathbb {1}}_{CF_i (T) <CN_i } (CN_i - CF_i(T))]\), then,

\(\theta (S) - \theta (S \cup \{a\}) = \sum _{i=1}^t \delta (S,a,i) \), and \(\theta (T) - \theta (T \cup \{a\}) = \sum _{i=1}^t \delta (T,a,i) \).

For any i, let us consider the following two cases:

(1) If \({\mathbb {1}}_{CF_i (S) <CN_i }=0\), then \({\mathbb {1}}_{CF_i (S \cup \{a\})<CN_i }={\mathbb {1}}_{CF_i (T)<CN_i }={\mathbb {1}}_{CF_i (T\cup \{a\}) <CN_i }=0\). Hence, \(\delta (S,a,i)=\delta (T,a,i)=0\).

(2) If \({\mathbb {1}}_{CF_i (S) <CN_i }=1\), we have the following cases:

(2a) If \({\mathbb {1}}_{CF_i (T) <CN_i }=0\), then we have \({\mathbb {1}}_{CF_i (T \cup \{a\}) <CN_i }=0\). Let us consider the value of \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }\):

If \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }=0\), then \(\delta (S,a,i) =(CN_i - CF_i(S \cup \{a\})) < 0 = \delta (T,a,i) \).

If \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }=1\), then \(\delta (S,a,i) = CF_i (S) - CF_i(S \cup \{a\}) < 0 = \delta (T,a,i)\).

(2b) If \({\mathbb {1}}_{CF_i (T) <CN_i }=1\), let us consider the value of \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }\):

If \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }=0\), then \({\mathbb {1}}_{CF_i (T \cup \{a\}) <CN_i }=0\), and then \(\delta (S,a,i) = -(CN_i - CF_i(S)) \le -(CN_i - CF_i(T)) = \delta (T,a,i) \) (using Claim 2).

If \({\mathbb {1}}_{CF_i (S \cup \{a\}) <CN_i }=1\), then for \({\mathbb {1}}_{CF_i (T \cup \{a\}) <CN_i }\):

If \({\mathbb {1}}_{CF_i (T \cup \{a\}) <CN_i }=1\), then \(\delta (S,a,i)= CF_i(S) - CF_i(S \cup \{a\})) \le CF_i(T) - CF_i(T \cup \{a\})) =\delta (T, a,i)\) (using Claim 2 that \(CF_i(S)\) is a submodular function).

Otherwise, \({\mathbb {1}}_{CF_i (T \cup \{a\}) <CN_i }=0\), and then since we have \(CF_i (T \cup \{a\}) \ge CN_i\), \(\delta (S,a,i)= CF_i(S) - CF_i(S \cup \{a\})) \le CF_i(T) - CF_i(T \cup \{a\})) \le CF_i(T) - CN_i = \delta (T, a,i)\) (using Claim 2).

Putting all cases together, we have \(\theta (S) - \theta (S \cup \{a\}) \le \theta (T) - \theta (T \cup \{a\})\). Hence, \(g(S \cup \{a\}) - g(S) \ge g(T \cup \{a\}) - g(T)\).

g(S) is a submodular function. \(\square \)

Proof of Lemma 4.9

Since we uniformly randomly allocate \(\mathbf {x}\), \(\rho _{G, \mathbf {M}_i}(\mathbf {x})\) can be written as \(\rho _{G, \mathbf {M}_i}(\mathbf {x}) = \sum _S \Pr (S) r_{G, \mathbf {M}_i}(S)\), where S is a node set sampled from the random process of distributing \(\mathbf {x}\) (\(|S| = ||\mathbf {x}||_1\)), and \(r_{G, \mathbf {M}_i}(S)\) is the number of nodes \(SI_{\mathbf {M}_i}\) can reach after removing S.

Since \(\zeta _{G, \mathbf {M}_i}(\mathbf {x}) = \sum _{S} \Pr (S) C_{G, \mathbf {M}_i}(S)\) and \(\rho _{G, \mathbf {M}_i}(\mathbf {x}) = \sum _S \Pr (S) r_{G, \mathbf {M}_i}(S)\), we need to show that \( r_{G, \mathbf {M}_i}(S) \le C_{G, \mathbf {M}_i}(S)\). \( r_{G, \mathbf {M}_i}(S)\) is the number of nodes S can save in \(\mathbf {M}_i\), we can show that given any node u that \(SI_\mathbf {M}\) can save, the credit u given to \(SI_\mathbf {M}\) must be 1. This is because if we can save u, it means every path from \(SI_\mathbf {M}\) to u has been removed when S is removed. Hence, all nodes within the paths from \(SI_\mathbf {M}\) have been removed. These nodes are all nodes that propagate u’s credit to \(SI_\mathbf {M}\), so all credits of u can be contributed to \(C_{G, \mathbf {M}_i}(S)\). Hence, \(C_{G, \mathbf {M}_i}(S)\) is at least equal to \(r_{G, \mathbf {M}_i}(S)\). On the other hand, other nodes that S cannot save also make contributions to the credit of \(C_{G, \mathbf {M}_i}(S)\). Hence, \(C_{G, \mathbf {M}_i}(S) \ge r_{G, \mathbf {M}_i}(S)\), which leads to \(\rho _{G, \mathbf {M}_i}(\mathbf {x}) \le \zeta _{G, \mathbf {M}_i}(\mathbf {x}) \). \(\square \)

Proof of Lemma 4.11

We use a similar technique as in [4] given the properties of \(P_1\), \(P_2\) and \(P_3\) of \(\zeta _{G, \mathbf {M}_i} (\mathbf {x})\). For brevity, we write \(\zeta _{G, \mathbf {M}_i} (\mathbf {x})\) as \(\zeta (\mathbf {x})\).

First, we show that if \(\mathbf {y}=(y_i, \ldots ,y_n)^T\) where \(\sum _j y_j=m\), then \(\zeta (\mathbf {x}+ \mathbf {y}) - \zeta (\mathbf {x}) \le \sum _j y_j (\zeta (\mathbf {x}+\mathbf {e}_j)-\zeta (\mathbf {x}))\).

Let \(\mathbf {y}\) can be recursively obtained from a sequence \(\mathbf {e}^{(1)}, \ldots , \mathbf {e}^{(m)}\) (\(\mathbf {e}^{(i)} \in \{\mathbf {e}_1,\ldots ,\mathbf {e}_n\}\)) such that \(\mathbf {y}=\mathbf {y}^{(m)}=\mathbf {y}^{(m-1)}+\mathbf {e}^{(m)}\), \(\mathbf {y}^{(i)}=\mathbf {y}^{(i-1)}+\mathbf {e}^{(i)}\) (\(i \le m\)) and \(\mathbf {y}^{0}=\mathbf {0}\).

Obviously, \(\sum _{i=1}^m \mathbf {e}^{(i)}= \sum _j y_j \mathbf {e}_j =\mathbf {y}\). Then,

$$\begin{aligned}&\zeta (\mathbf {x}+ \mathbf {y}) -\zeta (\mathbf {x}) \\&\quad = \sum _{i=1}^m \zeta (\mathbf {x}+ \mathbf {y}^{(i)})-\zeta (\mathbf {x}+ \mathbf {y}^{(i-1)}) \\&\quad = \sum _{i=1}^m \zeta (\mathbf {x}+ \mathbf {y}^{(i-1)}+\mathbf {e}^{(i)})-\zeta (\mathbf {x}+ \mathbf {y}^{(i-1)}) \\&\quad \le \sum _{i=1}^m \zeta (\mathbf {x}+\mathbf {e}^{(i)})-\zeta (\mathbf {x}) \ \mathbf {(Diminishing\ Return)} \\&\quad = \sum _{j=1}^n y_j (\zeta (\mathbf {x}+\mathbf {e}_j)-\zeta (\mathbf {x})) \end{aligned}$$

Now, let us prove that ImmuNaiveGreedy gives a \((1-1/e)\)-approximate solution. Suppose \(\mathbf {x}\) is the solution from ImmuNaiveGreedy, and \(\mathbf {x}^*\) is the optimal solution. Clearly, we have \(\sum _{j}x_j=\sum _{j}x^*_j=m\). Let us define \(\mathbf {x}^{(i)}\) as the solution got from the ith iteration of the greedy algorithm; hence, \(\mathbf {x}=\mathbf {x}^{(m)}\). And \(\mathbf {x}^*\) can be represented as \(\sum _j x^*_j \mathbf {e}_j\). We have

$$\begin{aligned} \zeta (\mathbf {x}^*)&\le \zeta ( \mathbf {x}^*+\mathbf {x}^{(i)}) \\&= \zeta (\mathbf {x}^{(i)}) + ( \zeta ( \mathbf {x}^*+\mathbf {x}^{(i)}) - \zeta (\mathbf {x}^{(i)}) )\\&\le \zeta (\mathbf {x}^{(i)}) + \sum _j {x^*_j} (\zeta (\mathbf {x}^{(i)}+\mathbf {e}_j)- \zeta (\mathbf {x}^{(i)}) ) \\&\le \zeta (\mathbf {x}^{(i)}) + \sum _j {x^*_j} ( \zeta (\mathbf {x}^{(i+1)}) - \zeta (\mathbf {x}^{(i)})) \\&= \zeta (\mathbf {x}^{(i)}) +m( \zeta (\mathbf {x}^{(i+1)}) - \zeta (\mathbf {x}^{(i)})) \end{aligned}$$

Hence, \(\zeta (\mathbf {x}^{(i+1)}) \ge (1-\frac{1}{m})\zeta (\mathbf {x}^{(i)}) +\frac{1}{m} \zeta (\mathbf {x}^*)\). Recursively, we can get \(\zeta (\mathbf {x}^{(i)}) \ge (1-(1-\frac{1}{m})^i) \zeta (\mathbf {x}^*)\). Therefore, \(\zeta (\mathbf {x})=\zeta (\mathbf {x}^{(m)}) \ge (1-(1-\frac{1}{m})^m) \zeta (\mathbf {x}^*) \ge (1-1/e) \zeta (\mathbf {x}^*)\). \(\square \)

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Zhang, Y., Ramanathan, A., Vullikanti, A. et al. Data-driven efficient network and surveillance-based immunization. Knowl Inf Syst 61, 1667–1693 (2019). https://doi.org/10.1007/s10115-018-01326-x

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Keywords

  • Graph mining
  • Social networks
  • Immunization
  • Diffusion