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Controlling Epidemics Through Optimal Allocation of Test Kits and Vaccine Doses Across Networks Mingtao Xia, Lucas Bottcher,¨ Tom Chou

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Controlling Epidemics Through Optimal Allocation of Test Kits and Vaccine Doses Across Networks Mingtao Xia, Lucas Bottcher,¨ Tom Chou 1 Controlling epidemics through optimal allocation of test kits and vaccine doses across networks Mingtao Xia, Lucas Bottcher,¨ Tom Chou Abstract—Efficient testing and vaccination protocols are crit- optimal vaccination strategies have been derived for a rapidly ical aspects of epidemic management. To study the optimal spreading disease in a highly mobile urban population using allocation of limited testing and vaccination resources in a PMP [12]. Complementing these control-theory-based inter- heterogeneous contact network of interacting susceptible, re- covered, and infected individuals, we present a degree-based ventions, a recent work [13] developed methods relying on testing and vaccination model for which we use control-theoretic reinforcement learning (RL) to identify infectious high-degree methods to derive optimal testing and vaccination policies using nodes (“superspreaders”) in temporal networks and reduce control-theoretic methods. Within our framework, we find that the overall infection rate with limited medical resources. optimal intervention policies first target high-degree nodes before The application of optimal control methods and PMP to shifting to lower-degree nodes in a time-dependent manner. Using such optimal policies, it is possible to delay outbreaks a heterogeneous node-based susceptible-infected-recovered- and reduce incidence rates to a greater extent than uniform susceptible (SIRS) model with applications to rumor spreading and reinforcement-learning-based interventions, particularly on was studied in [14]. certain scale-free networks. The machine-learning-based interventions of [13] showed Index Terms—Infection networks, epidemics, testing, vaccina- that RL is able to outperform intervention policies derived tion, optimal control, reinforcement learning from purely structural node characterizations that are, for instance, based on centrality measures. However, the methods I. INTRODUCTION of [13] were applied to rather small networks with a maximum Limiting the spread of novel pathogens such as SARS-CoV- number of nodes of about 400. Here, we focus on a com- 2 requires efficient testing [1], [2] and quarantine strategies [3], plementary approach by formulating optimal control and RL- especially when vaccines are not available or effective. Even if based target policies for a degree-based epidemic model [15] effective vaccines become available at scale, their population- that is constrained only by the maximum degree and not by the wide distribution is a complex and time-consuming endeavor, system size (i.e., number of nodes). Early work by May and influenced by, for example, age-structure [4]–[6], vaccine Anderson [16] employed such effective degree models to study hesitancy [7], and different objectives [8]. the population-level dynamics of human immunodeficiency Until a sufficient level of immunity within a population is virus (HIV) infections. These degree-based models and later reached, distancing and quarantine policies can also be used adaptations [17]–[19] do not account for degree correlations. to help slow the spread and evolutionary dynamics [9] of Effective degree models for susceptible-infected-susceptible infectious diseases. Epidemic modeling and control-theoretic (SIS) dynamics with degree correlations were derived in [20] approaches are useful for identifying both efficient testing and and applied to SIR dynamics in [21]. A further generalization vaccination policies. For an epidemic model of SARS-CoV- of these methods to model SIR dynamics with networked and 2 transmission, Pontryagin’s maximum principle (PMP) has well-mixed transmission pathways was presented in [22]. For a been used to derive optimal distancing and testing strategies detailed summary of degree-based epidemic models, see [23]. that minimize the number of COVID-19 cases and intervention In the next section, we propose and justify a degree-based costs [10]. Optimal control theory has also been applied epidemic, testing, and quarantining model. An optimal control to a multi-objective control problem that uses isolation and framework for this model is presented in Sec. III and, given vaccination to limit epidemic size and duration [11]. Both limited testing resources, an optimal testing strategy is calcu- of these recent investigations describe the underlying infec- lated. We extend the same underlying disease model to include tious disease dynamics through compartmental models without vaccination in Sec. IV and find optimal vaccination strategies underlying network structure, meaning that all interactions that minimize infection given a limited vaccination rate. We among different individuals are assumed to be homogeneous. summarize and discuss our results and how they depend on For a structured susceptible-infected-recovered (SIR) model, network and dynamical features of the model in Sec. V. For comparison, we also present in the Appendix a reinforcement- Mingtao Xia is in the Dept. of Mathematics at UCLA. learning-based algorithm that is able to approximate optimal E-mail: [email protected] testing strategies for the model introduced in Sec. II. Lucas Bottcher¨ is in the Dept. of Computational Medicine at UCLA and at the Frankfurt School of Finance and Management. E-mail: [email protected] Tom Chou is in the Depts. of Computational Medicine and Mathematics at II. DEGREE-BASED EPIDEMIC AND TESTING MODEL UCLA E-mail: [email protected] For the formulation of optimal testing policies that allocate Manuscript received July 18, 2021; revised August 31, 2021. testing resources to different individuals in a contact network, 2 we adopt an effective degree model of SIR dynamics with testing in a static network of N nodes. Nodes represent individuals, and edges between nodes represent corresponding contacts. Therefore, the degree of a node represents the number of its contacts. If K is the maximum degree across all nodes, we can divide the population into K distinct subpopu- lations, each of size Nk (k = 1; 2;:::;K) such that all nodes th PK in the k group have degree k. Therefore, N = k=1 Nk. In our epidemic model, we distinguish between untested u ∗ and tested infected individuals. Let Sk(t), Ik (t), Ik (t), and Rk(t) denote the numbers of susceptible, untested infected, tested infected, and recovered nodes with degree k at time t, respectively. Since these subpopulations together represent the entire population (the total number of nodes N), both N and Nk are constants in our model. Their values satisfy u ∗ the normalization condition Sk + Ik + Ik + Rk = Nk. The corresponding fractions are u u sk(t) = Sk(t)=N; ik(t) = Ik (t)=N; ∗ ∗ (1) ik(t) = Ik (t)=N; rk(t) = Rk(t)=N; P u ∗ Fig. 1. Degree distribution of a Barabasi–Albert´ network and a stochastic such that k(sk + ik + ik + rk) = 1. Using an effective- block model. (a) The degree distribution of a Barabasi–Albert´ network with degree approach [16], [22], we describe the evolution of the 99,939 nodes. Each new node is connected to m = 2 existing nodes (i.e., the above subpopulations by degree of each node is at least 2) using preferential attachment. Then nodes with degrees larger than 100 [26] are removed from the network. The grey K solid line is a guide-to-the-eye with slope -3 [27]. The inset shows a realization dsk(t) X P (`jk) = − ks (t) βuiu(t) + β∗i∗(t); (2) of a Barabasi–Albert´ network with 100 nodes. Node size scales with their dt k P (`) ` ` `=1 betweenness centrality. (b) The conditional probability P (`jk) associated with the Barabasi–Albert´ network generated in (a). (c) The degree distribution of a K diu(t) X P (`jk) stochastic block model with four blocks and 100,000 nodes. The inset shows k =ks (t) βuiu(t) + β∗i∗(t) (3) dt k P (`) ` ` a realization of a stochastic block model with 800 nodes, but using the same `=1 block probability matrix. (d) The conditional probability P (`jk) associated u u fk(t) u with the SBM. In both (b) and (d), all elements that are strictly zero are − γ ik(t) − ik(t); uncolored. Nk ∗ dik(t) ∗ ∗ fk(t) u = − γ ik(t) + ik(t); (4) dt Nk lead to quarantining and do not affect the disease dynamics. dr (t) u u ∗ u k u u ∗ ∗ However a fraction I =(Sk + I + I + Rk) ≡ I =Nk of =γ ik(t) + γ ik(t); (5) k k k k dt these fk(t)∆t tests will be administered to untested infecteds. where P (`) = N`=N is the degree distribution and P (`jk) is Once infected nodes have been identified by testing, they can the probability that a chosen node with degree k is connected be quarantined and removed from the disease transmission to a node with degree `. Our degree-based formulation of SIR dynamics. If infected individuals who already have been tested dynamics with testing, Eqs. (2)–(5), is an approximation of strictly avoid future testing, more tests will be available for the full node-based dynamics assuming that nodes of the same the other subpopulations, increasing the rate at which the degree are equally likely to be infected at any given time [15]. remaining untested infecteds will be tested. In this case, the Susceptible individuals become infected through contact fraction of tests administered to untested infecteds is modified: u u u u ∗ with untested and tested infected individuals at rates β Ik =(Sk +Ik +Rk) ≡ Ik =(Nk −Ik ). After normalizing by the ∗ u and β , respectively. Untested and tested infected individuals total population N, we arrive at the testing terms −fk(t)ik=Nk u ∗ u u recover at rates γ and γ , respectively. Differences in the (Eqs. (2) and (3)) or −fk(t)ik=[Nk(1−Ik =Nk)], respectively. recovery rates γu and γ∗ reflect differences in disease severity Biased testing can also be represented by using a testing u b u b ∗ of and treatment options for untested and tested infected fraction of the form Ik e =(Ik e +Sk +Ik +Rk), where b > 0 individuals.
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