Intervention of resource allocation strategies on spatial spread of epidemic

10 Medical resources are crucial in mitigating the epidemic, especially during pandemics such as 11 the ongoing COVID-19. Thereby, reasonable resource deployment inevitably plays a significant role 12 in suppressing the epidemic under limited resources. When an epidemic breaks out, people can 13 produce resources for self-protection, or donate resources to help others. That is, the exchange of 14 resources also affects the transmission between individuals, thus, altering the epidemic dynamics. To 15 understand factors on resource deployment and the interplay between resource and transmission we 16 construct a metapopulation network model with resource allocation. Our results indicate actively or 17 promptly donating resources is not helpful to suppress the epidemic under both homogeneous 18 population distribution (HOD) and heterogeneous population distribution (HED). Besides, 19 strengthening the speed of resource production can significantly increase the recovery rate so that 20 reduce the final outbreak size. These results may provide policy guidance towards epidemic 21 containment.


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Curbing the spread of epidemics is vital to human society today. During the past decades, human 24 has experienced several major pandemics such as the Severe Acute Respiratory Syndrome (SARS) 25 in 2003 1-3 , the Middle East Respiratory Syndrome (MERS) in 2012 4,5 , the western Africa Ebola 6-8 , 26 and so on. Currently, the ongoing novel coronavirus disease (COVID-19) has diffused to almost every 27 country in a short period of time, which has been characterized as a Public Health Emergency of 28 International Concern (PHEIC) by WHO 9-12 . With the increasing of new infected cases, there is an 29 enormous demand for medical resources such as masks, protective clothes, ventilators, etc 13 . Some 30 countries (regions, or cities) have excessively consumed medical resources because of curing massive 31 infected cases, leading to healthcare systems being overwhelmed 14,15 . Importantly, medical resources 32 play a crucial role in curbing the epidemic, and greatly affect the spread process 16 . However, faced 33 with the outbreak of the epidemic, it is necessary that countries (regions, or cities) produce medical 34 resources for self-protection, as well as contributing/receiving resources to/from others cannot be 35 ignored. Thus, there is an urgent need to better deploy limited medical resources to restrain the spread 36 of epidemic. 37 The studies about the effects of medical resources on suppressing epidemic through network 38 science are widespread [17][18][19][20][21] . Some studies have investigated how best to allocate the limited resources 39 based on single layer networks 22,23 . In addition, considering different effects such as information 40 diffusion and epidemic spreading, a number of studies have investigated the coevolution dynamics 41 coupling resource allocation with multilayer networks [24][25][26][27] . But these studies are mainly carried out  42  via contact networks in which nodes represent individuals and links denote the contacts between  43  individuals. Nowadays, due to frequent spatial activities of humans and convenient traffic such as  44  airline networks, epidemics especially COVID-19 can rapidly diffuse through the migration of  45 population. 46 Metapopulation network model can be well used to investigate the spatial spread of epidemic 47 due to the mobility of individuals 28-33 . In this framework, nodes represent subpopulations (e.g., 48 regions, cities or countries), while links represent the migration routes between subpopulations. The 49 infection occurs by the interaction of individuals within a subpopulation and the diffusion corresponds 50 to their migration along the links between subpopulations, which is called reaction-diffusion (RD)  51 process 31,34,35 . Some studies have explored the effects such as mobility rate and non-uniform 52 intervention for suppressing epidemic 36-38 , whereas factors such as medical resources inevitably 53 playing the uttermost role in suppressing epidemic have been ignored. As the outbreak of COVID-19 54 in Wuhan, China, the government promptly deployed medical resources from other cities to Wuhan, 55 and consequently controlled the epidemic. Hence, when facing an epidemic, deploying limited 56 resources reasonably to locations plays a fundamental role in suppressing the epidemic. 57 Here, we construct a metapopulation network model with Migration-Interaction-Return (MIR) 58 to investigate the coevolution dynamics of epidemic spreading and resource allocation (Fig. 1). 59 Specifically, we model the spatial structure of realistic populations and the behavior of individuals in 60 virtual social networks through a reaction-diffusion process based on the classical susceptible-

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Once individuals moved, they will react in a well-mixed way according to the SIS model. Susceptible 68 individuals in subpopulation i get infected at rate λi(t) while infected individuals get recovered at rate μi(t). The 69 parameters λi(t) and μi(t) dynamically change with time due to the exchange of resources between 70 subpopulation i and its neighbors, ωi->j or ωj->i (dotted arrow). After reaction, they return to their resident 71 subpopulations and the next time step starts.

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Generally, infected individuals consume medical resources for treatment, while susceptible 73 individuals are responsible for producing resources, e.g., face masks, ventilators, etc. Hence, the 74 amount of resources produced by a subpopulation can be assumed proportional to the ratio of 75 susceptible individuals it contains. We designate a parameter θ (≥1) as resource production strength, 76 and the parameter θ times the current subpopulation's fraction of susceptible individuals is regarded 77 as productive resources in a time step. Apparently, a higher value of θ means a faster speed of resource 78 production. 79 With the emergence of infected individuals, subpopulations can perceive the risk from neighbors. 80 Usually, the more infected individuals emerge in a subpopulation, the more resources are supposed 81 to be donated to it. In other words, the amount of resource donation of one subpopulation increases 82 with the number of infected individuals that its neighboring subpopulations contain. In this paper, 83 considering factors that may affect a subpopulation's donation will, we can summarize them as how 84 many and when to donate resources, i.e., the donation awareness which restrains donating scale by 85 considering the need for self-protection, and its response speed to donate resources, which is often 86 related with the infected individuals surrounded with it. Therefore, we define the donation will mainly 87 controlled by two parameters, i.e., α ∈ [0, 1] represents the subpopulation's resource donation 88 awareness, and β (≥0) describes the subpopulation's resource donation sensitivity, respectively (See 89 Supplementary Information for details). The higher donation awareness means fewer resources can 90 be donated, and the lower donation sensitivity means a larger initial donation will and a steady growth 91 of it with the increasing of surrounded infected individuals, and vice versa. Differing from the 92 classical contact networks, in which each node denotes an individual and resources are uniformly 93 distributed to its infected neighbors. In our model, given that every subpopulation contains a crowd 94 of individuals, the resources are allocated proportional to the degree that the neighboring 95 subpopulations get infected. That is, a subpopulation with more serious infection would obtain more 96 resources, which agrees with the current policy guidance of governments. 97 Without a doubt, after exchange of resources between subpopulations, the infection rate and the 98 recovery rate will alter. Naturally, a subpopulation becomes risky to be infected after donating 99 resources, while it is helpful for treatment when holding more resources. Thus, we can formulate the 100 epidemiological process with the resource donation process. In other words, one subpopulation gets 101 a higher infection rate when it donates resources to others. Instead, it gets a higher recover rate when 102 holding more resources, i.e., it produces or receives more resources. In short, by producing and 103 donating resources to infected subpopulations, the epidemic process is coevolved with the resource 104 donation dynamics. 105 106

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To understand the coevolution process of resource donation dynamics and the epidemic 108 dynamics, we systematically explore the impacts of resources donations, such as donation awareness 109 α, donation sensitivity β, and resource production strength θ. We have performed an extensive set of 110 stochastic simulations on the scale free (SF) networks with 1000 nodes, whose average degree is 6.9. 111 The edge weights between nodes are uniformly distributed within the range of [1,50].

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To highlight the effects of each subpopulation's donation awareness α on the epidemic spreading, 131 we set β=1 for the general donation sensitivity, and θ=1 for the normal productive strength. Figure 2  132 shows the final prevalence ρ at steady state versus basic infection rate λ for various values of α under 133 the conditions of HOD and HED, respectively (There is a perfect agreement between the iterations 134 of Markov equations and Monte Carlo (MC) simulations). In addition, there is no resource exchange 135 between subpopulations when α=1, and full donation will when α=0. From Fig. 2, the epidemic 136 thresholds under HOD are overall higher than HED. Because under HED some subpopulations with 137 more population would have more infected cases initially, the epidemic easily breaks out and quickly 138 spreads out by migration. But under HOD, the infected cases are uniformly distributed in each 139 subpopulations initially, inducing a slower spreading with a higher epidemic threshold. 140 For the case of HOD, we see that lower awareness (smaller α) or stronger will of resource 141 donation among subpopulations would promote the epidemic spread with a reduced epidemic 142 threshold and a larger outbreak size. When an outbreak occurs in a subpopulation, neighboring 143 subpopulations with high donation will would donate more resources to it, leading to high infection 144 rates in them ( Supplementary Fig. S2 (d), (e), and (f)). The dynamics of the donation will 145 (Supplementary Fig. S2 (a)), as well as the infection ratio of neighboring subpopulations 146 ( Supplementary Fig. S2 (c)) for all the subpopulations in the networks are collected and support the 147 results. It seems that actively donating resources to the infected subpopulation is not helpful to 148 suppress the epidemic. 149 In contrast, for the case of HED, we can see that lower awareness (smaller α) or stronger will of 150 donation among subpopulations would delay the outbreak of the epidemic with a higher epidemic 151 threshold, but induce a larger final outbreak size. Because under HED the epidemic would easily 152 break out in subpopulations with more population, a lower awareness indicates that neighboring 153 subpopulation would donate more resources to it, leading to the containment of the epidemic 154 ( Supplementary Fig. S4). But a lower awareness also induces a larger outbreak size as previous 155 situation. It suggests that actively donating resources just delays the outbreak at the early time, but is 156 not helpful to suppress the epidemic while continuously donating resources. 157 Effects of the resource donation sensitivity on the epidemic spread

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In order to understand the impacts of the donation sensitivity β on the epidemic, we set α=0 with 167 no awareness namely full will of resource donation, and θ=1 for the normal speed of resource 168 production. Figure 3 shows the final prevalence ρ at steady state versus the basic infection rate λ 169 under various values of β (There is a perfect agreement between the iterations of the Markov 170 equations and MC simulations). Particularly, the donation will is constant, i.e. 0.5, if β=0 171 ( Supplementary Fig. S1). From Fig. 3, the epidemic thresholds under HOD are overall higher than 172 HED. 173 Under the condition of HOD, we see that higher donation sensitivity (larger β) can delay the 174 epidemic with a higher epidemic threshold, but induce a higher final break size. Because a larger β 175 means a lower initial donation will (see Supplementary Fig. S1), neighboring subpopulations donate 176 fewer resources when the epidemic breaks out in one subpopulation, leading to low infection rates in 177 them (shown as Supplementary Fig. S5 (d), (e), and (f)). However, a lager β induces higher final 178 outbreak size because the rapid growth of donation will leads to high infection rates in neighboring 179 subpopulations ( Supplementary Fig. S6 (d), (e), and (f)). It seems that donating resources or quickly 180 response to the infected subpopulation would induce high infected scale instead. 181 Under the condition of HED, we can see that higher donation sensitivity (larger β) can promote 182 the epidemic outbreak with a lower epidemic threshold and induces a larger final outbreak size. 183 Because under HED the epidemic would easily break out in subpopulations with more population, 184 they can receive more resources from neighbors to suppress the epidemic due to a higher initial 185 donation will with a lower donation sensitivity (lower β), leading to a higher epidemic threshold 186 ( Supplementary Fig. S8). However, a lager β also induces higher final outbreak size as the rapid 187 growth of donation will leads to high infection rates ( Supplementary Fig. S7 (d), (e), and (f)). It 188 suggests that donating a big amount of resources earlier just delays the epidemic, and promptly 189 increasing resource donation is also not conducive to reduce final infected scale. 190 The coupling effects of donation awareness and donation sensitivity

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To shed light on the interplay between the donation awareness α and donation sensitivity β, we 196 plot the epidemic prevalence under the cases of HED and HOD respectively when close to threshold 197 as shown in Fig.4. For the case of HOD, it shows that that high donation awareness and high donation 198 sensitivity can delay the epidemic with a low prevalence, which clearly indicates that donating more 199 resources (lower α) may promote the epidemic, and donating resources promptly (higher β) can 200 validly suppress the epidemic. Therefore, these results indicate that we need to increase resource 201 donation steadily, avoiding donating a large amount of resources initially without protecting ourselves. 202 Instead, under the case of HED, low donation awareness and low donation sensitivity can 203 suppress the epidemic outbreak, which indicates that we need to immediately donate plenty of 204 resources to infected subpopulations, especially with more population, to rapidly suppress the 205 epidemic. But promptly increasing the resource donation (higher β) with the outbreak size is not a 206 valid strategy. 207 Effects of the resource production strength on the epidemic spread

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In the real word, individuals can recover to susceptible state by plenty of medical resources, so 216 the more resources they hold, the higher recovery rate they get. The production strength θ is the 217 parameter used to measure the ability of resource production in one time step. In order to interpret 218 the impacts of production strength on the epidemic spread, we set α=0 for no awareness namely full 219 will of resource donation, and β=1 for the general donation sensitivity. Figure 5 shows the final 220 epidemic prevalence ρ versus basic infection rate λ under various values of θ (There is a perfect 221 agreement between the iterations of the Markov equations and MC simulations). It can be regarded 222 as a normal speed of resource production for θ=1, and higher speed when θ is greater than 1. 223 From Fig.5, we see that higher production strength (larger θ) delays the epidemic with a higher 224 epidemic threshold and reduces final outbreak size under both HOD and HED. This is because 225 subpopulations can produce more resources each time step so that they averagely hold more resources, 226 which leads to higher recovery rates and lower effective infection rates (as Supplementary Fig. S9 (e) 227 shown). In addition, in SF networks with a heterogeneous topology, since hub subpopulations 228 generally have more population with more infected cases initially, the epidemic can easily break out 229 with a lower threshold under HED (Fig. S10). 230 As a consequence, higher resource production strength can effectively delay the epidemic spread 231 and reduce final infected ratio regardless of HOD and HED, but this effect is limited because the 232 average recovery rate approaches to 1 if θ reaches 20 or greater. Therefore, properly reinforcing the 233 strength of resource production for suppressing is necessary. 234 235

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Facing with epidemics, especially pandemics such as COIVD-19, medical resources 237 undoubtedly play a significant role in suppressing epidemics, so the reasonable deployment about 238 resources become an important issue that we need to further investigate. While most of the advances 239 previously described have been focused on capturing the resource deployment based on contact 240 networks, less attention has been paid on the metapopulation networks. Due to current convenient 241 traffic, the human interactions induce the spatial spreading of epidemics by individuals' movement 242 between regions (or cities, countries), which ignores the role of resource deployment on the so-called 243 metapopulation networks. 244 In this work, we construct a metapopulation network model to study the resource deployment 245 on epidemic evolution. The results indicate that properly donating resources can delay the epidemic 246 with a high epidemic threshold under heterogeneous population distribution, but actively or promptly 247 donating resources to the infected subpopulation is not helpful to suppress the epidemic regardless of 248 homogeneous or heterogeneous population distribution. Besides, strengthening the speed of resource 249 production can significantly increase the recovery rate so that reduce the final infection ratio. 250 Therefore, facing with the outbreak of the epidemic, we should not blindly help others while 251 neglecting self-protection. Meanwhile, strengthening the speed of resource production is an effective 252 measure, which can clearly increase recovery rate so that promote recovery of infected cases. 253 However, our current work inevitably has several limitations. First of all, we simply consider 254 that the amount of resources generated by one subpopulation is proportional to its ratio of susceptible 255 individuals. However, under heterogeneous population distribution, these subpopulations with more 256 population tend to generate more resources than those with fewer population, so we can further 257 consider the factor of individuals on resource production. Besides, we assume resources are produced 258 by each subpopulations itself, but ignore global resources that can be allocated to each one. The 259 generation and allocation patterns of resources are expected to be further explored and compared in 260 the future. 261 No.20ZR1419000. 267