Infectious diseases are diseases caused by pathogenic microorganisms, such as bacteria, viruses, parasites or fungi, which can be spread directly or indirectly from person to person. In recent years, newly emerging diseases have been identified with an unprecedented rate of one or more per year. Meanwhile, highly developed transportation systems have intensified the large-scale spreading trend of the epidemics thus highlights the urgent demand for the research on effective control strategies. For many directly person-to-person transmitted diseases, such as HIV/AIDS, SARS and the influenza, network models could capture the specific contact patterns which could possibly lead to a successful transmission of the diseases between individuals and thus provide an effective way to forecast the potential epidemic dynamics and explore the intervention measures. In network models, the individuals are described as nodes (vertices) and the contacts between these individuals are described as edges (links). Seeded in a random selected node, the epidemic can be spread along the interconnections among vertices. The initial infected nodes will proceed to the infectious period after a short time after being infected, during when they could spread the infection to their contact nodes, and the newly infected nodes could then infect their linked vertices, and so on.
Effective control of infectious diseases requires quantitative comparisons of several interventions, such as quarantine, infection control precautions, case identification and isolation, and immunization. The success of an intervention depends on the infectivity of the diseases and the contact patterns of the population. Network models define the detailed contact structure of the population, which would be conducive to the effectiveness assessment of different interventions, therefore they are widely applied to the study of immunization strategies. Several immunization strategies have been proposed based on network models. These strategies include both deterministic algorithms based on global network information, such as maximum-degree node immunization–, maximum-betweenness node immunization, and long-range travelers immunization,, and stochastic algorithms based on local network information, such as ring vaccination–, acquaintance immunization and community-bridge immunization. Although studies have shown that deterministic immunization strategies regularly outperform stochastic immunization strategies in controlling the epidemics, the realistic global network information is usually or even constantly unavailable in the infectious diseases control practice. Therefore the stochastic immunization strategies and their optimizations have long been the research hotspots. However, the effectiveness assessment of the stochastic immunization strategies mentioned above is almost entirely made based on classical network models, such as scale-free networks and small-world networks. Although these network models have partly described the characteristics of realistic networks, there are still many differences among them, thus the real effectiveness of stochastic immunization strategies mentioned above has not been performed clearly.
In this article, the optimal immunization ratio , for acquaintance immunization and community-bridge immunization was determined and the influences of several primary factors on the effectiveness of ring vaccination, i.e., the case detection rate, the contact trace escape rate and the contact trace lag were analyzed based on the realistic urban social network of Portland. Then the effectiveness of these three different stochastic immunization strategies in controlling the spreading of infectious diseases were compared.
Social contact networks
As the carrier of the diseases, daily activities of individuals underlie the spread of the epidemics in the population and people's choices about when and where to perform their activities are constrained by the transportation infrastructure. Based on these assumptions and the sociodemographic data, researchers from Los Alamos National Laboratory (LANL) generate a synthetic population of Portland along with the detailed activity arrangement for each individual during one day. The obtained synthetic population data were applied into the TRANSIMS and EpiSims systems,. The activities of the population during 24 hours and exacted the contact network of Portland were obtained from these data.
The final social contact network is an undirected weighted network with the individual attributes of the synthetic population recorded by each node and the contact duration of any two individuals in 24 hours recorded by the weight of the link between them. Moreover, the link between each pair of nodes has also recorded the contact type of these two nodes, involved nine different contact types. The nodes and edges attributes of the contact network were listed in Table 1.
A simple extension of the SIR model,–, will lead to the SVIDR model by adding three compartments, detected (D), vaccinated (V) and recovered detected (RD), see Figure 1. At any time, individual will be in and only in one specific state. A susceptible individual (S) suffers from an instantaneous infection rate at time t, where if j is in state I, otherwise 0. Therefore after a short time , the probability of the individual being infected is . Susceptible individuals who are infected proceed to class I, and then will recover to RI at a rate . Alternatively, infectious individuals could be directly detected at probability r. Following direct detection, infectious individuals are moved into class D and recover to class RD at the a , and meanwhile trigger the interventions. The interventions will dominate the probability of susceptible individuals proceeding to class V and this probability varies with interventions. Moreover, the probability of is also influenced by the interventions in addition to the direct detection probability r, which is called second-order detection (infectious individuals who escape the direct detection but captured by the interventions triggered by other individuals), see section 2.3. Individuals in class V and R (includes both RD and RI) will acquire fully immunity to the diseases and lose the susceptibility. Individuals in class D will be isolated and lose the capability to infect others.
AI applies to the scenario where large heterogeneity in contact structure is observed. The analysis of Portland contact network shows both the degree distribution and the vertex strength distribution are highly heterogeneous (see Figure 2), so AI could be used to control the epidemics in this network. AI strategy is defined as follows: first, selecting n random nodes from the network; second, for each chosen node, immunizing a random neighbor. A node with k connections will be targeted as the immunization node with the probability
, where is the average degree. Therefore, for any susceptible individual, it will enter into the controlled set and proceed to class V with the probability q. For each infectious individual, it will enter into the controlled set and proceed to class D with the probability (second-order detection). Notice that the probability of direct detection is r, so class I individual will proceed to class D with the total probability . For individuals in other states (V, R and D), no influences were demonstrated on the spreading of the epidemics, so no processing was made.
Assume that a node will be targeted as the community-bridge node under the community-bridge find algorithm with the probability q. The process of the targeted nodes is the same as it in AI, so it will not be repeated.
Suppose individual j has triggered an intervention, then the neighbors of node j in the contact network will enter into the controlled set with probability , which varies with the contact type (i). The value of indicates the availability of the contact trace relate to type (i). For example, indicates that all the neighbors of j will enter into the controlled set, i.e., the contact trace escape rate related to type (i) is 0. Therefore, for every susceptible node, it will enter into the controlled set with the probability and proceed to class V, where iff j is in class D, otherwise 0. For any infectious node, it will enter into the controlled set with probability and proceed to class D. For individuals in other states, no other processing is required.
Let be the probability that individual is in class at time t. Assume that the above mentioned process is Markovian on the relevant time scales, the dynamics of this probability is governed by the following master equations:
(1) where the computation of q varies with interventions.
Scenarios and parameterizations
At the beginning, 50 infectious individuals are seeded in the fully susceptible population. The fraction of class I individuals in the population is about 3.12 per 100000. In epidemiology, the basic reproductive number, is defined as the number of cases one case generates on average over the course of its infectious period in an otherwise totally susceptible population. is one of the most important contributions of mathematics to epidemiology and it has provided a metric to evaluate the risk of the outbreaks of the epidemics in the population. Only infectious diseases with could possibly lead to a potential outbreak. The infectivity levels of infectious diseases may differ considerably. For example, for the pandemic influenza in 1918, is between 2 and 3; for SARS in 2003, is between 1.8 and 4.2; for H1N1 in 2009, is between 1.8 and 3.2; for smallpox, could reach 6.0. Based on the analysis on the pandemic data from the 2009 H1N1 outbreak, , were chosen to parameterize our model for the baseline scenario in this research. To make the investigation more generally applicable and reliable, the scenarios when varies from 2.0 to 6.0 were analyzed in the sensitivity analysis. The relevant parameters are given in Table 2.
Contact network structures
Portland social contact network consists of more than 31 million contacts of 1.6 million individuals during 24 hours. Each contact is represented by an undirected weighted link, where the weight denotes the contact duration in hours. The average degree is , which is more than the precedent empirical results from questionnaire survey. This difference is understandable because of the report of the respondents might miss some regular contacts and most random contacts. The comparison to the social networks from Facebook shows that their average degree is close. The average clustering coefficient is and the modularity is . High heterogeneity was observed when taking the individual contact frequency (degree) into consideration, shown in Figure 2a. For weighted networks, the vertex strength could be defined in addition to the degree: . Figure 2b shows that the vertex strength distribution is also highly heterogeneous. The degree correlation could be measured based on the quantity , i.e., the average degree of the nearest neighbors of nodes with degree k
. Figure 2c shows that the slope of is positive, so the contact network is assortative mixing. The distribution of is given in Figure 2d.
Optimizing immunization ratios
In this work, the optimization of AI and CBI was studied firstly. For AI and CBI, the immunization ratio f is defined as the fraction of the immunized susceptible individuals after the interventions are triggered, i.e., . We then defined an objective function to investigate the optimal strategy. An intuitive and straightforward form of the objective function is given by number of individuals in class R and class V after the extinction of the epidemics, . This definition follows the minimum affected principle and assigns equal weights to class R individuals and class V individuals. Other more complex definitions could be obtained by weighting the two quantities.
The binary search method was adopted to explore the optimal immunization ratio in interval [0, 1]. The search process ends when search step . One notable thing is that the immunization ratio f for AI could be any value between 0 and 1: f = 0 indicates no interventions and f = 1 implies that all the individuals are immunized after the detection of the epidemics. However, for CBI, the immunized individuals are those targeted as community-bridge nodes by community-bridge find algorithm, which might be finite in the network. Take the sub network in Figure 3c as an example, according to the community-bridge find algorithm, node 1 and 5 are community-bridge nodes, i.e., potential immunization targets, while node 2, 3 and 4 are impossible to be targeted as immunization nodes. Therefore for any specific social contact network, there will be a ceiling immunization ratio for CBI. implies all the community-bridge nodes will be immunized. The realistic immunization ratio will always be less than .
Figure 3a and Figure 3b show the convergent trajectories of the optimal immunization ratio obtained by binary search method. For AI, a small f will lead to the unsuccessful control of the epidemics, while a large f will lead to a large amount of susceptible individuals immunized. will display a “valley” shape and the optimal immunization exists. For CBI, there is two possible different shapes of the convergent trajectory: one is the same as that in AI, where ; the other is the “slope” shape, which arises when , however, due to the finiteness of the community-bridge nodes in the network, the immunization ratio will be .
Analysis of key parameters of RV
RV is targeted locally in a ring around the identified sources of infection and is extensively investigated in theoretical studies on immunization strategies in responding to the outbreak of the smallpox and foot-and-mouth disease– as well as the disease control practice. The success of RV depends on several crucial factors, such as the rapid identification of cases and the efficient contact trace of the identified cases, which could be captured by the following parameters: (i) the probability of diagnosis per day of infectious individuals, denoted by r; (ii) the probability of successful trace for the contacts of type i, denoted by ; (iii) the time lag in tracing for the contacts of type i, denoted by . For close contacts, i.e., contacts in household, school and workplace, assume that the contact escape rate and the time lag are 0, that is, all close contacts could be traced and the immunization could be carried out immediately. The values of the parameters and related statements are listed in Table 2. For simplicity, the q was utilized instead of and instead of for casual contacts, where i = 2∼6.
Figure 4 shows the dynamics of the epidemics varies with the parameters. With the increase of r and q, the contour lines of the fraction of susceptible individuals slope upwards, indicating that the effectiveness of RV are strengthened with the increase of the probability of diagnosis of class I individuals and the probability of successful traces for casual contacts, see Figure 4a and Figure 4b. Figure 4c shows the contour lines slope downwards, indicating that the increase of the time lag in tracing for casual contacts will lead to more affected individuals (individuals with states R or V).
Effectiveness comparison and sensitivity analysis
The effectiveness of AI, CBI and RV in controlling the spread of infectious diseases has been compared. All these three immunization strategies were demonstrated to have successfully decreased the coverage of the epidemics compared to the scenario with no control measures (no control measures) in Figure 5a. A further comparative analysis of the difference in the fraction of susceptible individuals with respect to time steps between each pair of strategies were made in Figure 5b. Take the comparison between AI and NO as an example, the intersections between the horizontal line related to (AI – NO) and the contour lines show that (AI – NO) is negative in the early stage of the epidemics because of the dramatic decrease of the susceptible individuals resulting from the immunization after the outbreak of the diseases for AI. This phenomenon could also be observed in the scenario for CBI. With the evolution of the epidemics, the susceptible individuals will decrease more rapidly for NO than for AI, so (AI – NO) will become positive. The difference peaked around and then decreased, and was eventually positive, indicating that AI had decreased the coverage of the epidemics compared to NO. Similar analytics could be applied to other pair strategies (CBI – AI), (RV – CBI) and (RV – AI). Figure 5b shows AI outperforms CBI and RV significantly outperforms CBI and AI under the definition of the objective function .
The optimal immunization ratios for AI and CBI decrease with the increase of r, see Figure 6d. Figure 6a and Figure 6b give the dynamics of the epidemics with different r. Both of the contour lines of the fraction of susceptible individuals for AI and CBI slope upwards, indicating that the coverage of the epidemics decrease with the increase of r. The effectiveness of AI and CBI was further compared in Figure 6c, which shows AI always outperforms CBI for r ranging from 0.1 to 0.9 and the predominance will strengthen with the increase of r.
The effectiveness of RV is greatly denominated by r, q and . The effectiveness of RV compared to AI and CBI with different r, q and was analyzed in Figure 7. The contour lines in Figure 7a to Figure 7d slope upwards, indicating that the effectiveness of RV improves significantly with the increase of r and q compared to AI and CBI. It is notable that the dark colors are related to lower fractions of susceptible individuals, so the contour lines in Figure 7e and Figure 7f slope upwards slightly implies that the effectiveness of RV decreases with the increase of , yet still significantly outperforms AI and CBI.
Although recent large-scale outbreaks of the epidemics suggest is usually between 2.0 and 4.0, studies on smallpox shows that its could reach 6.0. Here the effectiveness of AI, CBI and RV for ranging from 2.0 to 6.0 was analyzed. Figure 8a to Figure 8c show the epidemic dynamics with different . All the contour lines slope downwards, indicating the coverage of the epidemics increases with the increase of for all the three strategies. For RV, the declining rate of the contour lines decreases, indicating the number of susceptible individuals decreases more slowly. Therefore, RV is more effective compared to AI and CBI. More straightforward comparisons of the effectiveness of these three strategies with different were given in Figure 8d to Figure 8h. With the increase of , the affected individuals will increase for all the strategies, however, the decrease of susceptible individuals for RV is significantly slower than for AI and CBI, i.e., RV notably outperforms AI and CBI.
In this paper the effectiveness of three stochastic immunization strategies in controlling the spreading of the epidemics based on realistic social contact networks was analyzed. We found that there exists an optimal immunization ratio for AI and CBI for a specific r which leads to a minimum number of individuals infected and immunized. This optimal immunization ratio decreases with the increase of r and could be determined by the binary search method. For RV, the case detection rate, the contact trace escape rate and the contact trace lag are three most important factors. The effectiveness of RV improves with the increase of the case detection rate, while decreases with the increase of the contact escape rate and the contact trace lag. The comparison of the effectiveness of these three strategies shows AI, CBI and RV have decreased the coverage of the epidemics compared to a baseline scenario (no control measures, NO), however the effectiveness of AI and CBI are very limited, between which AI outperforms CBI. RV is very effective in controlling the epidemics and its effectiveness significantly outperforms AI and CBI.
The sensitivity analysis shows the effectiveness of RV decrease with the decrease of case detection rate and the increase of contact trace escape rate and the contact trace time lag, yet still remarkably outperforms AI and CBI on equal terms. With the increase of the basic reproductive number , the coverage of the epidemics will increase for all these three strategies, however the increase of the number of class R and class V individuals for RV is much less than for AI and CBI, indicating that RV notably outperforms AI and CBI. Even when reaches 6.0, the effectiveness of RV is prominent, implying that RV could be applied to controlling the epidemics with a wide infectivity spectrum. We found that RV is a newly confirmed case-triggered and locally controlled strategy, so RV is more acceptable in infectious disease control practice, which will improve the effectiveness of RV in return. However, AI and CBI are epidemic-triggered and global controlled strategies. These control strategies call for large control resources due to a relatively high immunization ratio as analyzed in the main text, also the allocation and the logistics of the resources might be a problem. Moreover, the global control strategies might encounter the resistance of the public, so the effectiveness could be worse. Although the social contact network used in this research is just one instance for Portland urban population, it is the first time the effectiveness of several classical immunization stochastic immunization strategies are evaluated based on realistic contact networks. These results could have significant importance for epidemic control research and practice.