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What community-based public health approaches in West Africa for COVID-19 epidemic? A reflection based on the African socio-cultural context
other
Author(s):
Ndèye Marème Sougou
1
,
2
,
& ,
Jean-baptiste Diouf
3 ,
Mamadou Makhtar Mbacké Leye
1
,
2 ,
Oumar Bassoum
1
,
2 ,
Adama Faye
1
,
2 ,
Ibrahima Seck
1
,
2
Publication date (Electronic):
25 June 2020
Journal:
The Pan African Medical Journal
Publisher:
The African Field Epidemiology Network
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Abstract
The social and cultural dimensions of health influence the course of disease and condition the success of health interventions. In Africa, previous epidemics such as Ebola have shown the importance of contextualizing health interventions. This literature review contributes to the reflection on the analysis of community-based interventions in the context of the particularities of West Africa in the fight against the pandemic in COVID-19.
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Targeted Social Distancing Designs for Pandemic Influenza
Robert Glass, Laura Glass, Walter E. Beyeler … (2006)
At the start of an influenza pandemic, effective vaccine and antiviral drugs may not be available to the general population ( 1 , 2 ). If the accompanying illness and death rates of the virus strain are high, how might a community respond to protect itself? Closing roads, restricting travel, and community-level quarantine will enter discussions ( 3 , 4 ). However, within a community, influenza spreads from person to person through the social contact network. Therefore, understanding and strategically controlling this network during a period of pandemic is critical. We describe how social contact network–focused mitigation can be designed. At the foundation of the design process is a network-based simulation model for the spread of influenza. We apply this model to a community of 10,000 persons connected within an overlapping, stylized, social network representative of a small US town. After study of the unmitigated transmission of influenza within the community, we change the frequency of contact within targeted groups and build combinations of strategies that can contain the epidemic. Finally, we show how infectivity of the strain and underlying structure of the infectious contact network influence the design of social distancing strategies. In the absence of vaccine and antiviral drugs, design for specific communities would defend against highly virulent influenza. Methods The design process first creates an explicit social contact network in which persons are linked to others in a community. Spread of influenza within the network is then simulated by imposing behavioral rules for persons, their links, and the disease. These rules are modified to implement targeted mitigation strategies within the community, the effectiveness of which is evaluated ( 5 ). Contact Network A network is created by specifying groups of given sizes (or range of sizes) within which persons of specified ages interact (e.g., school classes, households, clubs). The average number of links per person within the group is also specified because cliques form or are imposed (e.g., seating in a classroom). This number is used to construct a within-group network that can take various forms. We used fully connected, random, or ring networks for each group. Random networks are formed by randomly choosing 2 persons within the group and linking them. This process is repeated until the number of links within the group yields the specified average (each person will have a different number of links). The ring is formed by first placing persons next to neighbors and linking them to form a complete circle. Additional links are then made to next nearest neighbors symmetrically around the ring. Finally, links within a group are given an average frequency of contact per day. With this approach, a network can be built from the experience of community members to exhibit the clustered yet small-world characteristics ( 6 ) and overlapping quality of a structured community ( 7 , 8 ). Our network represented a stylized small US town and took advantage of the diverse backgrounds of the authors (1 of whom is a teenager). The population of 10,000 conformed to the 2000 Census ( 9 ) and consisted of children ( 65 years of age, 12.5%). All persons belonged to multiple groups, each associated with a subnetwork of links that reflected their lives within the community (Figure 1, Table 1). Households were composed of families (adults with children or teenagers), adults, or older adults. The age-class makeup and size of households conformed to the 2000 Census ( 9 ). All persons within each household were linked to each other with mean link contact frequencies of 6/day. Every person also belonged to 1 multiage extended family (or neighborhood) group (mean membership 12.5, mean link contact frequency 1/day). Figure 1 Typical groups and person-to-person links for model teenager. The teenager (T1) belongs to a household (fully connected network, mean link contact frequency 6/day), an extended family or neighborhood (fully connected network, mean link contact frequency 1/day), and 6 school classes (ring network with connections to 2 other teenagers on each side as shown in black; purple links denote connections of other teenagers within the class; mean link contact frequency 1/day). Two random networks are also imposed, 1 within the age group (teenager random, average of 3 links/teenager, mean link contact frequency of 1/day), and 1 across all age groups (overall random average of 25 links/person [not all links shown], mean link contact frequency of 0.04/day). Table 1 Groups, membership, networks, and mean frequencies of contact per link Group (no. groups in community) Membership Average no. links per member Network type and parameters Mean frequency of contact per link per day Households without older adults (2,730) 1–2 adults, 0–4 children, 0–4 teenagers, mean size 3.13 2.13 Fully connected 6 Households with older adults (742) 1–2 older adults, mean size 1.75 0.75 Fully connected 6 Extended families or neighborhoods (800) 0–2 older adults, 0–8 adults, 0–8 teenagers, 0–8 children, mean size 12.5 11.5 Fully connected 1 Child classes (69) 1 class per child, 20–35 children in each 4 Ring network, 2 neighbors on either side 6 Child random (1) All children 3 Random network link density 3/1,769 1 Teenager classes (264) 6 classes per teenager, 20–35 teenagers in each 4 Ring network, 2 neighbors on either side 1 Teenager random (1) All teenagers 3 Random network link density of 3/1,129 1 Adult work (351) 1 work group per adult, 10–50 adults in each 6 Ring network, 3 neighbors on either side 1 Adult random (1) All adults 3 Random network link density of 3/5,849 1 Older adult gathering (156) 1 gathering per person, 5–20 persons in each 4 Ring network, 2 neighbors on either side 1 Older adult random (1) All older adults 3 Random network link density of 3/1,249 1 Overall random (1) All age classes 25 Random network link density of 25/9,999 1/25 a day All children and teenagers attended preschool or school; children attended 1 class/day, while teenagers attended 6 (classes of 20 to 35 children or teenagers). All adults went to work daily, where they interacted with other adults (work group size 10–50), and all older adults attended gatherings with other older adults (gathering size 5–20). For links within school classes, work, and gatherings of older adults, we assumed the simplest subnetwork that imposes local clustering: a ring lattice in which a person is linked to 2 (for children or teenager classes and gatherings of older adults) or 3 (adult work) neighboring persons on each side along the ring. Mean link contact frequencies for children in a class are 6/day. Teenager classes, adult work, and gatherings of older adults have mean link contact frequencies of 1/day. To represent additional within-age class interactions, such as extracurricular activities, playgrounds, bowling leagues, or friends, persons are randomly linked to an average of 3 other persons of the same age class (mean link contact frequency 1/day). Finally, to emulate a somewhat patterned set of random contacts from commercial transactions and other ventures into public spaces, we impose a random overall network across all age classes with a mean of 25 links/person to yield 1 contact/person/day (mean link contact frequency 0.04/day). Behavioral Rules The spread of influenza within the contact network is modeled as a series of 2 classes of events: transition of a person between disease states and person-to-person transmission of influenza. Disease state transitions follow the natural history of influenza (Figure 2). After the latent state, an infected person transitions to an infectious presymptomatic state or an infectious asymptomatic state with probability pS or 1 – pS, respectively. Those with symptoms either stay home with probability pH, thus influencing their contacts, or continue to circulate with probability 1 – pH. Infected asymptomatic persons continue interacting without behavioral changes. Persons who are symptomatic die or become immune with probability pM or 1 – pM, respectively, and asymptomatic persons become immune. Because this final transition does not influence the spread of the disease, we use pM = 0. Figure 2 Natural history of influenza in our model. Duration of each state for a given person is chosen from an exponential distribution. State relative infectivity (IR) and mean state duration were chosen to reflect the infectivity variation of Ferguson et al. ( 10 , 11 ) (see Figure 3). Transition probabilities between presymptomatic and postsymptomatic states are also noted. For symptomatic persons who stay at home, link frequencies outside the household are reduced by 90%. Person-to-person transmission events are evaluated at the beginning of each period during which a person is infectious. Assuming contact events are statistically independent, a transmission time for each infectious person's links within the contact network is chosen from an exponential distribution with a mean of the link's contact frequency scaled by ID × IR × IA × SP × SA, where ID is the infectivity of the disease, IR is the relative infectivity of the disease state, SP is the susceptibility of people to the disease (here taken as 1.0), IA is the relative infectivity of the person who is transmitting, and SA is the relative susceptibility of the person receiving. If the transmission time is less than the period during which the person will be in an infectious state (also chosen from an exponential distribution with the prescribed means; Figure 2), transmission is scheduled at the chosen time. Otherwise, transmission along that link does not occur during that period. All transmission parameters and contact frequencies may be modified in each of the states, as well as varied among age classes by relative scaling factors such as IR. In this way, disease representations and mitigation strategies are implemented. Most influenza-specific parameters used here reflect those of ( 10 , 11 ). We approximated normal influenza viral shedding data ( 15 ) with a time varying infectivity through choice of state periods and relative infectivity scaling factors (Figure 2 and Figure 3). The latent period is a constant (0.75 days) followed by a variable period (mean 0.5 days). The presymptomatic period (mean 0.5 days) has an IR of 0.25 after which it increased to 1.0 for the first part of the symptomatic period (mean 0.5 days), when viral shedding is maximum and coughing begins. IR is then reduced to 0.375 for the remainder of the infectious symptomatic period (mean 1 day). For infectious asymptomatic persons, IR was set at 0.25 for a mean period of 2.0 days, making these persons half as infective as those with symptoms. We chose pS as 0.5, pH as 0.5 for adults and older adults and pH as 0.9 for children and teenagers. When a person is in the symptomatic stay-home state, we reduce the frequency of all nonhousehold connections by 90%. Because children and teenagers have closer contact with others and are less likely to wash hands or control coughs ( 16 ), they are more infective and susceptible: IA and SA are both 1.5 for children, 1.25 for teenagers, and 1.0 for adults and older adults. Finally, ID is adjusted to yield specified attack rates within the community. Figure 3 Functional behavior of IR with time. Although infectivity of an asymptomatic person is constant with time (IR 0.25), infectivity of a symptomatic person changes from infectious presymptomatic (IR 0.25) to early infectious symptomatic (IR 1.0) to late symptomatic (IR 0.375). A symptomatic person with mean state periods as denoted in Figure 2 is shown in gray (asymptomatic with dashed line). Because state periods are different for each person (given by exponential distributions) and half of the infected persons are asymptomatic, the average population scale IR in time is smoothed as shown in blue. Both disease state periods and IR values were chosen to honor the clinically derived natural history of influenza ( 12 – 14 ), selected viral shedding data shown as open red squares ( 15 ), and the model of Ferguson et al. ( 10 , 11 ). Results We first show the spread of influenza within our unmitigated base case defined with parameters specified above and with ID chosen to yield an infected attack rate ≈50% to reflect the 1957–58 Asian influenza pandemic ( 10 ). Unless otherwise noted, we report infected attack rates and refer to them as simply attack rates rather than reporting the illness attack rate which is half of this value (pS = 0.5). We then demonstrate the design of effective local mitigation strategies for the base case that focus on targeted social distancing. Finally, we extend these results to design strategies for more infectious strains and for changes to the underlying infectious contact network that deemphasize the role of children and teenagers. All simulations are initialized by infecting 10 randomly chosen adults with the assumption that adults are first to be infected through business travel or interaction with visitors from outside the community ( 5 ). Some of these initial infections instigate others and grow into an epidemic. Results vary across multiple realizations of the community network and random choice of initially infected adults (controlled by random number seed) not all of which yield an epidemic, defined when the number infected is >1% of the population. For every set of parameters, we conducted >100 simulations with different random number seeds and collected statistics for all simulations and for only those that result in epidemics (Table 2). Table 2 Results for base case and miigation strategies* Strategy Averages for all simulations Averages for simulations with epidemics No. simulations Total infected Total time (d) Peak infected Time to peak (d) No. epidemics Total infected Total time (d) Peak infected Time to peak (d) Case 1: Base case pandemic influenza Average 1,000 4,908 81 688 35 978 5,018 82 703 36 SD 748 14 121 8 153 11 66 6 Case 2: Schools closed after 10 symptomatic cases, compliance 90% Average 100 3,877 113 326 48 99 3,916 114 329 48 SD 468 22 64 13 259 19 56 12 % reduction from base case 21 -40 53 -36 22 -39 53 -34 Case 3: Schools closed after 10 symptomatic cases, nonschool contacts doubled, compliance 90% Average 100 5,604 76 850 34 95 5,898 79 894 35 SD 1,293 18 206 9 122 10 72 6 % reduction from base case -14 6 -24 4 -18 4 -27 2 Case 4: Schools closed after 10 symptomatic cases, children and teenagers kept home, household contacts doubled, compliance 90% Average 100 341 60 43 16 93 361 62 45 17 SD 209 25 20 12 203 24 19 12 % reduction from base case 93 26 94 53 93 25 94 52 Case 5: Schools closed after 10 symptomatic cases, children and teenagers kept home, household contacts doubled, compliance 50% Average 100 1,551 135 90 47 95 1,630 141 94 49 SD 692 49 40 31 614 42 37 30 % reduction from base case 68 -67 87 -33 68 -72 87 -36 Case 6: Schools closed after 10 symptomatic cases, children kept home, household contacts doubled, compliance 90% Average 100 2,539 116 199 49 96 2,642 120 206 51 SD 661 30 66 17 433 23 56 14 % reduction from base case 48 -44 71 -38 47 -46 71 -40 Case 7: All with symptomatic cases stay at home, compliance 90% Average 100 3,692 91 408 41 94 3,926 95 433 43 SD 1,031 25 130 14 458 17 85 10 % reduction from base case 25 -12 41 -16 22 -16 38 -20 *Cases 2–7 are targeted social distancing strategies. Negative percent reductions reflect percent increases. Epidemics are defined as >100 infected. SD, standard deviation. Unmitigated Base Case The sequence of infected persons can be represented as an expanding network of infectious transmissions (Figure 4). The number of secondary infections produced by an infected person, or branching factor, is easily visualized within the infectious contact network. The average branching factor depends on the person's age class and generation during the epidemic (Figure 5A). The maximum value within the first 10 generations is 2.05 (standard deviation [SD] 0.57) for children, 2.09 (SD 0.72) for teenagers, 1.11 (SD 0.43) for adults, 0.81 (SD 0.47) for older adults, and 1.54 (SD 0.36) for the entire population. Variability (large SD, especially for specific age classes) reflects the heterogeneity inherent within community contact networks of this size (Figure 5B). Figure 4 Initial growth of an infectious contact network. Colored rectangles denote persons of designated age class, and colored arrows denote groups within which the infectious transmission takes place. In this example, from the adult initial seed (large purple rectangle), 2 household contacts (light purple arrows) bring influenza to the middle or high school (blue arrows) where it spreads to other teenagers. Teenagers then spread influenza to children in households who spread it to other children in the elementary schools. Children and teenagers form the backbone of the infectious contact network and are critical to its spread; infectious transmissions occur mostly in the household, neighborhood, and schools. Figure 5 Branching factor and the approximation of the reproductive number Ro . A) Overall and age class–specific branching factors as a function of generation averaged over 100 simulations. The standard deviations of these averages can be large ( 100 infected). Standard deviations for variation of threshold are 100 infected). RO , reproductive number; ID , disease infectivity; SD, standard deviation. Figure 8 Unmitigated age-specific attack rate results for disease infectivity (ID) factors of 1.0 and 2.0 and base case, variation 1 (removal of relative infectivity and susceptibility), variation 2 (increase in work group frequency of contact to give all children, teenagers, and adults the same overall contact frequencies), and variations 1 and 2 combined. Illness attack rates shown in Figure 6 are half these values. To find effective targeted social distancing strategy combinations across the range of disease infectivity and infectious contact networks, we formulated 5 strategies and applied them individually and in combination: 1) school closure (S) where the contact frequency within schools was reduced 90% and children and teenager's household contacts were doubled; 2) children and teenagers social distancing (CTsd) where their contact frequencies in all nonhousehold and nonschool groups were reduced 90% and their household contacts doubled; 3) adult and older adult social distancing (AOAsd) where their contact frequencies in all nonhousehold and nonwork groups were reduced 90% and household contacts doubled; 4) liberal leave (LL), where all children and teenagers and 90% of adults withdraw to the home when symptomatic; and 5) work social distancing (Wsd) where the contact frequency within work groups was reduced 50%. For each combination, we implemented the strategy(ies) the day after 10 symptomatic cases and conducted 100 simulations. As ID increases, more strategies must be combined to keep the attack rate 100 infected). Average standard deviation across the entire set of simulations was 2.2% with a maximum of 7.6%. Discussion Results for our stylized small town suggest that targeted social distancing strategies can be designed to effectively mitigate the local progression of pandemic influenza without the use of vaccine or antiviral drugs. For an infectivity similar to that of the 1957–58 Asian influenza pandemic, targeting children and teenagers, by not only closing schools but also by keeping these age classes at home, was effective. However, given uncertainty in the infectivity of the influenza strain, underlying social contact network, or relative infectivity/susceptibility of the young versus adults, planning to implement strategies that also target adults and the work environment is prudent. To mitigate a strain with infectivity similar to that of the 1918–19 Spanish influenza pandemic, simulations suggest that all young and adults must be targeted regardless of the likely enhanced transmission by the young. Implementation of social distancing strategies is challenging. They likely must be imposed for the duration of the local epidemic and possibly until a strain-specific vaccine is developed and distributed. If compliance with the strategy is high over this period, an epidemic within a community can be averted. However, if neighboring communities do not also use these interventions, infected neighbors will continue to introduce influenza and prolong the local epidemic, albeit at a depressed level more easily accommodated by healthcare systems. Our design approach explicitly implements disease-host interaction within the social contact network where the disease spreads. Measuring contact networks within communities for the spread of infectious diseases requires focused research that combines sociology, public health, and epidemiology. Such networks will likely differ across cultures, between urban and rural communities, and with community size. With the aid of detailed demographic data, expert elicitation of social scientists and community members, behavioral surveys, and possibly experiments, a network could be constructed for any community of interest. Configurations that consider, for example, college campuses or military reservations may be of use given that the highest death rate of any group in the 1918–19 Spanish influenza pandemic was in young adults ( 22 ).
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Game Theory of Social Distancing in Response to an Epidemic
Timothy Reluga (2010)
Introduction Epidemics of infectious diseases are a continuing threat to the health of human communities, and one brought to prominence in the public mind by the 2009 pandemic of H1N1 influenza [1]. One of the key questions of public health epidemiology is how individual and community actions can help mitigate and manage the costs of an epidemic. The basic problem I wish to address here is how rational social-distancing practices used by individuals during an epidemic will vary depending on the efficiency of the responses, and how these responses change the epidemic as a whole. Social distancing is an aspect of human behavior particularly important to epidemiology because of its universality; everybody can reduce their contact rates with other people by changing their behaviors, and reduced human contact reduces the transmission of many diseases. Theoretical work on social distancing has been stimulated by studies of agent-based influenza simulations indicating that small changes in behavior can have large effects on transmission patterns during an epidemic [2]. Further research on agent-based models has argued that social distancing can arrest epidemics if started quickly and maintained for a relatively long period [3]. Compartmental epidemic models have also been used to study social distancing by including states that represent individuals employing specific behaviors. For instance, Hyman and Li [4] formulate and begin the analysis of flu disease transmission in SIR models where some individuals decrease their activity levels following infection. Reluga and Medlock [5] uses this approach to show that while social distancing can resemble immunization, it can generate hysteresis phenomena much more readily than immunization. Rather than treating behaviors as states, some models treat behaviors as parameters determined by simple functions of the available information. Reluga et al. [6] studies dynamics where contact rates can depend on the perceived disease incidence. Buonomo et al. [7] investigates the impact of information dynamics on the stability of stationary solutions in epidemic models. Chen [8] considers a similar system but allows individuals to learn from a random sample of neighbors. Funk et al. [9] considers the information dynamics associated with social distancing in a network setting by prescribing a reduction in contacts based on proximity to infection. Related work by Epstein et al.[10] explicitly considers the spatial and information dynamics associated in response to an ongoing epidemic. Building on the ground-breaking work of Fine and Clarkson [11], there has been substantial recent interest in the application of game theory to epidemiology [12]–[17]. The games studied so far have primarily considered steady-state problems, and have not allowed for dynamic strategies. One notable exception to this is the work of Francis [18], which determines the time-dependent game-theoretical solution of a vaccination problem over the course of an epidemic. In another, van Boven et al. [19] studies the optimal use of anti-viral treatment by individuals when they take into account the direct and indirect costs of treatment. To study the best usage of social distancing, we apply differential-game theory at a population-scale. Differential games are games where strategies have a continuous time-dependence; at each point in time, a player can choose a different action. For instance, a pursuit-game between a target and a pursuer is a two-player differential game where each player's strategies consist of choosing how to move at each successive time until the target is caught by the pursuer or escapes. Geometrically, one might think of differential games as games where strategies are represented by curves instead of points. Two-player differential-game theory was systematically developed by Isaacs [20] as an extension of optimal control theory [21]–[23]. Here, we employ an extension of differential game theory to population games of the form described by Reluga and Galvani [24]. The analysis in this paper will be limited to the simplest case of the Kermack–McKendrick SIR model with strong mixing [25]. In the Model section, we formulate an epidemiological-economics model for an epidemic, accounting for the individual and community costs of both social distancing practices and infection. We then use differential game theory and numerical methods to identify the equilibrium strategies over the course of an epidemic. Numerical methods are used to investigate the finite-time problem where vaccines become available after a fixed interval from the start of the epidemic and the infinite-horizon problem without vaccination. Fundamental results on the value and timing of social distancing are obtained. Model In this article, social distancing refers to the adoption of behaviors by individuals in a community that reduce those individuals' risk of becoming infected by limiting their contact with other individuals or reducing the transmission risk during each contact. Typically, social distancing incurs some costs in terms of liberty, social capital, time, convenience, and money, so that people are only likely to adopt these measures when there is a specific incentive to do so. In addition to the personal consequences, the aggregate effects of social distancing form an economic externality, reducing the overall transmission of disease. This externality needs to be accounted for in the determination individuals optimal strategies, but, by definition, depends on the choice of strategy. To resolve this interdependence, we formulate our analysis as a population game where the payoff to each individual is determined by the individual's behavioral strategy and the average behavioral strategy used by the population as a whole. The model is related to that previously studied by Chen [26]. We will use to represent one specific individual's strategy of daily investment in social distancing. The population strategy is the aggregate daily investment in social distancing by the population. The overbar notation is used to indicate that the aggregate investment should be thought of as an average investment aggregated over all individuals in the population. In the limit of infinitely large populations, and can be thought of as independent because changes in one person's behavior will have little affect on the average behavior. Similarly, the epidemic's dynamics depend on the population strategy but are independent of any one individual's behavior . The effectiveness of social distancing is represented by a function , which is the relative risk of infection given a daily investment in social distancing practices. If there is no investment, the relative risk . As the daily investment increases, the relative risk decreases, but is bounded below by . We expect diminishing returns with increasing investment, so we will also make the convenient assumption that is convex. Consider a Susceptible-Infected-Recovered (SIR) epidemic model with susceptible ( ), infected and infectious ( ), and removed ( ) states. Suppose an epidemic starts with cases in a community of total individuals (taking ) and proceeds until time , at which point all the individuals in the susceptible state are vaccinated. This epidemic is fast relative to demographic processes and we do not distinguish among the possible states of individuals leaving the infectious state, so the population size can be treated as constant. Between time and time , the dynamics are described by (1a) (1b) (1c) where is the transmission rate and is the removal rate. This SIR model assumes the population is homogeneous, strongly mixed, and that the duration of infections is exponentially distributed. At the start of the epidemic when there are few cases of infection ( ), the basic reproduction number . The total cost of the epidemic to the community, , is the sum of the direct costs plus the indirect costs of any economic repercussions from the epidemic. To keep our analysis tightly focused, we will only consider direct costs of the epidemic, including the daily costs from infection, daily investments in social distancing, and the costs of vaccination. Mathematically, (2) where is the daily cost of each infection, is the cost of vaccination per person, and is the discount rate. Note that while the cost of infection is a constant, the investment in social distancing is a function of time. The last term in Eq. (2) is called a salvage term and represents the cumulative costs associated with individuals who are sick at the time the vaccine is made available ( ). The assumption that the entire remaining susceptible population is vaccinated at time and that vaccination takes effect instantly is, of course, unrealistic, but does provide an approximation to the delayed release of a vaccine. To simplify our studies, we will work with the dimensionless version of the equations by taking: (3) Under this choice of units, time will be measured in terms of disease generations, social distancing costs will be measured relative to the daily cost of infection, and population sizes will be measured relative to the critical population size necessary to sustain an epidemic. Epidemics usually start with one or a few index cases, so we focus on scenarios where . The dynamics can be described in terms the shape of , the discount rate , and a single initial-condition parameter (4) From this, it follows that . Since epidemics are often much faster than human demographic processes governing the discount rate [27], we will also take in all calculations. Henceforth, we will drop the hat-notation and work with the dimensionless parameters. The dimensionless equations are (5a) (5b) (5c) with the constraint that . Note that we drop the function notation when necessary to simplify the presentation. For our further analysis, we will assume (6) with the maximum efficiency of social distancing . Eq. (6) is nicely behaved for numerical solutions because of its relatively fat tail. The Social Distancing Game We now formulate a differential game for individuals choosing their best social distancing practices relative to the aggregate behavior of the population as a whole. The following game-theoretic analysis combines the ideas of Isaacs [20] and Reluga and Galvani [24]. The premise of the game is that at each point in the epidemic, people can choose to pay a cost associated with social distancing in exchange for a reduction in their risk of infection. The costs of an epidemic to the individual depend on the course of the epidemic and the individual's strategy of social distancing. The probabilities that an individual is in the susceptible, infected, or removed state at time evolve according to the Markov process (7) where is the individual's daily investment as a function of the epidemic's state-variables and the transition-rate matrix (8) Note that both and change over time. Along the lines discussed above, and represent different quantities in our analysis; represents one individual's investment strategy and the population strategy represents an aggregated average of all individual investments. We also note that there are several different ways and can be parameterized. They may be parameterized in terms of time, as and , or in implicit feedback form and , or in explicit feedback form and . The form used will be clear from the context. Since the events in the individual's life are stochastic, we can not predict the exact time spent in any one state or the precise payoff received at the end of the game. Instead, we calculate expected present values of each state at each time, conditional on the investment in social distancing. The expected present value is average value one expects after accounting for the probabilities of all future events, and discounting future costs relative to immediate costs. The expected present values of each state evolve according to the adjoint equations (9) where . The components , , and represent the expected present values of being in the susceptible, infected, or removed state at time when using strategy in a population using strategy . The expected present values depend on the population strategy through the infection prevalence . The adjoint equations governing the values of each state are derived from Markov decision process theory. They are (10a) (10b) (10c) with the constraints that for all time . Solution of (10)b and (10)c gives (11) If it is impossible to make a vaccine, the equations must be solved over an infinite horizon. Over an infinite horizon, , assuming becomes constant. In the case of no discounting ( ), we still have provided for sufficiently large . In the case where a perfect vaccine is universally available at terminal time , the value of the susceptible and removed states differs by the cost of vaccine for . To avoid complications with the choice of whether-or-not to vaccinate, we take so . This is reasonable in scenarios where the cost of the vaccine is covered by the government. The dynamics are independent of , so we need not consider removed individuals further. Taking and , we need only study the reduced system (12a) (12b) (12c) with boundary conditions (12d) The other conditions must be calculated from the solution of the boundary-value problem and provide useful information. will be the expected total cost of the epidemic to the individual. The final size of the epidemic is given by . Game Analysis Solving a game refers to the problem of finding the best strategy to play, given that all the other players are also trying to find a best strategy for themselves. In some games, there is a single strategy that minimizes a player's costs no matter what their opponents do, so that strategy can very reasonably be referred to as a solution. In many games, no such strategy exists. Rather, the best strategy depends on the actions of the other players. Any strategy played by one player is potentially vulnerable to a lack of knowledge of the strategies of the other players. In such games, it is most useful to look for strategies that are equilibria, in the sense that every player's strategy is better than the alternatives, given knowledge of their opponent's strategies. A Nash equilibrium solution to a population game like that described by System (12) is a strategy that is a best response, even when everybody else is using the same strategy. i.e. given , is a Nash equilibrium if for every alternative strategy , . A Nash equilibrium strategy is a subgame perfect equilibrium if it is also a Nash equilibrium at every state the system may pass through. I will not address the problem of ruling out finite-time blowup of the Hamilton–Jacobi equation and establishing existence and uniqueness of subgame perfect equilibria. But numerical and analytical analyses strongly support the conjecture that the stategies calculated here are the unique global subgame perfect equilibria to the social distancing game. The equilibria of System (12) can be calculated using the general methods of Isaacs [20]. The core idea is to implement a greedy-algorithm; at every step in the game, find the investment that maximizes the rate of increase in the individual's expect value . We represent strategies as functions in implicit feedback form. is the amount an individual invests per transmission generation when the system is at state . If is a subgame perfect equilibrium, then it satisfies the maximum principle (13) when everywhere. So long as behaves well, in the sense that it is differentiable, decreasing, and strictly convex, then is uniquely defined by the relations (14) Figure 1 shows the interface in phase space separating the region where the equilibrium strategy will include no investment in social distancing ( ) from the region where the equilibrium strategy requires investment in social distancing ( ). 10.1371/journal.pcbi.1000793.g001 Figure 1 Contour plots of relative risk surface for equilibrium strategies. The relative risk is presented in feedback form with implicit coordinates (left) and transformed to explicit coordinates (right) for the infinite-horizon problem with maximum efficiency . The greater the value of the susceptible state ( ), the greater the instantaneous social distancing. We find that increasing the number of susceptible individuals always decreases the investment in social distancing, and the greatest investments in social distancing occur when the smallest part of the population is susceptible. Note that in the dimensionless model, the value of the infection state . Two cases are immediately interesting. The first is the infinite-horizon problem – what is the equilibrium behavior when there is never a vaccine and the epidemic continues on until its natural end? The second is the finite-horizon problem – if a vaccine is introduced at time generations after the start of the epidemic, what is the optimal behavior while waiting for the vaccine? In both of these cases, it is assumed that all players know if and when the vaccine will be available. The infinite-horizon and finite-horizon problems are distinguished by their boundary conditions. In the finite-horizon case, we assume all susceptible individuals are vaccinated at final time , so , , , while and are unknown. In the limit of the infinite-horizon case ( ), we solve the two-point boundary value problem with terminal conditions , , and initial conditions , while and are unknown. But these conditions are insufficient to specify the infinite-horizon problem. The plane is a set of stationary solutions to Eq. (12), so we need a second order term to uniquely specify the terminal condition when we are perturbed slightly away from this plane. Using Eq. (12), we can show solutions solve the second-order terminal boundary condition (15) for as . Most of the equilibria we calculate are obtained numerically. Some exceptions are the special cases where , . Under these conditions, solutions can be obtained in closed-form. First, . While , and (16) When matched to the terminal boundary condition, we find that if we write in feedback form as a function of rather than , (17) is a solution so long as for all . Inspecting the inequality condition, we find that this holds as long as . Results A problem with solving Eq. (12) under Eq. (14) is that it requires to be known from past time and to be known from future time. This is a common feature of boundary-value problems, and is resolved by considering all terminal conditions . Using standard numerical techniques, identifying an equilibrium in the described boundary-value problem reduces to scalar root finding for to match the given . The special form of the population game allows the solution manifold to be calculated directly by integrating backwards in time, rather than requiring iterative approaches like those used for optimal-control problems [23]. Code for these calculations is available from the author on request. Before presenting the results, it is helpful to develop some intuition for the importance of the maximum efficiency of investments in social distancing. Given for an arbitrary relative risk function , then in the best-case scenarios, where diminishments on returns are weakest, one would have to invest atleast of the cost of infection per disease generation to totally isolate themselves. The units here are derived from dimensional analysis. This could be invested for no more than generations, before one's expenses would exceed the cost of becoming infected. When returns are diminishing, fewer than generations of total isolation are practical. Thus, the dimensionless efficiency can be thought of as an upper bound on the number of transmission generations individuals can afford to isolate themselves before the costs of social distancing outweigh the costs of infection. For the infinite-horizon problem, an example equilibrium strategy and the corresponding dynamics in the absence of social distancing are shown in Figure 2. We can show that if social distancing is highly inefficient (the maximum efficiency ), then social distancing is a waste of effort, no matter how large . If social distancing is efficient, then there is a threshold value of below which social distancing is still impractical because the expected costs per day to individuals is too small compared to the cost of social distancing, but above which some degree of social distancing is always part of the equilibrium strategic response to the epidemic (Figure 3). 10.1371/journal.pcbi.1000793.g002 Figure 2 Epidemic solutions with equilibrium social distancing and without social distancing. Social distancing reduces the epidemic peak and prolongs the epidemic, as we can see by comparing a time series with subgame-perfect social distancing (top left) and a time series with the same initial condition but no social distancing (bottom left) (parameters , ). In the phase plane (right), we see that both epidemics track each other perfectly until , when individuals begin to use social distancing to reduce transmission. Eventually, social distancing leads to a smaller epidemic. The convexity change appearing at the bottom the phaseplane orbit with social distancing corresponds to the cessation of social distancing. 10.1371/journal.pcbi.1000793.g003 Figure 3 Social distancing threshold. This is the threshold that dictates whether or not equilibrium behavior involves some social distancing. It depends on both the basic reproduction number and the maximum efficiency , and is independent of the exact form of . As rough rules of thumb, if or , then equilibrium behavior involves no social distancing. The exact window over which social distancing is used depends on the basic reproduction number, the initial and terminal conditions, and the efficiency of distancing measures. The feedback form of equilibrium strategies, transformed from coordinates to the coordinates of the phase-space is represented with contour plots in Figure 1. Among equilibrium strategies, social distancing is never used until part-way into the epidemic, and ceases before the epidemic fully dies out. The consequences of social distancing are shown in Figure 4. The per-capita cost of an epidemic is larger for larger basic reproduction numbers. The more efficient social distancing, the more of the epidemic cost can be saved per person. However, the net savings from social distancing reaches a maximum around , and never saves more than % of the cost of the epidemic per person. For larger 's, social distancing is less beneficial. 10.1371/journal.pcbi.1000793.g004 Figure 4 Total costs and savings. Plots of the total per-capita cost of an epidemic (left) under equilibrium social distancing for the infinite-horizon problem with several efficiencies under Eq. (6), and the corresponding per-capita savings (right). Savings in expected cost compared to universal abstention from social distancing are largest for moderate basic reproduction numbers, but are relatively small, even in the limit of infinitely efficient social distancing. The case corresponds to infection of the minimum number of people necessary to reduce the reproduction ratio below . We can also calculate solutions of the finite-time horizon problem where a vaccine becomes universally available at a fixed time after the detection of disease (Figure 5). If mass vaccination occurs soon enough, active social distancing occurs right up to the date of vaccination. Using numerical calculations of equilibria over finite-time horizons, we find that there is a limited window of opportunity during which mass vaccine can significantly reduce the cost of the epidemic, and that social distancing lengthens this window (Figure 6). The calculations show that increases in either the amount of time before vaccine availability or the basic reproduction number increase the costs of the epidemic. Smaller initial numbers of infections allow longer windows of opportunity. This is as expected because the larger the initial portion of the population infected, the shorter the time it takes the epidemic to run its full course. 10.1371/journal.pcbi.1000793.g005 Figure 5 Solutions when vaccine becomes available after a fixed time. These are time series of an equilibrium solution for social distancing when mass vaccination occurs generations (left) and generations (right) after the start of the epidemic. Investments in social distancing begin well after the start of the epidemic but continue right up to the time of vaccination. Social distancing begins sooner when vaccine development is faster. For these parameter values ( ), individuals save % of the cost of infection per capita (left) and % of the cost of infection (right). 10.1371/journal.pcbi.1000793.g006 Figure 6 Windows of Opportunity for Vaccination. Plots of how the net expected losses per individual ( ) depend on the delay between the start of social-distancing practices and the date when mass-vaccination becomes universally available if individuals use a Nash equilibrium strategy. The more efficient social distancing, the less individuals invest prior to vaccine introduction. The blue lines ( ) do not use social distancing, as the efficiency is below the threshold. The dotted lines represent the minimal asymptotic epidemic costs necessary to stop an epidemic. Discussion Here, I have described the calculations necessary to identify the equilibrium solution of the differential game for social distancing behaviors during an epidemic. The benefits associated with the equilibrium solution can be interpreted as the best outcome of a simple social-distancing policy. We find that the benefits of social distancing are constrained by fundamental properties of epidemic dynamics and the efficiency with which distancing can be accomplished. The efficiency results are most easily summarized in terms of the maximum efficiency , which is the percent reduction in contact rate per percent of infection cost invested per disease generation. As a rule-of-thumb, is an upper bound on the number of transmission generations individuals can isolate before the costs of social distancing outweigh the costs of infection. Social distancing is not practical if this efficiency is small compared to the number of generations in the fastest epidemics ( ). While social distancing can yield large reductions in transmission rate over short periods of time, optimal social-distancing strategies yield only moderate reductions in the cost of the epidemic. Our calculations have determined the equilibrium strategies from the perspective of individuals. Alternatively, we could ask what the optimal social distancing practices are from the perspective of minimizing the total cost of the epidemic to the community. Determination of the optimal community strategy leads to a nonlinear optimal control problem that can be studied using standard procedures [23]. Yet, practical bounds on the performance of the optimal community strategy can be obtained without further calculation. The optimal community strategy will cost less than the game-theoretic solution per capita, but must cost more than , as that is the minimum number of people who must become sick to reduce the effective reproduction number below the epidemic threshold. Preliminary calculations indicate that optimal community strategies and game equilibrium strategies converge as grows, and significant differences are only observable for a narrow window of basic reproduction numbers near . The results presented require a number of caveats. I have, for instance, only considered one particular form for the relative risk function. Most of the analysis has been undertaken in the absence of discounting ( ), under the assumption that the epidemic will be fast compared to planning horizons. Discounting would diminish importance of long term risks compared to the instant costs of social distancing, and thus should diminish the benefits of social distancing. The benefits of social distancing will also be diminished by incorporation of positive terminal costs of vaccination ( ). Realistically, mass vaccination cannot be accomplished all-at-once, as we assume. It's much more likely that vaccination will be rolled out continuously as it becomes available. This could be incorporated into our analysis, for instance, by including a time-dependent forcing. Other approaches include extending the model to incorporate vaccination results of Morton and Wickwire [28], or to allow an open market for vaccine purchase [18]. The simple epidemic model is particularly weak in its prediction of the growths of epidemics because it assumes the population is randomly mixed at all times. We know, however, that the contact patterns among individuals are highly structured, with regular temporal, spatial, and social correlations. One consequence of heterogeneous contact structure is that epidemics proceed more slowly than the simple epidemic model naively predicts. Thus, the simple epidemic model is often considered as a worst-case-scenario, when compared with more complex network models [29], [30] and agent-based models [31]–[33]. In the context of social distancing, it is not immediately clear how weaker mixing hypotheses will affect our results. Weakened mixing will prolong an epidemic, increasing the window over which social distancing is needed. But under weakened mixing, individuals may be able to use local information to refine their strategies in ways analogous to the ideas of Funk et al. [9] and Perisic and Bauch [34]. In general, the analysis of aggregate games with stochastic population dynamics require a significant technical leaps, and are the subjects of active research. One of the fundamental assumptions in our analysis is that there are no cost-neutral behavior changes that can reduce contact rates. In fact, life-experience provides good evidence that many conventional aspects of human behavior are conditional on cultural norms, and that different cultures may adopt alternative conventions. The introduction of a new infectious disease may alter the motivational pressures so that behavioral norms that were previously equivalent are no longer, and that one norm is now preferred to the others. In such cases, there are likely to be switching costs that retard the rapid adoption of the better behaviors that conflict with cultural norms. The rate of behavior change, then, would be limited by the rate of adoption of compensatory changes in cultural norms that reduce the cost of social distancing. Another deep issue is that behavior changes have externalities beyond influencing disease incidence, but we have not accounted for these externalities. People's daily activities contribute not just to their own well-being but also to the maintenance of our economy and infrastructure. Social distancing behaviors may have serious negative consequences for economic productivity, which might feed back into slowing the distribution of vaccines and increasing daily cost-of-living expenses. We can extend our analysis to include economic feedbacks by incorporating capital dynamics explicitly. Individuals may accumulate capital resources like food, water, fuel, and prophylactic medicine prior to an epidemic, but these resources will gradually be depleted and might be difficult to replace if social distancing interferes with the economy flow of goods and services. Further capital costs at the community and state scales may augment epidemic valuations. These factors appear to have been instrumental in the recent US debate of school-closure policies. One feature of a model with explicit capital dynamics is the possibility of large economic shocks. This and related topics will be explored in future work. These calculations raise two important mathematical conjectures which I have not attempted to address. The first is that the social distancing game possesses a unique subgame-perfect Nash equilibrium. There is reasonable numerical evidence of this in cases where the relative risk function is strictly convex, and stronger unpublished arguments of this in cases of piecewise linear . I believe this will also be the case for non-convex but monotone relative risks under some allowances of mixed-strategies. A second conjecture, not yet addressed formally, is that increases in the efficiency of social distancing always lead to greater use of social distancing, all other factors being equal. This seems like common sense, but the precise dependence of Figure 1 on the efficiency has yet to be determined mathematically. As with all game-theoretic models, human behavior is unlikely to completely agree with our equilibria for many reasons, including incomplete information about the epidemic and vaccine and strong prior beliefs that impede rational responses. On the other hand, our approach is applicable to a large set of related models. We can analyze many more realistic representations of pathogen life-cycles. For instance, arbitrary infection-period distributions and infection rates can be approximated using a linear chain of states or delay-equations [24]. Structured populations with metapopulation-style mixing patterns may also be analyzed. I hope to apply the methods to a wider variety of community-environment interactions in the future.
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Facemasks for the prevention of infection in healthcare and community settings.
C. MacIntyre, Abrar Ahmad Chughtai (2015)
Facemasks are recommended for diseases transmitted through droplets and respirators for respiratory aerosols, yet recommendations and terminology vary between guidelines. The concepts of droplet and airborne transmission that are entrenched in clinical practice have recently been shown to be more complex than previously thought. Several randomised clinical trials of facemasks have been conducted in community and healthcare settings, using widely varying interventions, including mixed interventions (such as masks and handwashing), and diverse outcomes. Of the nine trials of facemasks identified in community settings, in all but one, facemasks were used for respiratory protection of well people. They found that facemasks and facemasks plus hand hygiene may prevent infection in community settings, subject to early use and compliance. Two trials in healthcare workers favoured respirators for clinical respiratory illness. The use of reusable cloth masks is widespread globally, particularly in Asia, which is an important region for emerging infections, but there is no clinical research to inform their use and most policies offer no guidance on them. Health economic analyses of facemasks are scarce and the few published cost effectiveness models do not use clinical efficacy data. The lack of research on facemasks and respirators is reflected in varied and sometimes conflicting policies and guidelines. Further research should focus on examining the efficacy of facemasks against specific infectious threats such as influenza and tuberculosis, assessing the efficacy of cloth masks, investigating common practices such as reuse of masks, assessing compliance, filling in policy gaps, and obtaining cost effectiveness data using clinical efficacy estimates.
Author and article information
Journal
Journal ID (nlm-ta): Pan Afr Med J
Journal ID (iso-abbrev): Pan Afr Med J
Journal ID (publisher-id): PAMJ
Title:
The Pan African Medical Journal
Publisher:
The African Field Epidemiology Network
ISSN
(Electronic):
1937-8688
Publication date
(Electronic):
25
June
2020
Publication date Collection: 2020
Volume: 35
Issue: Suppl 2
Electronic Location Identifier: 91
Affiliations
Author notes
[&
]
Corresponding author: Ndèye Marème Sougou, Department of Preventive Medicine and Public Health, Cheikh
Anta Diop University of Dakar, Senegal,
nmsougou@
123456hotmail.com
Article
Publisher ID: PAMJ-SUPP-35-2-91DOI: 10.11604/pamj.supp.2020.35.2.24415
PMC ID: 7875794
SO-VID: da54e770-d3b7-4877-8ecb-e1e9b2832d68
Copyright © © Ndèye Marème Sougou et al.
License:
The Pan African Medical Journal - ISSN 1937-8688. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
History
Date
received
: 19
June
2020
Date
accepted
: 24
June
2020
Categories
Subject:
Essay
ScienceOpen disciplines: Medicine