Modelling the influence of progressive social awareness, lockdown and anthropogenic migration on the dynamics of an epidemic
MModelling the influence of progressive socialawareness, lockdown and anthropogenic migrationon the dynamics of an epidemic
R. Bhattacharyya ∗ and Partha Konar † Physical Research Laboratory, Ahmedabad - 380009, Gujarat, India
Abstract
The basic Susceptible-Infected-Recovered (SIR) model is extended to includeeffects of progressive social awareness, lockdowns and anthropogenic migration.It is found that social awareness can effectively contain the spread by loweringthe basic reproduction rate R . Interestingly, the awareness is found to be moreeffective in a society which can adopt the awareness faster compared to theone having a slower response. The paper also separates the mortality fractionfrom the clinically recovered fraction and attempts to model the outcome oflockdowns, in absence and presence of social awareness. It is seen that staggeredexits from lockdowns are not only economically beneficial but also helps to curbthe infection spread. Moreover, a staggered exit strategy with progressive socialawareness is found to be the most efficient intervention. The paper also exploresthe effects of anthropogenic migration on the dynamics of the epidemic in a two-zone scenario. The calculations yield dissimilar evolution of different fractionsin different zones. Such models can be convenient to strategize the division of alarge zone into smaller sub-zones for a disproportionate imposition of lockdown,or, an exit from one. Calculations are done with parameters consistent with theSARS-COV-2 pathogen in the Indian context. Keywords:
Mathematical model, Susceptible-Infected-Recovered (SIR), Epidemicmigration ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . s o c - ph ] A ug Introduction
The mathematical modelling of infectious disease is necessary to understand its spreadamong a population as the individuals interact among themselves. Additional to var-ious transmission mechanisms and properties of the pathogen, the spread can also bea function of societal properties which can include social habits, travel patterns, socialdistancing and personal hygiene. The models—stand-alone or combined with statisti-cal techniques—provide insights related to the severity of infection by predicting thenumber of infected persons, the rate at which they are getting infected and the mortal-ity rate; among others. The information can further be employed to strategize variousinterventions in advance to contain the spread. For example, in the ongoing COVID19pandemic [1] in India, interventions in the form of early screenings and isolations alongwith the ultimate lockdown—claimed by WHO to be ”timely and toughest” [2]— areimplemented.Effective, but mathematically straightforward, are the compartmental models whichassign individuals of a population at a particular stage of the epidemic to designatedcompartments [3]. Governed by ordinary differential equations (ODEs), individuals arethen allowed to move from one compartment to another as they pass through variousstages of the epidemic. The number of compartments, their coupling and the inter-compartmental flow is decided by various properties of the concerned pathogen; in-cluding its incubation period and the duration of immunity in the recovered patients—along with other external factors like the availability of a vaccine or the number ofbirths and deaths during the evolution. The models inherently assume individualsin a particular compartment to be characteristically identical. Such an assumption ispossible only when the population is large enough to make the probability of distribut-ing identical individuals in a compartment statistically significant. Consequently, thecompartmental models are expected to work well for systems having large populations.The simplest of the compartmental models are the SIR model, first used by Kermackand McKendrick in 1927 [4] and subsequently applied to a variety of diseases, espe-cially airborne childhood diseases with lifelong immunity upon recovery—like measles,mumps, rubella and pertussis (see [5] and references therein). In its basic form, themodel lacks vital dynamics, i.e. does not take into account the births/deaths alongwith the incubation period of the pathogen and the recurrence of susceptibility in com-pletely recovered individuals. Further extensions of this model (Susceptible - Exposed- Infectious - Recovered (SEIR) and Susceptible - Exposed - Infectious - Recovered- Susceptible (SEIRS) are made to include the long incubation periods of certainpathogens ( like chickenpox and dengue) during which an individual can be infectedbut not infectious. A comprehensive list of these models along with their governingODEs can be found in [6], the hosting site of the Epidemiological MODeling software2EMOD)—developed and maintained by The Institute for Disease Modeling (IDM),an institute within the Global Good Fund—a collaboration between Intellectual Ven-tures and Bill and Melinda Gates; idem [7]. Although the models described above arehighly sophisticated, but we believe extensions are required to integrate societal andbehavioural changes in response to an epidemic. Toward the objective, we consider theSIR model as the baseline for its mathematical simplicity. Additionally, we also addinter- zone dynamics due to migration and a procedure to calculate mortality amongthe recovered individuals. The organization of the paper is as follows: Section II intro-duces Initial Value Problems (IVPs) with their governing ODEs, Section III documentsthe simulations and analyze the results while Section IV summarizes the importantfindings.
II The Initial Value Problem
Since the proposed IVPs are based on the SIR model, in the following, we introducethe model ODEs to lay the basis for their attempted advancements [4]. With N asthe total population, variables S , I and R denote the number of individuals who areSusceptible (not infected), Infected and Recovered at an instant t . The correspondingfractions are s ( t ) = SN , (1) i ( t ) = IN , (2) r ( t ) = RN , (3)which can also be interpreted as probabilities satisfying s + i + r = 1 , (4)in the absence of any external forcing, i.e. no change in population because ofbirth/death or migration. The rate of change of S is directly proportional to thefraction s ( t ) and the total number of infected I ( t ), yielding. dSdt = − b s ( t ) I ( t ) , (5)where b is the proportionality constant. Realizing, the infected ultimately get recov-ered (or removed, because of death) dRdt = − k I ( t ) , (6)3rovided the recovered individuals acquire a permanent immunity to the pathogenand, there is no delay between the exposure and getting infected. Dividing both sideswith N , a convenient form is dsdt = − b s ( t ) i ( t ) , (7) drdt = k i ( t ) . (8)The ratio b/k ≡ R which is recognised as basic reproduction rate, quantifies theexpected number of secondary infections from a single infection in a population whereall individuals are susceptible. Taking derivatives on both sides of the Equation (4),the i equation is obtained as didt = [ R s ( t ) − i ( t ) . (9)At t = 0, didt | t =0 = [ R s (0) − i (0) , (10)which shows, for an infection to become epidemic, the condition ( R s (0) − > R time-dependent. To fix ideas, notable is the efficacy of a spread depends on thesocial awareness about the epidemic along with the properties of the pathogen. Suchsocial back-reactions have already been recognized [9]. Funk et.al. [8] have developed amathematical model which studies the dynamics of an epidemic in the presence of socialawareness through either direct observations or rumour. The results document theepidemic dynamics to complement human behaviour and vice versa. Arguably, socialawareness can lead to a proactive observance of hygiene—like regular hand washing,avoidance of physical contact, and maintaining social distancing. Importantly, theawareness is progressive, i.e. increases with time as the epidemic unfolds. For example,individuals may not be aware or fail to recognize the importance of the above preventivemeasures until the epidemic significantly develops. Also, an aggressive campaign byauthorities can implement some of the above deterrents effectively. For example, Govt.of India campaigned to raise awareness about the COVID19 by setting an informationnugget as a default caller tune across all cell phone service providers. The campaignwas particularly effective in rural areas where Internet access is rudimentary, butalmost everyone has cell phones. Contrarily, it is not practically feasible to implementdeterrents 100% effectively in a finite time. The reason may either be the consequentrecession or resistance of the populace to the changing lifestyle. To model such a4esponse, we consider b ( t ) = b exp ( − t/τ ) (11)where the time constant τ determines how fast and effectively a population can as-similate preventive interventions. Notably, a monotonically decreasing b ( t ) such as theabove, only takes into account the social back reactions which arrests the epidemic.Contrarily, the back reaction can have a negative impact also. For example, propaga-tion of rumors and other misinformations can inhibit the progressive social awareness,making b ( t ) non-monotonic—a scenario excluded in the present analysis. The modifiedSIR equations are dsdt = − b exp ( − t/τ ) s ( t ) i ( t ) , (12) drdt = k i ( t ) , (13) didt = k [ R ( t ) s ( t ) − i ( t ) . (14)Importantly, a time dependent decaying R opens up the possibility of satisfying[ R ( t ) s ( t ) − < r ) from the recovered (ˆ r ) one, by simply assuming thefatality rate ( m ), based on the virulence strain of pathogen and also existing treatmentfacility for the age distribution of particular demography. However, a co-morbidity cansignificantly raise the fatality rate which is not considered by the model. For example,a recent paper concludes that patients older than 65 years have more than two timeshigher risk of dying from COVID-19 while a similar risk exists if the patient is male[10]. Then r ( t ) = m ˇ r ( t ) + (1 − m ) ˆ r ( t ) . (16)One can extract the rate from the existing data across the different system. It is alsoexpected that the gradual understanding would enable us to rationalize the rate in asubsequent epoch.Another important extension is the inclusion of lockdown phase mimicked by asudden reduction of effective b ( b in ) value for a certain time (lockdown time) like afinite square well. That is implemented in the model with two sets of continuous and5ifferentiable Sigmoid functions, such as, b modellockdown = b in + ( b start − b in )[1 + exp ( t − t start )] + ( b end − b in )[1 + exp ( t end − t )] . (17). Here, b start/end and b in are corresponding values of effective b before (and after)the lockdown and during the lockdown respectively. Similarly, t start and t end are thestarting point and end point (days) of such lockdown. Above mentioned single stagelockdown for an extended period of days is neither feasible or recommended consideringthe substantial social and economical cost. It is followed with a multi step lockdown ora staggered removals of lockdown by gradual removal of restrictions. These scenariois studied by using two or three staged finite square well developed with Sigmoidfunctions in a phased manner. Examples are, b modellockdown,staggered [2] = b (1) in + ( b start − b (1) in )[1 + exp ( t − t start )] + ( b (2) in − b (1) in )[1 + exp ( t (1) in − t )] , + ( b end − b (2) in )[1 + exp ( t end − t )] , (18)with the similar notation as before except two different b values ( b ( i =1 , in ) in two stagesof lockdown. However, one can also express in economical use of variables in terms oflockdown periods (∆ ( i ) t ) in multi stage cases. Three stage model is thus expressed as, b modellockdown,staggered [3] = b (1) in + ( b start − b (1) in )[1 + exp ( t − t start )]+ ( b (2) in − b (1) in )[1 + exp ( t start + ∆ (1) t − t )]+ ( b (3) in − b (2) in )[1 + exp ( t start + ∆ (1) t + ∆ (2) t − t )]+ ( b end − b (3) in )[1 + exp ( t start + ∆ (1) t + ∆ (2) t + ∆ (3) t − t )] . (19)Finally, we extend SIR equations to allow for anthropogenic migration from one zoneto another. Markedly, the zones can either be separated geographical locations or ahypothetical separation of the same location into two subzones. The model is devel-oped with the continuous-time approach which results in ODEs where the variables areinherently continuous and rely on Mathematica’s accuracy in solving such equationswith a finite time step ∆ t . The outcome is expected to match with the reality only inthe limit ∆ t →
0. In contrast, mathematical models with discrete-time can be used tosolve the SIR equations [11]. In a similar work, Zakary et al. devised a discrete-time6IR model that describes the propagation of a disease in a population of individualswho travel between multiple regions [12].To develop the governing differential equations, two zones: zone 1 and zone 2 aredefined such that the total number of susceptible, infected and recovered individualssatisfy S = S + S , (20) I = I + I , (21) R = R + R . (22)Further, S + I + R = N , (23) S + I + R = N , (24)where N and N are the total populations in regions 1 and 2 while N is the over-all population. We further assume the N and N to be significantly large such thatreasonable inter-zonal migration does not affect them: in other words, the total popu-lations N and N are assumed to be independently constant. Corresponding fractionsare defined as s i = S i /N i , i i = I i /N i , r i = R i /N i , (25)where i = 1 ,
2. With α S ( t ) being the number of susceptible individuals migratingfrom zone 2 to zone 1 and α S ( t ) form zone 1 to zone 2 dS dt ∝ (1 − α ) s ( t ) + α s ( t ) , (26) dS dt ∝ (1 − α ) s ( t ) + α s ( t ) , (27)and dS dt ∝ (1 − β ) I ( t ) + β I ( t ) , (28) dS dt ∝ (1 − β ) I ( t ) + β I ( t ) , (29)finally leading to dS dt = − b [(1 − α ) s ( t ) + α s ( t )] [(1 − β ) I ( t ) + β I ( t )] . (30)Dividing by N , we get the s -equation for the zone 17 s dt = − b [(1 − α ) s ( t ) + α s ( t )] [(1 − β ) i ( t ) + β n n i ( t )] , (31)where n = N /N , n = N /N and N + N = N . A similar derivation for s gives ds dt = − b [(1 − α ) s ( t ) + α s ( t )] [(1 − β ) i ( t ) + β n n i ( t )] . (32)Similarly, the zonal equations for r s are found to be dr dt = k (cid:20) (1 − β ) i + β n n i (cid:21) (33) dr dt = k (cid:20) (1 − β ) i + β n n i (cid:21) (34)To obtain the i equation, we employ the conservation relations (23) and (24) in theirfractional form i.e ( s + i + r ) = 1 , (35)( s + i + r ) = 1 , (36)to generate (cid:32) ds i dt + di i dt + dr i dt (cid:33) = 0 , (37)where i = 1 ,
2. Using the s ( t ) and r ( t ) equations along with the conditions (35) and(36) form, the i ( t )-equations for the two zones are obtained as di dt = b [(1 − α ) s ( t ) + α s ( t )] [(1 − β ) i ( t ) + β n n i ( t )] − k [(1 − β ) i ( t ) + β n n i ( t )] , (38)and di dt = b [(1 − α ) s ( t ) + α s ( t )] [(1 − β ) i ( t ) + β n n i ( t )] − k [(1 − β ) i ( t ) + β n n i ( t )] . (39)The six ODEs (31),(32), (38), (39), (33) and (34) form a closed set for the six variables s , i , r , s , i , r . Expectedly, in the absence of migrations from zone 1 to 2 and8 � ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ .Figure 1: Demonstration of susceptible rate s(t) in blue, infected rate i(t) in red andremoved rate r(t) in green which includes recovered rate (ˆ r ( t ) which is not shownseparately) and the mortality rate ˇ r ( t ) in orange. Parameters considered as b = 0 . k = 0 . m = 0 .
05. .vice versa ( α = α = β = β = 0), the above equations reduce to the original SIRequations. III Simulations and Results
The relevant ODEs are solved by using NDSolve function of the Mathematica withthe appropriate initial condition. To benchmark, the following provides results for SIRsimulations with initial conditions, s | t =0 = 1 , (40) r | t =0 = 0 , (41) i | t =0 = 1 × − . (42)The initial values are chosen in line with the spread of COVID19 in India. With theapproximate total population of India ( ≈ . × ) normalized to unity, 100 infectionsper day yields a normalized value of i (0) = 1 × − , which we use in our calculations.Notably, the 100 infections per day were achieved during the middle of March 2020.The constant parameter k represent the rate at which the fraction of infected convertsinto recovered. Assuming an average period of 10 days the pathogen takes to spreadthe infection, we can choose an approximate k with a fraction of 1 /
10. The solutionsare illustrated in the Figure 1 with choice of parameters b = 0 . k = 0 .
1, amountingto R = 3. The histories of s ( t ), i ( t ), r ( t ) are represented by lines of colors blue, red,green respectively. The curve in orange represents the mortality rate. Notably, the9 � ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (a) �� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (b) �� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (c) ( t ) r ( t ) s ( t ) τ = � = ���� = ��� (d) Figure 2: Demonstration of the basic model is infused with progressive social awarenessby τ = (a) 1000 days, (b) 200 days and (c) 150 days respectively. Color scheme,notations and rest of the parameters remains same as Figure 1. At inset of each plotcorresponding variations of b is indicated. In plot (d) all three plots are described ina ternary diagram.distribution of i ( t ) is Gaussian. An increase in b (and hence R ) decreases the FullWidth at Half Maxima (FWHM) and the peak of the Gaussian (not shown), indicatinga faster spread of the infection. The sum of the mortality and the clinical recoveryrates is equal to the recovery rate in standard SIR plots. To distinguish, hereafter, werefer recovery rate in the standard SIR model as the ”removed” rate; removed, sincethe individuals in this category can not be further infected. The dashed line representsthe sum ( s + i + r ) and is equal to 1, as expected from the conservation (4).Figure 2 represent plots where the basic model is infused with the progressive socialawareness characterized by b ( t ) = b exp ( t/τ ); the time constant τ signifies the rate atwhich the b ( t ) and hence R ( t ) falls, quantifying how fast the society adapts variousinterventions. The inset diagram in 2(a) plots the evolution of b ( t ) with b = 0 . τ = 1000 days, which is near constant for all practical purpose, indicating very littlesocial awareness (or zero intervention) with time. The s , i and r plots are, expectedly,identical to the constant b case. The influence of the progressive social awarenessis evident in the next two plots 2(b) and 2(c) in the same Figure where aggressive10nterventions are imposed with τ = 200 and 150 days respectively. The insets for boththe plots, again, show the time variations of b . Evidently, the one with the fastest decayas in Figure 2(c) exhibits the infection curve to be most flattened and having muchlowest peak value. The above findings qualitatively agree with the recent simulationsby [13] which show delayed onset of successively diminished peaks in the total infectedpopulation with a stricter adherence to “social distancing”.To further explore the influence of social awareness on infection fraction, Figure2(d) presents a ternary diagram of the variables s ( t ), i ( t ) and r ( t ) in the ( s, i, r )space for different τ values to quantify their interrelationship. In the plot, the timeis implicit and ( s, i, r ) ∈ { , } satisfying s ( t ) + i ( t ) + r ( t ) = 1 at all t . To elucidatefurther, we consider any of the one curve in the plot and note that at the initialpoint s ( t = 0) = 1, i ( t = 0) = 1 × − , r ( t = 0) = 0 representing lower-leftcorner at the plot. All the three variables evolve implicitly with time, and after asufficiently large time interval, all the curves terminate at i = 0. The curve havingthe largest time constant, τ = 1000 days (minimally progressing social awareness), isthe highest peaked—having the largest FWHM. The opposite is true for the smallest τ = 150 days curve. Notably, the three curves with τ = 150 , , τ values. Contextual to the paper, such non-linear dependencyimplies that a society capable of developing social awareness at a moderately fasterpace during an epidemic gets far more benefited by additional campaigns than the onewhere the awareness develops at a slower rate. We believe, incorporation of the basicprogram on epidemic awareness at school curriculum can better prepare a society fora faster response. The program will mostly be beneficial for epidemics like COVID19where in the absence of vaccination and antiviral drugs; social interventions like socialdistancing, basic respiratory hygiene/cough etiquette, appropriate hand washing etc.,are the only available deterrents.A complete or a partial lockdown implies an enforced social measure to breakthe chain of infection by maximizing the social distancing and hence, minimizing thespread. In the following, we discuss the effects of lockdown on the infected fractionwithout and with progressive social awareness. For the purpose, notable is the re-alization that the lockdown effectively lowers the value of R for some finite period.Hence, the simplest lockdown is mimicked with a sudden reduction of b given by afinite square well. Such condition is modelled with a pair of sigmoid functions as de-scribed in Equation 17 and shown as insets in Figure 3. Here, in the time evolutionof b , the larger value represents no-lockdown, and the smaller value signifies the lock-down. The plots 3(a) and 3(b) represent histories of the variables for b in = 0 . .
2, respectively while k = 0 .
1. The no-lockdown value of b in both cases is fixed at0 .
3. The lockdown period (∆ t ) is 50 days, spanning between the day 50 to day 100.11 � ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (a) �� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (b) Figure 3: Demonstration of the basic model infused with simplest lockdown mimickedwith a b given by a finite square well as shown in the inset of plots with lockdownvariables b in = 0 . .
2. and the histories of the variables are shown respectively.Color scheme, notations and rest of the parameters remains same as Figure 1.Note that effective R in0 drops to the values 1 and 2 respectively during these examplelockdown periods. The choice of low R in Figure 3(a) effectively stops the infectionand idealizes the lockdown to be perfect. The Figure illustrates two dissimilar peaks in i ( t ) where the first peak (barely visible for this parameter choice) is in response to thelockdown and is centred at t ≈
52. The second peak in i ( t ) onsets after the lockdownis over and is located at t ≈ ≈ .
28 and spread t ∈ { , } days.A visual comparison with Figure 1 which documents a similar spread of the i ( t ) curve t ∈ { , } and a peak value of ≈ .
28 suggests a standalone lockdown with low R in0 can only delay the peak, providing additional preparation time for the authorities.Second plot in Figure 3(b) consider a rather pessimistic lockdown with comparativelyhigher side of R in0 during lockdown. Unlike the previous case, here the trend is moreadmixture with the no-lockdown scenario, as in Figure 1 except alleviating the curve,reducing the peak and broadening the spread. One realizes that the impact of lock-down period and the choice of b in ( R in0 ) produces an interplay between double Gaussianin i ( t ) and their interference which is demonstrated in our next discussion.The Figures 4(a) and 4(b) depict instances ( t i max day) of maximum i ( t ) and itsmagnitude i max as a function of ‘lockdown period’ and the parameter b in in contourplots. Plot 4(a) shows the occurrence of the peak value of i ( t ) is delayed with increasinglockdown period, as expected for low value of b in e.g. in Figure 3(a). However for a large b in one gets broader distribution with the peak value remaining mostly adjacent to thestarting point of the lockdown, as realized in Figure 3(b). Figure 4(b) demonstratesthe fact that the peak values i max mostly remain same irrespective of lockdown periodfor a fixed b in . However we encounter the same i max twice staying in constant lockdownperiod. These two peaks correspond to the transition from one to another of the doubleGaussian we discussed before. Interestingly, for particular range of b in ∈ { b α , b β } , thepeak value is minimum for lockdown perion ∆ L > ∆ ∗ , giving an optimal range of b in
20 40 60 80 100 120 1400.000.050.100.150.200.250.30
Lockdown period ( days ) b i n (a)
20 40 60 80 100 120 1400.000.050.100.150.200.250.30
Lockdown period ( days ) b i n (b) Figure 4: Contour plots to demonstrate the (a) day ( t i max ) when infection rate i ( t ) hitsthe global maxima and corresponding (b) instantaneous magnitude i max for differentchoices for lock down period and the parameter b in . Rest of the parameters remainssame as Figure 1.where a lockdown can be greatly effective.Further investigations are made to see the effects of staggered removals of lockdown.Such removals can be beneficial for the overall economy and also helps the daily wa-gers to earn their livelihoods together with the fine balance in keeping infection ratesmanageable. Figure 5 illustrates the effects of lockdowns having differently staggeredremovals. Panels 5(a) and 5(c) shows the square-step-well functions representing thesquare-well form of b for a two and three-stage exist after modelling the scenario as de-scribed in Equations (18) and (19). Characteristic values are { b (1)in = 0 . b (2)in = 0 . } and { b (1)in = 0 . b (2)in = 0 . b (3)in = 0 . } for two-stage and three-stage staggeredremovals respectively. As before, b is fixed at 0.3 outside the lockdown period. The i -curves in the panels a and b are almost similar, the peak for the two-staged staggeredexit being slightly delayed than the one for the three-staged exit. However, the peakvalue of the three-stage curve is also somewhat milder than the two-staged one. Thesecharacteristics also manifest in the corresponding ternary diagram, where we added afew additional cases for demonstration purpose. Lowest peaked ones (in solid lines) areselected through scanning the parameter space on choosing a set of b values providingminimum peak. One also notices the overlapping lines in perfect lockdown (where b ’sare zero during lockdown) with ones with no lockdown (where b ’s don’t reduce dur-ing lockdown). This demonstrates our earlier argument that perfect lockdown simplydelays the curve and show up as overlapping lines in the ternary diagram, where thetime axis is implicit in them.We further add progressive awareness in the above multistage scenarios. Figure 6depicts the effect. Once again, the profile of b is in the inset whereas panels (a) and13 � ��� ��� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (a) ( t ) r ( t ) s ( t ) { , } { . , . } { . , . } { . , . } � ��� = ���� � = ����������� = �� + �� ������������ ����� �� ���� (b) �� ��� ��� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (c) ( t ) r ( t ) s ( t ) { , , } { . , . , . } { . , . , . } { . , . , . } � ��� = ���� � = ����������� = �� + �� + �� ������������ ����� �� ���� (d) Figure 5: Demonstration of the basic model infused with two step (upper plots) andthree step (upper plots) staggered removals of lockdown mimicked with a b given bya finite step well as shown in the inset of plots. Lockdown variables are picked as b in = { . , . } ( { . , . , . } ) for {
50 + 50 } days ( {
40 + 30 + 30 } days) intwo (three) stage staggered removal as demonstrated as inset plots in (a) and (c)respectively . As before, b is always 0.3 outside the lockdown period. Histories ofthe variables are shown respectively. In plot (b) and (d) described the running of thevariables for the same set of parameters in a ternary diagram along with some otherparameters showing for the consistency.(b) represent the evolution of the variables for τ = 400 and 800 days, respectively. Inboth cases, the infection curves are found to be more flattened compared to the onewithout social awareness. This indicates a staggered exit from the lockdown alongwith measures to increase social awareness is not only good for an early restart of theeconomy but also beneficial in flattening the infection curve.The section is completed with a discussion on the modeling of the effects of migra-tion between two zones. Notably, such calculations can easily be extended to includeany number of zones. The s, i, r plots for the two zones (the subscripts pointing thespecific zone) are depicted in Figure 7. For simplicity the exchange constants are keptequal and tiny: α = β = α = β = 10 , amounting to an equally small numberof individuals traveling from zone 1 to zone 2 and vice versa providing internal mixing14 � ��� ��� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (a) �� ��� ��� ��� ��� ��� � ( ��� ) ��������������� �� �� � ∨ + � ∧ tb (b) ( t ) r ( t ) s ( t ) Solid: τ = τ = { }{ }{ }{ } � ��� = ���� � = ����������� = �� + �� + �� ������������ ����� �� ���� (c) Figure 6: Demonstration of the basic model infused with three-step (upper plots)staggered removals of lockdown mimicked with a b given by a finite step well as shownin the inset of plots in case of infused with progressive social awareness by τ = (a) 400days and (b) 800 days. Lockdown variables are picked as b in = { . , . , . } for {
40 + 30 + 30 } days in three stage staggered removal as demonstrated as inset plots in(a) and (b) respectively for these two values of τ . As before, b is always 0.3 outside thelockdown period. Histories of the variables are shown respectively. Plot (c) describethe running of the variables for the same set of parameters in a ternary diagram alongwith some other parameters showing for the consistency.among zones. The initial i ( t = 0) in the zones 1 and 2 are selected to be { − , } respectively. The histories of the different variables are shown in Figures 7a and 7b insolid and dashed curves respectively. Expectedly, the infection and recovery initiallybegun at zone 1 and as individuals inter-migrated between the zones—intensified inzone 2 also (Figure 7a). The combined evolution is depicted in Figure 7b which plots s = n s + n s , i = n i + n i and ˆ r + ˇ r = n ˆ r + n ˆ r + n ˇ r + n ˇ r . Futureplan involves testing and modeling different combinations to yield targeted favorableoutcomes. For example, finding the optimal set of parameter to effectively contain thespread within a minimal sized zone. which we keep as a future study.15 � ��� ��� ��� � ��������������� � � � � � � � � ∨ + � � ∧ � � � � � � � � ∨ + � � ∧ (a) �� ��� ��� ��� � ��������������� � �� � � �� � � �� ∨ + � �� ∧ (b) Figure 7: Demonstration of the
SIR dynamics in presence of anthropogenic mi-gration between two equally populated zones having small inter-zone transfer rates: α = β = 10 − and α = β = 10 − . Initial small infection rates of i ( t = 0) = 10 − was introduced only in the first zone. As expected both infection and recovery startedinitially in first zone (represented with solid lines). However very soon it was intro-duced in second zone too, and corresponding rates are shown with dashed lines in leftplot. Combined effect in both zones together is represented in right plot. IV Summary
The paper recognizes the importance of social back-reaction on the dynamics of anepidemic. In this work, the basic SIR framework is extended to explore the effectsof progressive social awareness which is mathematically modelled by a decaying expo-nential. It is found that the awareness lowers the effective R and reduces the peakinfection rate while delaying its appearance. Additionally, the progressive awareness ismore effective in societies having some seed knowledge about the various social deter-rents. Consequently, its inclusion in basic school curriculum can be effective in curbingfuture epidemics like COVID19 where social interventions remain only available de-terrents for a significant amount of time.The extended model also studies the effects of lockdowns, mimicked by square-well functions generating different effective R s and having different staggered exitstrategies. It is found that the simplest lockdown with single-phase implementationand exit neither flattens the infection curve nor decreases its peak but only delay itsappearance. The additional time can be utilized by the authority to prepare logistics.A staggered exit from a lockdown is better as the strategy flattens the infection curveas well as reduces and delays the peak. Also, such exist strategies are better from theeconomical perceptive also. The most efficient is an exit strategy planned with a jointincrease in social awareness. Such manoeuvrings can minimize the peak and flattensthe infection curve most—which we believe can be beneficial in future epidemics wherelockdowns will be necessary.We have also extended the SIR model to include two-zone anthropogenic migration.16he example presented is a basic one where an initially equal number of individuals areassumed to populate the two zones. A small number of people are allowed to migratebetween the zones while zone 2 is absolutely infection-free, and zone 1 is characterizedwith i (0) = 10 − . With time, both the regions get substantially infected with i havingthe same peak and spread while the peak for i appearing earlier than i .The paper lays a groundwork where different scenarios to arrest the spread of anepidemic along with the importance of social awareness is explored. Although, in thepresent work, the above scenarios are mostly examined individually; nevertheless, werecognize the natural synergy between progressive social awareness, lockdowns andanthropogenic migration to control the spread of epidemics. Such a study is left as afuture exercise. V Acknowledgement
The work is supported by Physical Research Laboratory (PRL), Department of Space,Government of India. All the computations were performed using the HPC resources(Vikram-100 HPC) and TDP project at PRL.
References446