title: Modelling the influence of progressive social awareness, lockdown and anthropogenic migration on the dynamics of an epidemic
authors: R. Bhattacharyya Partha Konar
text: 
  
   
  
  ]R. Bhattacharyyaramit@prl.res.in 
  ]Partha Konarkonar@prl.res.in 
  []Physical Research Laboratory, Ahmedabad - 380009, Gujarat, India
                       
  
  
  
  Modelling the influence of progressive social awareness, lockdown and anthropogenic migration on the dynamics of an epidemic
  
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  The basic Susceptible-Infected-Recovered (SIR) model is extended to include effects of progressive social awareness, lockdowns and anthropogenic migration. 
  It is found that social awareness can effectively contain the spread by lowering the basic reproduction rate  R_0. Interestingly, the awareness is found to be more effective in a society which can adopt the awareness faster compared to the one having a slower response. The paper also separates the mortality fraction from the clinically recovered fraction and attempts to model the outcome of lockdowns, in absence and presence of social awareness. It is seen that staggered exits from lockdowns are not only economically beneficial but also helps to curb the infection spread. Moreover, a staggered exit strategy with progressive social awareness is found to be the most efficient intervention.
  The paper also explores the effects of anthropogenic migration on the dynamics of the epidemic in a two-zone scenario.
  The calculations yield dissimilar evolution of different fractions in different zones. 
  Such models can be convenient to strategize the division of a large zone into smaller sub-zones for a disproportionate imposition
  of lockdown, or, an exit from one. Calculations are done with parameters consistent with the SARS-COV-2 pathogen in the Indian context.
  
  
  Keywords:
      Mathematical model,
      Susceptible-Infected-Recovered (SIR),
      Epidemic migration
  
  
  
  
  § INTRODUCTION
  
  
  The mathematical modelling of infectious disease is necessary to understand its spread 
  among a population as the individuals interact among themselves. Additional to various 
  transmission mechanisms and properties of the pathogen, the spread can also be
  a function of societal properties which can include social habits, travel patterns, social distancing and 
  personal hygiene. The models—stand-alone 
  or combined with statistical techniques—provide insights related to the severity
  of infection by predicting the number of 
  infected persons, the rate at which they are getting infected and the mortality rate; among others. The 
  information can further be employed to strategize various interventions in advance to contain the spread. For 
  example, in 
  the ongoing COVID19 pandemic <cit.> in India, interventions in the form of 
  early screenings and isolations along with the ultimate lockdown—claimed by WHO to 
  be "timely and toughest" <cit.>— are implemented.
  
   
  Effective, but mathematically straightforward, are the compartmental models which assign individuals of a population at a particular stage of the epidemic to designated compartments <cit.>. 
  Governed by ordinary differential equations (ODEs), individuals are then allowed to move from one compartment to another as they pass through various stages of the epidemic. The number of compartments, their coupling and the inter-compartmental flow is decided by various properties of the concerned pathogen; including 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 of births and deaths during the evolution. The models inherently assume individuals in a particular compartment to be characteristically identical. Such an assumption is possible only when the population is large enough to make the probability of distributing identical individuals in a compartment statistically significant. Consequently, the compartmental models are expected to work well for systems having large populations. The simplest of the compartmental models are the SIR model,
  first used by Kermack and McKendrick in 1927 <cit.> and subsequently applied to a variety of diseases, 
  especially airborne childhood diseases with lifelong immunity upon recovery—like measles, mumps, 
  rubella and pertussis (see <cit.> and references therein). In its basic form, the model lacks vital dynamics, i.e. does not take into account the births/deaths
  along with the incubation period of the pathogen and the recurrence of susceptibility in completely 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 certain pathogens ( like chickenpox and dengue) during which an individual can be infected but not infectious. A comprehensive list of these models along with their governing ODEs can be found in <cit.>, 
  the hosting site of the Epidemiological MODeling software (EMOD)—developed and maintained by 
  The Institute for Disease Modeling (IDM), an institute within the Global Good Fund—a collaboration between 
  Intellectual Ventures and Bill and Melinda Gates; idem <cit.>.
  Although the models described above are highly sophisticated, but we believe extensions are required to integrate societal and behavioural
  changes in response to an epidemic. Toward the objective, 
  we consider the SIR model as the baseline for its mathematical simplicity. Additionally, we also add inter-
  zone dynamics due to migration and a procedure to calculate mortality among the recovered individuals. 
  The organization of the paper is as follows: Section <ref> introduces Initial
  Value Problems (IVPs) with their governing ODEs, Section <ref> documents the simulations
  and analyze the results while Section <ref> summarizes the important findings. 
  
  
  
  
  § THE INITIAL VALUE PROBLEM
  
  
   
  Since the proposed IVPs are based on the SIR model, in the following, we introduce the model ODEs to lay the basis for their attempted advancements <cit.>. With N as the total population, 
  variables S, I and R denote the number of individuals who are Susceptible (not infected),
  Infected and Recovered at an instant t. The corresponding fractions are
  
  
      s(t)=S/N,
            i(t)=I/N,
            r(t)=R/N,
  
  which can also be interpreted as probabilities satisfying 
  
  
      s+i+r=1,
  
  in the absence of any external forcing, i.e. no change in population because of birth/death or
  migration. The rate of change of S is directly proportional to the fraction s(t) and the total number of infected I(t), yielding. 
  
  
      dS/dt = -b   s(t)   I(t)  ,
  
  where b is the proportionality constant. 
  Realizing, the infected ultimately get recovered 
  (or removed, because of death) 
  
  
      dR/dt = -k   I(t)  ,
  
  provided the recovered individuals acquire a permanent immunity to the pathogen
  and, there is no delay between the exposure and getting infected. Dividing both
  sides with N, a convenient form is 
  
  
      ds/dt = -b   s(t)   i(t)  ,
  
  
  
      dr/dt = k   i(t)  .
   
  The ratio b/k≡ R_0 which is recognised as basic reproduction rate, 
  quantifies the expected number of secondary
  infections from a single infection in a population where all 
  individuals are susceptible. Taking derivatives on both sides of the
  Equation (<ref>), the
  i equation is obtained as
  
  
      di/dt=[R_0   s(t)   - 1 ]i(t).
  
  At t=0, 
  
  
      di/dt|_t=0=[R_0   s(0)   - 1]i(0),
  
  which shows, for an infection to become epidemic, the condition (R_0s(0)-1)> 0 must be satisfied.
  Otherwise the infection does not spread but dies out. 
  
  In order to explore the societal/behavioural impact on the pathogen spread, we make R_0 time-dependent. 
  To fix ideas, notable is the efficacy of a spread depends on the social awareness about the epidemic along with the properties of the pathogen. Such social back-reactions have already been recognized <cit.>. Funk et.al. <cit.>
  have developed a mathematical model which studies the dynamics of an epidemic in the presence of social awareness through either direct observations or rumour. The results document the epidemic dynamics to 
  complement human
  behaviour and vice versa. Arguably, social awareness can lead to a proactive observance of hygiene—like 
  regular hand washing, avoidance of physical contact, and maintaining social distancing. Importantly, the awareness 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 preventive measures until the epidemic significantly develops. Also, 
  an aggressive campaign by authorities can implement some of the above deterrents effectively. For example, 
  Govt. of India campaigned to raise awareness about the COVID19
  by setting an information nugget as a default caller tune across all cell phone service providers. The campaign was particularly effective in rural areas where Internet access is rudimentary, but almost everyone has cell phones. Contrarily, 
  it is not practically feasible to implement deterrents 100% effectively in a finite time.
  The reason may either be the consequent recession or
  resistance of the populace to the changing lifestyle. 
   To model such a response, we consider
  
  
      b(t) = b_0  exp(-t/τ)
  
  where the time constant τ determines how fast and effectively a population 
  can assimilate preventive interventions. Notably, a monotonically decreasing
  b(t) such as the above, only takes into account the social back reactions 
  which arrests the epidemic. Contrarily, the back reaction can have a negative impact also. 
  For example, propagation of rumors and other misinformations can inhibit the progressive 
  social awareness, making b(t) non-monotonic—a scenario excluded  in the present analysis.
   The modified SIR equations are
  
  
      ds/dt = - b_0  exp(-t/τ)  s(t)   i(t)  ,
            dr/dt = k   i(t)  ,
            di/dt = k  [R_0 (t)   s(t) - 1]   i(t)  .
  
  Importantly, a time dependent decaying R_0 opens up the possibility of satisfying 
  
  
      [R_0 (t)s(t)-1] < 1
  
  during evolution, after which the epidemic fizzles out.
  
  
  
  Notably, the SIR model does not differentiate between the clinically recovered population and the deceased but considers both as recovered in a sense that they are no-more susceptible or infected. In this present example, 
  it is straightforward to separate the fractional mortality (ř) from the recovered (r̂) one, 
  by simply assuming the fatality rate (m), based on the virulence strain of pathogen and also existing 
  treatment facility for the age distribution of particular demography. However, a co-morbidity 
  can significantly 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 times higher risk of 
  dying from  COVID-19 while a similar risk exists if the patient is male <cit.>.
  Then 
  
      r(t) = m  ř(t)  +  (1 - m)  r̂(t).
  
  One can extract the rate from the existing data across the different system. It is also expected that the gradual understanding would enable us to rationalize the rate in a subsequent epoch. 
  
  
  Another important extension is the inclusion of lockdown phase mimicked by a sudden reduction of effective b 
  (b_in) value for a certain time (lockdown time) like a finite square well. That is implemented in the 
  model with two sets of continuous and differentiable Sigmoid functions, such as,
  
  
      b_lockdown^model =  b_in + (b_start - b_in)/[1 + exp(t - t_start)]
                                                         + (b_end - b_in)/[1 + exp(t_end - t)] .
  .
  
  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 the starting point and end point 
  (days) of such lockdown.  Above mentioned single stage lockdown for an extended period of days is neither 
  feasible or recommended considering the substantial social and economical cost. It is followed with a multi step lockdown or a staggered removals of lockdown by gradual removal of restrictions. These scenario is studied by 
  using two or three staged finite square well developed with Sigmoid functions in a phased manner. Examples are,
  
      b_lockdown, staggered[2]^model  =   b_in^(1)   +   (b_start - b_in^(1) )/[1 + exp(t - t_start)]
               + (b_in^(2) - b_in^(1) )/[ 1 + exp(t_in^(1)  - t)],
         +   (b_end - b_in^(2) )/[1 + exp(t_end - t)],
  
  with the similar notation as before except two different b values (b_in^(i=1,2)) in two stages of lockdown. 
  However, one can also express in economical use of variables in terms of lockdown periods (Δ_t^(i)) in 
  multi stage cases. Three stage model is thus expressed as,
  
      b_lockdown, staggered[3]^model  =   b_in^(1)   +   (b_start
      - b_in^(1) )/[1 + exp(t - t_start)]
         +   (b_in^(2)  - b_in^(1) )/[1 +
      exp(t_start + Δ_t^(1)  - t)]
         +   (b_in^(3)  - b_in^(2) )/[1 + exp(t_start + Δ_t^(1)  +
      Δ_t^(2)  - t )]
         +   (b_end - b_in^(3) )/[  1 + exp(t_start + Δ_t^(1) 
      + Δ_t^(2)  + Δ_t^(3)  - t)] .
  
  
  Finally, we extend SIR equations to allow for anthropogenic migration from one zone to another. Markedly, the zones can either be separated geographical locations
  or a hypothetical separation of the same location into two subzones. The model is developed with the continuous-time approach which results in ODEs where the variables are inherently continuous
  and rely on Mathematica's accuracy in solving such equations with a finite time step 
  Δ t. The outcome is expected to match with the reality only in the
  limit Δ t → 0.  In contrast, mathematical models with discrete-time
  can be used to solve the SIR equations <cit.>.
  In a similar work, Zakary et al. devised
  a discrete-time SIR model that describes the propagation of
  a disease in a population of individuals who travel between
  multiple regions <cit.>.
  
  To develop the governing differential equations, two zones: zone 1
  and zone 2 are defined such that the total number of susceptible, infected and recovered individuals
  satisfy
  
      S=S_1+S_2, 
             I=I_1 + I_2, 
             R=R_1+R_2.
  
  Further, 
  
      S_1+I_1+R_1=N_1, 
             S_2+I_2+R_2=N_2,
  
  where N_1 and N_2 are the total populations in 
  regions 1 and 2 while N is the overall population. We 
  further assume the N_1 and N_2 to be significantly large such 
  that reasonable inter-zonal migration does not affect them: 
  in other words, the total populations N_1 and N_2 are assumed to be 
  independently constant.
  Corresponding fractions are defined as
  
      s_i=S_i/N_i,    
       i_i=I_i/N_i,    
       r_i=R_i/N_i,
  
  where i=1,2.
  With α_21 S_2(t) being the number of susceptible individuals migrating from zone 2 to zone 1 and
  α_12 S_1(t) form zone 1 to zone 2 
  
  
      dS_1/dt∝(1-α_12)s_1(t)+ α_21 s_2(t),
            dS_2/dt∝(1-α_21)s_2(t)+ α_12 s_1(t),
  
  and
  
      dS_1/dt∝(1-β_12)I_1(t)+ β_21 I_2(t),
            dS_2/dt∝(1-β_21)I_2(t)+ β_12 I_1(t),
  
  finally leading to 
  
      dS_1/dt   =   -b[(1-α_12)s_1(t)+ α_21 s_2(t)]
      
      [(1-β_12)I_1(t)+ β_21 I_2(t)].
  
  
  
  Dividing by N_1, we get the s-equation for the zone 1
  
  
  
      ds_1/dt   =   -b  [(1-α_12)s_1(t)+ α_21 s_2(t)]
      
      [(1-β_12)i_1(t)+ β_21n_2/n_1 i_2(t)],
  
  where n_1=N_1/N, n_2=N_2/N and N_1+N_2=N. 
  A similar derivation for s_2 gives
  
  
  
      ds_2/dt   =   -b  [(1-α_21)s_2(t)+ α_12 s_1(t)]
      
      [(1-β_21)i_2(t)+ β_12n_1/n_2 i_2(t)].
  
  Similarly, the zonal equations for rs  are found to be 
  
  
  
      dr_1/dt=k[(1-β_12)i_1+β_21n_2/n_1i_2]
            dr_2/dt=k[(1-β_21)i_2+β_12n_1/n_2i_1]
  
  To obtain the i equation, we employ the conservation relations (<ref>) and (<ref>) in
  their fractional form i.e
  
      (s_1+i_1+r_1)=1, 
            (s_2+i_2+r_2)=1,
  
  
  to generate
  
  
      (ds_i/dt+di_i/dt+dr_i/dt)=0,
  
  where i=1,2. Using the s(t) and r(t) equations along with the conditions  
  (<ref>) and (<ref>)
  form, the i(t)-equations for the two zones are obtained as
  
  
      di_1/dt   =   b  [(1-α_12)s_1(t)+ α_21 s_2(t)]
      
      [(1-β_12)i_1(t)+ β_21n_2/n_1 i_2(t)]
            -   k[(1-β_12)i_1(t)+β_21n_2/n_1 i_2(t)],
  
  and 
  
      di_2/dt   =   b  [(1-α_21)s_2(t)+ α_12 s_1(t)]
      
      [(1-β_21)i_2(t)+ β_12n_1/n_2 i_1(t)]
            - k   [(1-β_21)i_2(t)+β_12n_1/n_2 i_1(t)].
  
  The six ODEs (<ref>),(<ref>), (<ref>), (<ref>), 
  (<ref>) and (<ref>)  form a closed set for the 
  six variables s_1, i_1, r_1, s_2, i_2, r_2. 
  Expectedly, in the absence of migrations from zone 1 to 2 and vice versa 
  (α_12=α_21=β_12=β_21=0), the above equations reduce to 
  the original SIR equations. 
  
  
  
  
  
  
  § SIMULATIONS AND RESULTS
  
  
  
  
  
  
  
  The relevant ODEs are solved by using NDSolve function of the Mathematica with the appropriate initial condition. To benchmark, the following provides results for SIR simulations with initial conditions, 
  
      s|_t=0=1,
            r|_t=0=0, 
            i|_t=0= 1×10^-7.
  
  The initial values are chosen in line with the spread of COVID19 in India. With the 
  approximate total population of
  India (≈ 1.00×10^9) normalized to unity, 100 infections per day 
  yields a normalized value of i(0)=1×10^-7, 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 converts into recovered. Assuming an average period of 10 days the pathogen takes to spread the infection, we can choose an approximate k with a fraction of 1/10.
  The solutions are illustrated in the Figure <ref> with choice of parameters b=0.3, k=0.1, amounting to R_0=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, the distribution
  of i(t) is Gaussian. An increase in b (and hence R_0) decreases the Full Width at Half Maxima (FWHM) and the peak of the Gaussian (not shown), indicating a faster spread of the infection. The sum of the mortality and the clinical recovery rates is equal to the recovery rate in standard SIR plots. To distinguish, hereafter, we refer recovery rate in the standard SIR model as the "removed" rate; removed, since the individuals in this category can not be further infected.  The dashed line represents the sum (s+i+r) and is equal to 1, as expected from the conservation (<ref>). 
  
  
  
  
  
  
  Figure  <ref> 
  represent plots where the basic model is infused 
  with the progressive social awareness characterized by b(t)=b_0 exp(t/τ); the time constant τ signifies the 
  rate at which the b(t) and hence R_0(t) falls, quantifying how fast the society adapts various 
  interventions. The inset diagram in <ref>(a) plots the evolution of b(t) with b_0=0.3 and τ=1000 days, which is near constant for all
  practical purpose, indicating very little social 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 awareness is 
  evident in the next two plots <ref>(b) and <ref>(c) in the same Figure where aggressive 
  interventions are imposed with τ = 200 and 150 days respectively. The insets for both the plots, again, 
  show the time variations of b. Evidently, the one with the fastest decay as in Figure <ref>(c) 
  exhibits the infection curve to be most flattened and having much lowest peak value. The above findings 
  qualitatively agree with the recent simulations by <cit.> which show delayed onset of 
  successively diminished peaks in the total infected population with a stricter adherence to “social 
  distancing".
  
  
  To further explore the influence of social awareness on infection fraction, 
  Figure <ref>(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 time is implicit and (s, i,r)∈{0,1} satisfying s(t)+ i(t)+r(t)=1 at all t. 
  To elucidate further, we consider any of the one curve in the plot
  and note that at the initial point s(t=0)=1, i(t=0)=1 × 10^-7, r(t=0) =0 representing lower-left corner at the plot. All the three
  variables evolve implicitly with time, and after a sufficiently large time interval, all the curves terminate at i=0. The  
  curve having the largest time constant, 
  τ=1000 days (minimally progressing social awareness),  is the highest peaked—having the largest FWHM. The opposite is true for the smallest τ=150 days curve. Notably, 
  the three curves with  τ=150, 200,1000 days have three different peak values. 
  Importantly, the spacing between the termination points is more for the lower τ values.
   Contextual to the paper, 
  such non-linear dependency implies that a society capable of developing social awareness
  at a moderately faster pace during an epidemic gets far more benefited by additional campaigns than the one where the awareness develops at a slower rate. We believe, incorporation of the basic program on epidemic awareness at school curriculum can better prepare a society for a faster response. 
  The program will mostly be beneficial for epidemics like COVID19 where in the absence of vaccination and antiviral drugs; social interventions like social distancing, 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 break the chain of infection by maximizing the social distancing and hence, minimizing the spread. In the following, we discuss the effects of lockdown on the infected fraction without and with progressive social awareness. For the purpose, notable is the realization that the lockdown effectively lowers the value of R_0 for some finite period. Hence, the simplest lockdown is mimicked with a sudden reduction of b given by a finite square well. Such condition is modelled with a pair of sigmoid functions as described in Equation <ref> and shown as insets in Figure <ref>. Here, in the time evolution of b,  the larger value represents no-lockdown, and the smaller value signifies the lockdown. The plots <ref>(a) and <ref>(b) represent histories of the variables
  for b_in=0.1 and 0.2, respectively while k=0.1. 
  The no-lockdown value of b in both cases is fixed at 0.3. The lockdown period (Δ_t) is 50 days, spanning between the day 50 to day 100. Note that effective R_0^in drops to the values 1 and 2 respectively during these example lockdown periods. The choice of low R_0 in Figure <ref>(a) effectively stops the infection and 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 the lockdown and is centred at t≈ 52. The second peak in i(t) onsets after the lockdown is over and is located at t≈ 130, having a value 
  ≈ 0.28 and spread t∈{120,180 } days. 
  A visual comparison with Figure <ref> which documents a similar spread of the 
  i(t) curve t∈{50,130} and a peak value of ≈ 0.28 suggests a standalone lockdown with low R_0^in can 
  only delay the peak, providing additional preparation time for the authorities. Second plot in Figure <ref>(b) consider a rather pessimistic lockdown with comparatively higher side of R_0^in during lockdown. Unlike the previous case, here the trend is more admixture with the no-lockdown scenario, as in Figure <ref> except alleviating the curve, reducing the peak and broadening the spread. One realizes that the impact of lockdown period and the choice of b_in (R_0^in) produces an interplay between double Gaussian in i(t) and their interference which is demonstrated in our next discussion.
  
  
  
  
  
  
  The Figures <ref>(a) and  <ref>(b) depict instances (t_i_max day) of maximum i(t) 
  and its magnitude i_max as a function of `lockdown period' and the parameter b_in in contour plots.
  Plot <ref>(a) shows the occurrence of the peak value of i(t)  is delayed with increasing lockdown period, as expected for low value of b_in e.g. in Figure <ref>(a). However for a large b_in one gets broader distribution with the peak value remaining mostly adjacent to the starting point of the lockdown, as realized  in Figure <ref>(b). Figure <ref>(b)  demonstrates the fact that the peak values i_max mostly  remain same irrespective of lockdown period for a fixed  b_in . However we encounter the same i_max  twice staying in constant lockdown period. These two peaks correspond to the transition from one to another of the double Gaussian we discussed before. 
  Interestingly, for particular range of b_in∈{ b_α, b_β}, the peak value is minimum for lockdown perion Δ_L > Δ^*, giving an optimal range of b_in 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 wagers to earn their livelihoods together with the fine balance in keeping infection rates manageable.  
  Figure <ref> illustrates the effects of lockdowns having differently staggered removals. 
  Panels <ref>(a) and <ref>(c) shows the square-step-well functions representing the square-well form of b  for a two and three-stage exist after modelling the scenario as described in Equations (<ref>) and (<ref>).  Characteristic values are 
  {b_in^(1)=0.17, b_in^(2)=0.24} and
  {b_in^(1)=0.17, b_in^(2)=0.21, b_in^(3)=0.26} 
   for two-stage and three-stage staggered removals 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 staggered exit being slightly delayed than the one for the three-staged exit. However, the peak value of the three-stage curve is also somewhat milder than the two-staged one. These characteristics also manifest in the corresponding ternary diagram, where we added a few additional cases for demonstration purpose. Lowest peaked ones (in solid lines) are selected through scanning the parameter space on choosing a set of b values providing minimum peak. One also notices the overlapping lines in perfect lockdown (where b's are zero during lockdown) with ones with no lockdown (where b's don't reduce during lockdown). This demonstrates our earlier argument that perfect lockdown simply delays the curve and show up as overlapping lines in the ternary diagram, where the time axis is implicit in them.
  
  
  
  We further add progressive awareness in the above multistage scenarios. Figure <ref>
  depicts the effect. Once again, the profile of b is in the inset whereas panels (a) and (b)
  represent the evolution of the variables for τ=400 and 800 days, respectively. In both cases, the infection curves are found to be more flattened compared to the one without social awareness. This indicates a staggered exit from the lockdown along with measures to increase social awareness is not only good for an early restart of the economy but also beneficial in flattening the infection curve. 
  
  
  
  
  
  
  The section is completed with a discussion on the modeling of the effects of migration between
  two zones. Notably,
  such calculations can easily be extended to include any number of zones. The s, i, r plots for the two zones (the subscripts pointing the specific zone) are depicted in Figure <ref>.
  For simplicity the exchange constants are kept equal and tiny: α_12= β_12 =α_21= β_21 =
  10^4, amounting to an equally small number of individuals traveling from zone 1 to zone 2 and
  vice versa providing internal mixing among zones.  The initial  i(t=0) in the zones 1 and 2 are selected to be {10^-7, 0} respectively. 
  The histories of the different variables are shown in Figures <ref> and <ref> in solid and dashed curves respectively.
  Expectedly, the infection and recovery initially begun at zone 1 and as individuals inter-migrated between
  the zones—intensified in zone 2 also (Figure <ref>). The combined evolution is  depicted in 
  Figure <ref> which plots s_12=n_1 s_1 +n_2 s_2, i_12=n_1 i_1 +n_2 i_2 and r̂_12
  +ř_12=n_1 r̂_1 +n_1r̂_2+n_2 ř_1+n_2 ř_2. Future plan involves testing and modeling different combinations to yield targeted favorable outcomes. For example, finding the optimal set of
  parameter to effectively contain the spread within a minimal sized zone. which we keep as a future study. 
  
  
  
  
  
  
  
  
  § SUMMARY
  
  
  The paper recognizes the importance of social back-reaction 
  on the dynamics of an epidemic. In this work, the basic SIR framework is extended to explore the effects of progressive social awareness which is mathematically modelled by a decaying exponential. It is found that the awareness lowers the effective R_0 and reduces the peak infection rate while delaying its appearance. Additionally, the progressive awareness is more effective in societies having some seed knowledge about the various social deterrents. Consequently, 
  its inclusion in basic school curriculum can be effective in curbing future epidemics like COVID19
  where social interventions remain only available deterrents for a significant amount of time. 
  
  The extended model also studies the effects of lockdowns, mimicked by square-well functions generating different effective R_0s and having different staggered exit strategies. 
  It is found that the simplest lockdown with single-phase implementation and exit
  neither flattens the infection curve nor decreases its peak but only delay its appearance.
  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 curve as well as reduces and
  delays the peak. Also, such exist strategies are better from the economical perceptive also.
  The most efficient is an exit strategy planned with a joint increase in social awareness. 
  Such manoeuvrings can minimize the peak and flattens the infection curve most—which we believe can be beneficial in future epidemics where lockdowns will be necessary. 
  
  We have also extended the SIR model to include two-zone anthropogenic migration.  The example presented is a basic one where an initially equal number of individuals are assumed to populate the two zones. A small number of people are allowed to migrate between the zones
  while zone 2 is absolutely infection-free, and zone 1 is characterized with i(0)=10^-7. With time,
  both the regions get substantially infected with i having the same peak and spread while the peak for i_1 appearing earlier than i_2.
  
  The paper lays a groundwork where different scenarios to arrest the spread of an epidemic along with the importance of social awareness is explored. Although, in the present work, the above scenarios are mostly examined individually; nevertheless, we recognize the natural synergy between progressive social awareness, lockdowns and anthropogenic migration to control the spread of epidemics. Such a study is left as a future exercise.
  
  
  
  
  
  
  § 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. 
  
  
  
  
  999
  
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