Currently the world is very much under the threat of COVID-19 and the number of active cases are surging rapidly ever since December 2019. People with compromised immune systems and especially old individuals, are at highest risk from this virus. A global pandemic such as SARS-CoV-2 tests our capability to effectively control the situation and protect the population. Using estimates for seasonality and immunity, Kissler et.al [1] projected that prolonged social distancing may be necessary till 2022 and SARS-CoV-2 survillence should be maintained since a resurgence of the virus could be possible as late as 2024. The primary aim of any modeling is to devise strategies that reduce human loss and predict speedy recovery back to normal life. To that end, protection or prevention from the spread of the virus is of paramount interest.
In the absence of a vaccine or a possible cure yet, it becomes important to have some idea about its spread and some ways to control the situation. This can help the health and administrative authorities to better assess the situation and be prepared for any eventuality.
Since Kermark and Mckendrick in [2] built up a system to study epidemiology in 1927, the concept of "compartment modeling" is widely used until now. Since then several analytical work [3] [4] [5] has been carried out on epidemic diffusion focused on compartmental models. Generally, the population is divided into different classes of individuals which are allowed to mix homogeneously. In the present work, we study the SEIRS (Susceptible -Exposed -Infectious -Recovered -Susceptible) model to analyse a very specific problem at hand, namely, the diffusion of population between two cities in the backdrop of COVID attack. Population diffusion has been introduced in [6] and an earlier, more general, study using SIS model was studied by Liu et.al [7] , we use SEIRS model as it is more suitable to analyse the present COVID situation. We pose a few questions and suggest control measures for the benefit of the population. No effect of vaccines has been considered, as it is still not available. In our model, a small fraction of the population in a city migrates to another city. In an earlier work in SIR model [8] the susceptible individuals are allowed to migrate in or out of the cities. In the present work, we allow individuals of all types to migrate and allow some of the exposed individuals to turn infectious during transit as well.
One of the crucial problems arising from population mixing is the resurgence of infected and exposed individuals. The novelty of the new virus is that while an individual might not be infected, there is a good chance for him/her to be susceptible or even exposed. When one of the cities has a large influx of individuals, the chance of an outbreak increases and in the absence of any vaccine, the very first measure one can take is a strict implementation of the containment.
With the parameters at hand, what could be the best way to reduce the exposed and infected individuals is a major question. It turns out that this depends on (i) how early we take the action, and (ii) migration parameters as well as reinforcement of public health diktats. This also provides us with a long-term solution which can be implemented to bring the situation back to some kind of normalcy. For this case we analyse different scenerios depending upon the action taken by respective cities. In order for an action to be taken we monitor the population of infected individual in the system and put forward certain preventive measures to counteract the spread. Finally, we study the effect of punctuated lockdown implemented in the cities.
SEIR model has been used extensively to study the dynamics of epidemics. In this work we try to study the SARS-CoV-2 with two very specific problems at our hand. The first one, which has been studied earlier by Zhang et.al [3] , with the immigration of individuals in different compartments. The second, being a more practical problem as of now, that studies the immigration betweeen two different cities once the lockdown is lifted.
The first problem, while being a derivative of the previous work [3] , adds two parameters important for the present pandemic: the social distancing parameter and the ability of recovered individuals to become susceptible again. Thus effectively we work with SEIRS model. In section 2.1 we present the SEIRS model for a single city with constant immigration allowed and in section 2.2, modify the SEIRS model to take into account the migration of individuals in two cities following their own internal dynamics goverened by SEIR model.
While being in a lockdown is a short-term solution, we need to focus on the effects of lifting it. The incidence of a disease is the number of fresh cases per unit time, and this plays a significant role in the study of epidemiology. We assume the population has a homogeneous spatial distribution and the mixing of individuals follow the law of 'mass action'. We assume that the local density of the total population is a constant, albeit the total population size N(t)= S(t)+ E(t)+I(t)+ R(t) may vary with time. Here S(t), E(t), I(t) and R(t) denote the size of Susceptible(S), Exposed(E), Infected (I) and Recovered(R) classes at any time t, respectively. The sum E(t) +I(t) is the total infected population.
Many contact rate forms are used in the incidence term in the epidemic models, the most common being the standard incidence λSI N which starts with the assumption that the adequate contact rate is a constant λ. The bilinear incidence βSI = βN S N I implies that the adequate contact rate βN is linearly proportional to the total population size N. The drawback of this is that, it would be unrealistic for a population too large. Since we want to simulate some degree of social distancing, we use the saturated incidence rate. A constant inflow denoted by A adds to the susceptibles. Assuming that the natural deaths occur at a rate proportional to the population size N, the natural death term is d N, here d is the natural death rate constant. In order to simulate the effect of social distancing, we make use of the Michelis-Menten equation to get the saturated incidence. The term
signifies the decrease in fraction of population from susceptibles and entering into the group of exposed individuals. The nonnegative parameter η controls the effect of social measures like quarantine. 1 1+ηI measures the inhibition effect from the behavioral change of susceptible individuals when their number increases or from the crowding effect of the infective individuals. This incidence rate is more reasonable than the bilinear incidence rate because it includes the behavioral change and crowding effect of the infective individuals and prevents the unboundedness of the contact rate by choosing suitable parameters. All the parameters δ, σ, α and γ are all postive constants. δ is the rate constant at which recovered individuals become susceptible,σ is the rate at which exposed individuals becomes infected, hence 1 σ is the mean latent period. α is the disease related death rate and γ is the rate of recovery.The following flowchart shows the basic SEIRS model with varying population size, social distancing.
The ordinary differential equations can then be formulated as
Since the total population is not constant, we have Total population at time t as,
We use the SEIRS model described in section 2.1. We consider a model with state variables S 1(2) , E 1(2) , I 1(2) , R 1(2) that represent the number of susceptible, infected, exposed and recovered individuals in city 1(2), respectively. The differential equation governing the two-city model is:
In this section we describe the parameters used for numerical simulation.
• Constant flow of population, A: Constant inflow of individuals is allowed to enter the susceptible population. An arbitrary non zero number of individuals are considered in this study. We allow a small number (A=10) in the study as it directly affects the reproduction number.
• Migration parameter, m 1(2) : represents the migrating-out rate of susceptible individuals in city 1 (2) . (m 1(2) ≥ 0 and m 1 = m 2 )
• Contact rate, β: the effective per capita contact rate of the infective individuals and the bilinear incidence rate is βSI
• Social Distancing parameter, η: models the social distancing in the two cities. 1 η is the half-saturation constant of infected individuals.
• Return to susceptible rate: a fraction δ ≥ 0 of the recovered compartment loses their immunity and becomes susceptible in unit time.
• Natural death rate, d: constant per capita natural death rate d > 0 in each group.
• Transmission parameter, λm 1(2) : the transmission rate from individual from city 1 travel to city 2
• Death and Recovery rates: a fraction α ≥ 0, of the infective dies from the infection, and a fraction γ ≥ 0 of the infective recovers in unit time. Here we assume that the individual recovers in 14 days
• Incubation rate, σ: the rate of latent individuals becoming infected (average duration of incubation is 1/σ). Here Incubation period is considered to be 5 days.
Inorder to study the population migration between two cities by system (2), we consider certain basic assumptions which are as follows:
1. The cities are not identical. They start with different initial conditions and can implement different preventive measures.
2. An influx of population A joins the susceptible class per unit time. We consider a small number of individual entering in the susceptible population per unit time.
3. Disease is transmitted by saturated incidence rate
1+ηI 1(2) for city 1 (2) . It controls the rate of spread, which represents the probability of transmitting disease between a susceptible and an infectious individual.
4. Susceptible, Exposed, infected, and recovered individuals of city 1(2) migrate to city 2(1) at a per capita rate of m 1(2) .
5. When exposed individual from city 1(2) travel to city 2(1), disease is transmitted with the incidence rate
, with a transmission rate λm 1 (2) 6. We suppose that individuals who are traveling do not give birth and die. We also allow a small fraction of individuals from the pool of recovered to turn susceptible again.
Description of the cities: Other parameters such as contact rate (β), transmission rate(λ, σ, δ), death rate(d, α), recovery rate (γ) are same for both the cities. We make the following set of assumptions for the presented model.
1. The individuals in city 1(2) at time are denoted as X 1 (2) , where X represents susceptible(S), exposed(E), infected (I) and recovered(R). Thus the total individuals in city 1(2) are N 1(2) = X 1 (2) 2. A fraction of individuals from each compartment can migrate in this model, and m 1 (2) represents the migrating-out rate of susceptible individuals in city 1 (2) . (m 1(2) ≥ 0 and m 1 = m 2 in general)
3. N represents the total population in two cities and hence N= N 1 + N 2 = constant
4. An inflow of A from independent sources are allowed in both the cities. Table 1 Parameters used for simulation. Some parameters which are well known like σ and γ have been taken from Fairoza et.al [9] , while for others best estimates are used.
The basic reproduction number is a fundamental concept in epidemiology. It represents the number of secondary infections resulting from a single primary infection into susceptible population. When finitely many different categories of individuals are involved, the next-generation matrix(NGM) is the natural basis for the definition of reproduction number.
In compartmental models such as this, individuals are in a finite number of dicrete states which changes with time from one state to the other. As is the case the same symbol is used as a label for a state and to denote the corresponding population size. For the computation of R 0 we only regard the states that apply to infected individuals which for the present case are (E 1 , E 2 , I 1 , I 2 ). We decompose the jacobian matrix generated from the ODEs as T + Σ, where T is the transmission part, describing the production of new infections and Σ is the transition part, describing the changes in the state. At the infection free steafy state E 1 = E 2 = I 1 = I 2 = R 1 = R 2 = 0, hence S = A d One approach of finding the R 0 is to formulate a 4 × 4 matrix comprising of (E 1 , E 2 , I 1 , I 2 ) and use the description given by Diekmann et.al [10] to find a rather complicated formula. We here treat two cities individually and find their respective reproduction number. For this we set x= (E 1(2) , I 1(2) ) ′ , where the prime denotes transpose. We now want to write the linearized infection subsystem as,ẋ
and,
Then the basic reproduction number is the spectral radius of the matrix R 1(2) 0 = −T 1(2) Σ −1 1(2) . Let R 0 = max{R 1 0 , R 2 0 }. We therefore have,
For the analyis of the system (2), we vary the parameters such that we keep the reproduction number between 1 and 2.5. The major variation in the R 0 comes from β and the effect of varying β on diffeent population is shown in fig. 1(a) and 1(b) . The signficance of the value R 0 is that it gives the number of secondary infections resulting from a single primary infection into an otherwise susceptible population. A value of R 0 > 1 signifies the spread of epidemic in an otherwise suscpetible population i.e the cases are rising. In this section we present a very basic results for the SEIRS model for one city discussed in section (2.1). It is well known that contact rate β plays a huge role in spread of virus and social distancing factor (η) should control the spread of the virus. Using the parameter set described in the previous section and in Table 1 , the effect of contact rate is observed in fig. 1(a) and 1(b) . While the susceptible population does not show a considerable change, we observe as expected, a rise in exposed and infected population. As it has been advised worldwide to practice social distancing, we observe that social distancing parameter η has a considerable effect in the growth of population as seen in fig. 2(a) and 2(b) . In this section we present the results of population migration between cities . An important question in population migration concerns the effect of migration between cities. Individuals from all compartments are allowed to move, and individuals who were initially in exposed compartment are allowed to turn into infected during the transit. Susceptible population for both cities decreases due to migration and gets transferred to exposed population as shown in fig. 3(a) and 3(b) . The major effect of migration comes into the infected population where city which recieves higher number of population has large growth in infected individual as shown in fig. 4(a) and 4(a) . This gives us an idea that, migration of individuals under any circumstances is bound to put pressure on the city health care and number of infected patients will increase drastically. Exposed population 10 4 City-1 City-2 (b) Infected and exposed population for threshold of 10% This leads us to look for certain preventive measures to counteract the spread of increasing infected individuals. We mainly analyse two kinds of threshold here before which preventive measures like social distancing and removing the quarantine are implemented. If infected population exceeds the threshold of 1 % of initial population, we implement (η 1 , η 2 , m 1 , m 2 ) = (0.5, 0.25, 0.01, 0.02) and else social distancing is relaxed and larger migration is allowed (η 1 , η 2 , m 1 , m 2 ) = (0.05, 0.25, 0.4, 0.8). It is worth mentioning that similar results could be obtained by implementing greater social distancing and smaller migration for the whole period, but such a solution does not help restore normalcy in the cities. Hence preventive measures are to be undertaken only when threshold crosses is beneficial. We also note that restricting the migration alone does not help the situation and social distancing has be implemented.
With the above preventive measure into action we investigate the importance of threshold value in curbing the population. Finally from fig.5 (a) and 5(b) we see that if preventive measures are being implemented at larger threshold of 10%, a larger population gets affected.
In this section we study the effect of punctuated lockdown in both the cities. Instead of relaxing the lockdown completely, the restrictions are lifted for 15 days and then imposed back for the next 15 days. For the relaxed phase (η 1 , η 2 , m 1 , m 2 ) = (0.01, 0.01, 0.8, 0.3) and for lockdown phase (η 1 , η 2 , m 1 , m 2 ) = (0.9, 0.5, 0.05, 0.02) are used. Assuming a small fraction do turn positive again under Covid testing after recovery from the infection, a damped oscillatory behavior is observed. The infected populations reaches the peak for both the cities and then decreases to rise to certain values before decreasing again. The infected population reaches close to zero after 150 days (see fig. 6(b) ). The exposed population reaches zero only to rise again due to the fact that susceptible population is yet to reach zero (see fig. 6(a) ).
The oscillatory behavior in the population is the result of a small fraction of recovered individual becoming susceptible again. This leads to the idea that one should have close observation over even such individuals who are recovered. 
We considered the SEIRS epidemic diffusion model for COVID-19 with population migration between two cities. The migration rate between cities is unequal and individuals from all compartments are allowed to migrate between the cities. A fraction of exposed individual from either city turns infected during transit. Unless a vaccine is forthcoming in the immediate future, protective health measures are paramount to prevent the spread. In order to control the infected population, preventive measures should be implemented at lower thresholds. A punctuated lockdown is also studied in this context, which showed that infected population comes down to zero in an oscillatory fashion. The oscillatory behavior is observed when a small fraction of recovered individuals is allowed to become susceptible again. This raises the important point that the recovered individuals should be under observation to prevent increase of susceptibles. An increase in the number of susceptibles pushes up the fraction of infected inividuals leading to consequent peaks. The punctuated lockdown is an interesting deviation from the extant strategies and shows its usefulness while handling situations post first lockdown.
The model with different strategies emphasize the need for finding ways to minimise the spread till a cure/vaccine is in the market. Ideally a possible extension to more than two cities should be considered, but some of the essential features are already visible in the results we obtained here.