key: cord-0756545-3dx6deei
authors: Ali, M.; Imran, M.; Khan, A.
title: Analysis and Prediction of the COVID-19 outbreak in Pakistan
date: 2020-06-23
journal: nan
DOI: 10.1101/2020.06.21.20136341
sha: 69dcba1ed16a3cd4c1bbc40c161f0422fa966830
doc_id: 756545
cord_uid: 3dx6deei

In this study, we estimate the severity of the COVID-19 outbreak in Pakistan prior to and after lockdown restrictions were eased. We also project the epidemic curve considering realistic quarantine, social distancing, and possible medication scenarios. We use a deterministic epidemic model that includes asymptomatic, quarantined, isolated, and medicated population compartments for our analysis. We calculate the basic reproduction number R0 for the pre and post lockdown periods, noting that during this time, no medication was available. The pre-lock down the value of R0 is estimated to be 1.07, and the post lockdown value is estimated to be 1.86. We use this analysis to project the epidemic curve for a variety of lockdown, social distancing, and medication scenarios. We note that if no substantial efforts are made to contain the epidemic, it will peak in mid of September, with the maximum projected active cases being close to 700,000. In a realistic, best-case scenario, we project that the epidemic peaks in early to mid-July with the maximum active cases being around 120000.We note that social distancing measures and medication, if available, will help flatten the curve, however without the reintroduction of further lockdown, it would be very difficult to bring R0 below 1. Our study strongly supports the recent WHO recommendation of reintroducing lockdowns to control the epidemic.

Coronavirus disease (COVID-19) is a respiratory tract illness which originated in Wuhan, China in early December 2019. It spread to other countries in Asia, Europe and North America in early 2020 and was declared a pandemic by the WHO on March 11 2020. As of June 15 2020, the disease is prevalent in at least 212 countries and territories, with more than 8 million cases reported and around 435,000 fatalities [3] .

Considering the situation on 10 June the WHO issued an advisory strongly recommending an intermittent lock down for the following two weeks. In this study we model the COVID 19 outbreak using a variant of the model proposed by Imran et.al [24] for the transmission dynamics of the disease. This model incorporates compartments for quarantined, isolated, and asymptomatic individuals, pathways considered important in transmission of the disease. We also include a compartment for individuals taking medication as we would like to study the possible effects of medication in controlling the disease. The dynamics of the COVID 19 outbreak in Pakistan have been qualitatively different before and after the lock down was eased. We estimate the value of R0 in Pakistan for the two different phases of the outbreak, noting that during this time no medication was available. We then project the disease curve considering minimal intervention as well as various control strategies. At the moment the viable strategies for controlling the outbreak are quarantine, isolation, social distancing and possibly medication. In fact, there have been emergency approvals for different treatments for the disease in various countries including Pakistan. We project the epidemic curve considering various quarantine, social distancing and medication scenarios. We note that the most effective control measure is a lock down but a very strict lock down does not seem to be an option that is being considered by the government, however a moderate and intermittent lock down may be possible. Further, by making sure social distancing protocols issued by the government are followed we can reduce the contact rate to varying degrees also flattening the disease curve. Finally, we consider various medication scenarios, as mentioned medication to date has been shown in varying degrees, to reduce the time of infection as well as alleviating symptoms, we primarily consider medication in the context of shortening the time of infection and thereby helping to bring the disease numbers down. We look at various levels of quarantine, social distancing and medication to study how each of these would affect the disease numbers as well as when would the infected numbers peak under them. We conclude the study by summarizing our findings and suggesting realistic control measures with their impact on the epidemic.

The COVID-19 transmission model we consider is based upon the SEIR model and takes into account the effects of quarantine, isolation, medication, and asymptomatic individuals. The total population N (t) is divided into eight mutually exclusive subpopulations, susceptibles S, these are individuals who can fall ill by coming in contact with an infected individual, quarantined susceptibles QS, these individuals are removed from the susceptible group at rate , either through self quarantined or lock-down measures, they, however, go back to the susceptible group at rate ξ. The susceptibles move to the exposed class by coming in contact with any infectious individual, at rate ρλ, some exposed individuals will not show symptoms and are accounted for in the model by the movement to the asymptomatic class at rate (1 − ρ)λ. Exposed individuals E become infected at rate σ. Infected individuals I can be isolated or given medication at a rate of τ and αI whereas a fraction of isolated QI are given medication at rate αQ. The individuals taking medication are represented by a separate compartment M , these include individuals from the isolated and infected classes who are under treatment. Based on clinical trials thus far, the main effect of medication we incorporate is that, it shortens the duration of the disease [6, 8, 14] . The recovery time for infected I, asymptomatic A, isolation QI and medication M is given by 1/γ,1/θA,1/θ and 1/θM respectively. The schematic of the transmission pathways is given in Fig.1 . The total population N (t) is given by the sum of the sub-populations.

The governing equations are given below (2).

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Here β is the effective contact rate, where as η1,η2 and η4 are the associated relative infectiousness parameters for the QI ,A and M sub-populations. Table.2 and Table. 3 represent the description of variables and parameters of the model, and are given in the appendix.

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The state variables described in model (2) shows non-negative solutions for all time t ≥ 0 with non-negative initial conditions. Lemma 3.1. For any given non-negative initial conditions, there exist a unique solution S, QS, E, I, A, QI , M, R respectively, for all t ≥ 0. Moreover, it satisfy the following inequality of boundedness. Proof is attached in the appendix A.

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The copyright holder for this preprint this version posted June 23, 2020. The model (2) achieves the Disease Free Equilibrium (DFE) whenever there is no induction of infection by the disease i.e. the force of infection is zero,λ = 0. Mathematically, this can be donr by equating the right hand side of (2) to zero with λ = 0. Let F0 represent the DFE of the model.

, π µ(µ + ξ + )

, 0, 0, 0, 0, 0, 0

The local stability of Disease free equilibrium (DFE) is quantified by the threshold quantity R0 which is found by means of the next generation operator method [27] .

The thresh hold quantity R represents the average number of new secondary infections produced by the single infection in the completely susceptible population. It is calculated by the spectral radius of the F V −1 matrix from the next generation method [27] . The associated F and V matrices of model (2) are as follows This Lemma interprets that a small influx of the infecties will not lead to cause bigger outbreaks and disease infection will become extinct in long run.

To have concrete sense, that the disease extinction is independent on the initially sub-populations in model (2), we have to show that the steady state (DFE) follows globally asymptotically stable (GAS) condition. Proof is given in appendix A

The equilibrium state of the system (2) in presence of the infection i.e. λ = 0 is known as the endemic equilibrium. Let F1 represents the arbitrary endemic equilibrium of the model (2).

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The copyright holder for this preprint this version posted June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136341 doi: medRxiv preprint Further,The force of infection λ can be written in terms of the equilibrium as

with N * * = S * * + Q * * S + E * * + I * * + A * * + Q * * I + M * * + R * * Solving for the system (2) at this specific fixed point, the endemic equilibrium becomes

We would like to study how R0 changes with the parameters of the model, in particular we are interested in the variation of R0 as the quarantine rate, the contact rate and the rate of medication are varied. These rates can be changed by the use of different control strategies, this in turn will be used to project the epidemic curve under various control regimes.

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The copyright holder for this preprint this version posted June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136341 doi: medRxiv preprint In Figure 2 (a) the contours of R0 are plotted against αI the rate of medication and β the contact rate. In our analysis we assume that the contact rate can be changed by following stricter social distancing protocols, the medication rate of course can be varied by wider use of available medication. We have also assumed that available medication reduced the duration of the disease by 25%, as reported in several studies. We note that while a higher medication rate (with this particular efficacy) reduces R0 medication by itself cannot make R0 < 1 for high contact rates. Figure 2 (b) is a contour plot of R0 with quarantine rate ( ) and the contact rate (β). The primary mechanism of quarantine is lock down, although self quarantine can also help. We note that R0 can be reduced significantly by either enforcing strict social distancing, thereby reducing β or by enforcing a lock down and reducing . In Figure 2 (c) contours of R0 are plotted against the medication rate and the quarantine rate. We note again that while medication does lower the value of R0, for very low quarantine rates just using medication as a control strategy cannot make R0 < 1.

The epidemic data of COVID-19 Pakistan for this study is sourced from the COVID-19 Data Repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University [32]. The epidemic data consist of three time series of confirmed cases, deaths and the recovered cases spanning over more than ten weeks till June 15,2020, including the lock down period of 5 weeks. The active cases time series shown in Fig 3 is calculated   8 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The transmissibility of an infection is quantified by a threshold quantity R0,which defined as average number of newly infected individuals by a single infected individual in complete susceptible population. For an ongoing epidemic of COVID-19, Estimates of R0 can give better insights about the transmissibility of infection in a certain country. These Estimates further depend upon the critical parameter such as contact rate β, incubation rate σ and other related parameters. Since there are control polices such as social distancing measures and quarantine being enforced in order to lessen effective contacts between susceptible and infected population groups. It is assumed that the effective contact rate is function of time. During lock down individuals will have less interaction as compared to after lock down so it is assumed to have two different contact rates of β, hence giving us a different transmissibility index.

where t * is the time when lock down was relaxed. For Pakistan, the epidemic curve has followed two very different trajectories before and after the lock down was eased. As we have described in detail in the Introduction the strict lock down was relaxed around the beginning of the fourth week of April, hence t * is taken to be 5 weeks starting from the March 16,2020.

Since the direct estimates of parameters are really difficult especially, when epidemic/pandemic is still going on. However, we can adopt an indirect method proposed by the [30, 31] to estimates these parameters from the infected data available. The Ordinary Least square optimization technique is used which assumed that the observed data has the constant variance error distribution.

Yj = f (tj, ΘOLS) + j 9 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The value of R0 post relaxation is significantly higher as can be expected, we will use this to project the epidemic curve under a variety of scenarios. We also note the significant reduction in the quarantine rate once the lock down was relaxed.

In this section we project the epidemic curve under different quarantine, social distancing and medication strategies. At the moment there is no government mandated lock down in Pakistan, however some people are observing self quarantine measures, there are however guidelines for proper social distancing in public spaces, and as the infected numbers rise the health authorities are enforcing those guidelines, we consider this to essentially reduce the effective contact rate, finally medication has been approved by the government to treat COVID 19, and we

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The copyright holder for this preprint this version posted June 23, 2020. . https://doi.org/10.1101/2020.06.21.20136341 doi: medRxiv preprint hope that in the coming days it will become widely available. We consider three possible quarantine scenarios, the first where no additional steps are taken by the government, we call this the 'minimal quarantine', the second 'moderate quarantine', where some additional lock down measures are enforced to increase the quarantine rate to be twice of the post lock down value and finally a 'strict quarantine' where measures are taken to bring the quarantine rate back to the pre lock down levels. For each one of these cases we consider high, medium and low levels of social distancing leading to reduced effective contact rates, both with medication and without the availability of medication. In Figure 5 we consider the scenario where no steps to lock down are taken and the quarantine rate remains at the post lock down level for different contact rates. Decreasing the contact rate not only lowers the peak of the epidemic curve but also shifts it to the left, which means that that the disease will peak earlier. For the case when no medication is available Figure 5 (a) we note that the the peak will be reached in mid to late September with the maximum number of active infected being around 700000 for low levels of social distancing, 400000 for moderate social distancing and 200000 if high levels of social distancing are enforced. We also note that in these scenarios the case related fatalities that can be expected by December 2020 are respectively 2.5, 2 and 1.5 million. In Figure  5 (b) for when medication is available we note that the disease peaks between August and early September and the maximum active infected number are now reduced to 275000, 150000 and 110000 in the three different social distancing scenarios. Around 1 million, 700000 and 450000 deaths can occur in these cases by the years end. We would like to note that this is not a very likely scenario as the rising numbers of infected and disease related deaths have prompted a response both in terms of some additional lock down measures and stricter implementation of the social distancing protocols.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. In Figure 6 we consider the outcome when a moderate lock down is imposed increasing the quarantine rate to twice the post lock down level. Without medication Figure 6 (a) we note that the the peak will be reached in mid to late July if social distancing protocols are enforced and in mid to late August otherwise, the maximum number of active infected in these cases is projected to be around 150000, 175000 and 300000. The total expected fatalities due to disease are around 500000, 800000 and 1.2 million respectively, through December. In Figure 6 (b) for the case medication we note that disease peaks between early to mid July and the maximum active infected number with a high level of social distancing is around 100000, with disease related fatalities through December projected to be around 225000. With moderate social distancing the maximum infected number is projected to be 120000 with 325000 expected deaths by the end of the year. Finally, with low social distancing efforts we expect 150000 active infected cases at the peak of the outbreak and 500000 total fatalities. In our opinion the most likely scenario to be played out over the next few month will involve moderate lock down and high social distancing efforts.

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. In Figure 7 above we consider the situation when a strict lock down is imposed increasing the quarantine rate to pre lock down level. Without medication Figure 7 (a) the peak will be reached in early to mid July (depending on the contact rate) with the maximum number of active infected being around 125000 − 200000. In this scenario 300000 − 700000 disease related deaths are expected by the end of the year. In Figure 7 (b) when medication is available the disease peaks between late June to early July with the maximum active infected number being around 90000 − 125000 (again depending on the social distancing measures), with around 150000 − 300000 deaths due the the infection. We would like to point out that this level of quarantine is not very realistic, considering the economic and social costs the government is unlikely to enforce such stringent lock down measures.

In this study we have estimated the severity of the COVID 19 outbreak in Pakistan, modelling the transmission dynamics of the disease using an extension of the SEIR model. The model includes important transmission pathways such as asymptomatics, quarantined and isolated individuals as well as the medicated population sub group. We also use this model to project the epidemic curve and give estimates of the disease burden under various realistic control strategies. An ordinary differential equation based model is used for the transmission dynamic of COVID 19 as described above. Disease free equilibrium (DFE) and an en equilibrium when the disease is endemic in the population is derived. We establish a threshold quantity R0 in terms of the parameters of the model, observing that the disease dies out whenever R0 < 1 and is endemic if R0 > 1. We study the variation of R0 on some of the model parameters, specifically on the parameters which can be varied by various control mechanisms, such as quarantine, social distancing and medication. We note that quarantine and social distancing and medication can help in lowering the value of R0 and bring the epidemic under control, with quarantine being the most effective mechanism followed by social distancing and medication. However there are practical difficulties in establishing a strict quarantine as is evident from the relaxation of lock downs in most countries, hence a mix of all these strategies is perhaps the best way forward. Noting that in Pakistan the epidemic progressed in two distinct phases, before and after the lock down restrictions were relaxed, we estimate the value of R0 separately for both these phases. For the pre relaxation phase we estimate R0 = 1.07 and for the post relaxation phase our estimate comes out to be R0 = 1.86. This is consistent with the growth dynamics and other estimates in the literature. These results are then used to project the epidemic curve. We explore various possible scenarios, with varying levels of lock down, social distancing and medication. The lock down and social distancing measures are captured by quarantine rate ( ) and contact rate (β) in our study. In the case where a strict lock down is enforced taking the quarantine rate back to pre-relaxation level, varying the contact and medication rates, we observe that in the best case scenario with high level of all controls the epidemic will peak around late June to early July with a maximum active infected number around 90000. For a moderate lock down which is modelled by taking the quarantine rate to be twice the post-relaxation rate, and varying the contact and medication rates we note that the epidemic will peak around mid July with the active infected numbers around 120000 at that point considering that social distancing measures are enhanced. Finally, we look at the case when no further lock down measures are taken and the quarantine rate remains at the value estimated on June 15, in this scenario we note that the epidemic will peak around late September with the maximum active infected number around 700000.

In this work we estimated the severity of the COVID 19 outbreak in Pakistan, noting that the epidemic followed two different trajectories before and after lock down restrictions were relaxed. The restrictions we observe kept the value of R0 close to 1, however once the restrictions were eased the number of cases increased at a high rate, reflected in R0 = 1.86. We also generated possible future trajectories based on intervention and control measures to varying degrees. In case no significant measures are taken at the peak of the epidemic, which will occur in September the maximum active cases will cross the 700000 mark while in a very optimistic scenario this will happen in early July with maximum active cases contained to around 150000. We strongly agree with the recent WHO recommendation that some form of moderate lock down needs to be reinforced, this along with social distancing and possible medication will result in a disease peak sometime in mid July and contain the maximum active infected numbers to around 120000. Inaction at this point of the epidemic is projected to have very serious consequences both in terms of disease burden and mortality.

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It follows that

Thus model (2) solutions exists for given initial conditions, and are eventually bounded on every finite time interval.

Proof. Using (8) and (9), it follows that as time approaches to infinity t → ∞, the population is bounded by the positive number so the set D N (t) ≤ Π µ therefore, the set D is positively invariant.

Proof. Consider the Lyapunov function for model (2).

where

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The Lyaponov derivativeL is given aṡ 

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