key: cord-0716547-yyeyshmj
authors: Adhikari, Khagendra; Gautam, Ramesh; Pokharel, Anjana; Uprety, Kedar Nath; Vaidya, Naveen K.
title: Transmission Dynamics of COVID-19 in Nepal: Mathematical Model Uncovering Effective Controls
date: 2021-03-24
journal: J Theor Biol
DOI: 10.1016/j.jtbi.2021.110680
sha: 5c5a45b376ef380742558f0fba422b3473679b66
doc_id: 716547
cord_uid: yyeyshmj

While most of the countries around the globe are combating the pandemic of COVID-19, the level of its impact is quite variable among different countries. In particular, the data from Nepal, a developing country having an open border provision with highly COVID-19 affected country India, has shown a biphasic pattern of epidemic, a controlled phase (until July 21, 2020) followed by an outgrown phase (after July 21, 2020). To uncover the effective strategies implemented during the controlled phase, we develop a mathematical model that is able to describe the data from both phases of COVID-19 dynamics in Nepal. Using our best parameter estimates with 95% confidence interval, we found that during the controlled phase most of the recorded cases were imported from outside the country with a small number generated from the local transmission, consistent with the data. Our model predicts that these successful strategies were able to maintain the reproduction number at around 0.21 during the controlled phase, preventing 442,640 cases of COVID-19 and saving more than 1,200 lives in Nepal. However, during the outgrown phase, when the strategies such as border screening and quarantine, lockdown, and detection and isolation, were altered, the reproduction number raised to 1.8, resulting in exponentially growing cases of COVID-19. We further used our model to predict the long-term dynamics of COVID-19 in Nepal and found that without any interventions the current trend may result in about 18.76 million cases (10.70 million detected and 8.06 million undetected) and 89 thousand deaths in Nepal by the end of 2021. Finally, using our predictive model, we evaluated the effects of various control strategies on the long-term outcome of this epidemics and identified ideal strategies to curb the epidemic in Nepal.

Since the first reported case in China in December 2019 as a case of pneumonia of unknown cause, the novel corona virus disease (COVID-19) has spread rapidly all over the world, and on March 11, 2020, the World Health Organization (WHO) declared COVID-19 a pandemic [58] . As of September 16, 2020, 5 more than 29 million cases of COVID-19 and more than 900 thousand deaths due to the disease have been reported worldwide [65] . In its global devastating effects on all aspects of human lives, the impact of the epidemic quite varies from country to country, thus the study focused on a specific country can provide better understanding of the disease and its control strategies. 10 In Nepal, the first case of COVID-19 was confirmed on January 23, 2020, which was also the first COVID-19 case in South Asia [22] . After this first case found to be an infected Nepali student who had recently returned from Wuhan, China [14] , no additional case was reported until March 23, 2020. On March 24, 2020, the Government of Nepal implemented a country-wide lockdown in-15 cluding business closures and restrictions on movement within the country and access to flights in and out of the country [50] . In addition, the Government of Nepal aggressively initiated a border screen policy to quarantine people trav-eling to Nepal from abroad, to test them, and to isolate them if the test is positive. Because of such timely and aggressively implemented control strate- 20 gies, the number of COVID-19 cases in Nepal remained relatively low (only 4% of the total cases from local transmission) until mid of July, 2020 [18], when these policies ended. Since these policies were lifted, the new cases began to grow dramatically, and as of September 16, 2020 the total of 58,327 cases (mostly from local transmission) have been reported [16] . We define the epi- 25 demic phase from March 22 to July 21 as the controlled phase, during which the daily recorded cases remained significantly low, and the epidemic phase from July 22 to September 16 (end of the study) as the outgrown phase, during which the daily recorded COVID-19 cases exponentially increased. A detailed study of the biphasic epidemic trend of COVID-19 appeared in Nepal provides us with 30 an opportunity to identify and evaluate effects of control strategies in the context of countries like Nepal, which is uniquely characterized by an open border provision with India, one of the highest COVID-19 affected countries.

Mathematical modeling using nonlinear systems has been an important tool for describing the dynamics of infectious diseases and evaluating the control 35 strategies to curb epidemics [24, 26, 29, 34, 37, 40, 51, 52, 70, 71] . Deterministic mathematical models, including the SEIR (Susceptible-Exposed-Infected-Recovered) model, have been widely used in quantitative studies of COVID-19 pandemics. While some models were used to estimate the parameters, such as incubation period and infectious period [3, 31, 33, 53] , others examined the 40 effectiveness of control strategies, such as lockdown, detection and isolation, border screening, and medical resources [9, 11, 19, 20, 25, 42, 47, 54, 68] . The quarantine for the traveler and suspected cases were also studied as the effective control measures for mitigating COVID-19 [1, 27, 32, 69] . Regarding in Nepal, previous studies [5, 7, 39, 66] have provided some insights into dou-history of recorded infectious people shows that more than 80% infectious cases 50 came from abroad, especially from India, during the early period of epidemics [17, 41] . Also, despite the Nepal government's effort of applying strategies, such as border screening, quarantine and isolation, poor handling policy at the border does exist, allowing many infected individuals enter the community without quarantine [43] . 55 In this study, we develop a deterministic mathematical model, which incorporates the imported as well as locally generated cases along with various policies implemented for the control of COVID-19 in Nepal. Using case data from both the controlled and outgrown phases of epidemics in Nepal, we estimated key parameters as well as the basic and effective reproductive numbers. Using our 60 model, we evaluated the control strategies implemented in Nepal. Furthermore, we applied our model to predict the long-term dynamics of COVID-19 in Nepal, and provided the simulations to demonstrate how these control strategies can curb the epidemics in Nepal. 

The data used in this study was obtained from the Ministry of Health and 

We consider the transmission dynamics model based on the SEIR framework.

We divide the whole population into five distinct compartments: S (susceptible), E (exposed), I R (recorded infectious), I N (non-recorded infectious), and R (recovered). In our model, susceptible individuals contract the virus when 85 they come in contact with the non-recorded infectious individuals at the rate β. These exposed individuals become infectious at per capita rate δ with the proportion θ being recorded and 1−θ remaining non-recorded. Individuals from both I R and I N classes get recovered with the rate η or die with the rate k.

µ and Λ represent the per capita rate of natural mortality and the natural re-90 cruitment rate into the susceptible class. We represent the entry of individuals from abroad, mainly across the Nepal-India border, by the time-dependent rate λ(t), among which the proportion ρ are infected and the remaining (1 − ρ) are susceptible.

The main control strategies implemented by the government of Nepal are:

(i) Border screening and quarantine, (ii) Lockdown, and (iii) Detection and isolation.

Border screening and quarantine. To model the border screening and quarantine strategy, we introduce a quarantined class, Q, to which φλ(t) of 100 individuals from abroad enter, where φ represents the rate of border screen. For these quarantined individuals PCR test is performed with rate τ and the tested individuals with positive result enter into the I R class and are isolated. As the expected rate of positive test in people entering into the country is ρ, we assume that ρ represents the portion of the tested population getting positive result, Lockdown. Lockdown strategy reduces the contact among individuals, and we assume the reduction of contact by ξ resulting in the transmission rate β → (1 − ξ)β. Since the strategy was altered in two different phases, the controlled and the outgrown, we consider two different reduction rates of contact as follows.

where t c represents the time when the epidemic phase changes corresponding to alteration of policies (July 21, 2020). As a result, the net infection rate becomes

β before and after t = t c , respectively.

Detection and isolation. As mentioned above, recorded infected individuals, I R , in our model are isolated. Therefore, the detection and isolation strategy can be incorporated into our model by altering the rate θ. We introduce a parameter ψ to represent the effect of the detection so that the rate of individuals in exposed class, who are detected and recorded, changes as θ → ψθ.

Since the strategy of testing for individuals in the general community was altered after the lockdown was lifted, we take two different detection rates for the controlled and outgrown phases as follows.

As a result, the net detection and isolation rate becomes θ c = ψ c θ and θ o = ψ o θ before and after t = t c , respectively.

Combining all the control strategies implemented in Nepal into the basic transmission dynamics model, we obtain the model as shown in Figure 1 . The is shown in black color while the implemented control policies are indicated by red color.

model is described by the following system of ordinary differential equations.

Here, the total population is given by N = Q + S + E + I R + I N + R.

Even though the first case of COVID-19 in Nepal was confirmed on January first case identified on January 23, 2020 had been recovered [22] by the beginning of our dynamics, and hence we take R(0) = 1. Since the initial time of our dynamic model is the beginning of the epidemic, we assume E(0) = 1, I R (0) = 0, and I N (0) = 1.

Since the infected individuals remain in the exposed class for about 5.2 days 135 until they become infectious [3, 45 , 59], we take δ = 1/5.2 = 0.1923 per day.

Also, the infectious individuals get recovered in about 17 days [58] , implying the average recovery rate η = 1/17 = 0.0588 per day. We estimate the rate of death due to COVID-19 using the data taken from the official website of the Nepal government [22] . Specifically, we take the average death rate from March 22 to 140 September 16, 2020, and obtain the death rate k = 0.000281 per day. We take µ and Λ in such a way that the natural birth rate and death rate remain equal for the period of this pandemic. In addition, we use the quarantine and PCR data along with the model to estimate parameters τ and γ, which are related to people leaving quarantine center. We estimate the remaining parameters 145 φ, β c , β o , θ c , θ o and ρ by using the least square fitting of the model to the daily recorded new case data.

We implemented the previous method [40] to perform the data fitting and to identify a reasonable confidence interval of the estimated parameters. In brief, 150 the method involves thorough process of consecutive reduction of number of parameters until the reasonable confidence intervals are identified. The process allowed us to identify the parameters φ, β c , β o , θ c , θ o , and ρ that can be reasonably estimated from the available data. Further reduction of the number of parameters from the current six parameters provided a poor fit (F-test, p-value

For the model fitting the data available is the daily new cases of recorded infectious people. Using our model, the recorded new infections generated at time t, L(t), can be computed using the following equation:

We solve the system of differential equations numerically using a fourth order 160 Runge-Kutta method. We use the solutions to obtain the best-fit parameters via a nonlinear least squares regression method that minimizes the following sum of the squared residuals:

where φ, β c , β o , θ c , θ o , and ρ are parameters to be estimated, and L(t k ) andL(t k )

are the new cases of recorded infectious people predicted by the model and those 165 given in the available data, respectively. Here, n represents the total number of data points used for the model fitting. To obtain the confidence limits for the estimated parameters, we compute the standard errors from the sensitivity matrix (S) by using the complex-step derivative techniques described previously [4, 6, 40] . 170 Furthermore, we use the sensitivity-based method [35] to analyze the identifiability of these parameters. In particular, we found the matrix S T S to be of the full rank (rank = 6), which confirms the identifiability of the estimated parameters [35] . In our study, all computations were carried out in MATLAB 2020a (The MathWorks, Inc.) using its various routines, including "ode45"

Given the open border of Nepal with India, one of the most COVID-19 infected countries, and related border screen and quarantine policies implemented 180 by the Nepal government, the rate of border screening and quarantine is important for accurate evaluation of the policy. However, the official data of this information is not available. We use our model to estimate the rate of border screen and quarantine, φλ(t), from the data of the active quarantine population,

, and the number of PCR-tests performed, P CR(t i ).

Using the fact that the natural death is negligible during the short period of epidemic (i.e., µ ≈ 0), we apply the model equation (1) at the data collected time t i to obtain the following approximation:

where t i − t i−1 = 1 day, as the data was recorded every day. In this expression, τQ(t i ) is given by P CR(t i ), and γQ(t i ) represents those leaving quarantine center without PCR test (no-P CR(t i )), implying

Since φλ(t i ) ≥ 0, we obtain the minimum estimate of no-P CR(t i ) as

Using data of active quarantine,Q(t i ), PCR tests, P CR(t i ), and the estimated population leaving quarantine center without PCR, no-P CR(t i ), we then estimated the daily number of people border screened and entered into the quarantine, φλ(t i ), until July 21, 2020 (the controlled phase). Our estimates 190 show that the rate of border screen and quarantine was relatively low (less than 2 thousand per day) until the mid of May, 2020, and then the rate increased rapidly reaching a peak of about 16 thousand per day around mid-June. After the peak, the rate began to fall and reached a low level by the end of the first phase of epidemic (Fig. 2) . Data shows that, after July 21 (the outgrown phase), 195 the active quarantined population continues to decrease indicating less impact of these individuals on the epidemic during the outgrown phase. Therefore, for simplicity, we assume that φλ(t i ) decreases linearly after July 21 (Fig. 2) .

Furthermore, we use our model to estimate the per capita rate of individuals leaving quarantine center with (τ ) and without (γ) PCR test. We can approximate these rates as follows

Our calculation shows that the individuals leave the quarantine center at the rate τ = 0.06 with PCR test and at the rate γ = 0.00975 without PCR test. of study and compared our estimates with the data (Fig. 3b) . Our model is capable of accurately predicting the cumulative cases of COVID-19 in Nepal for 210 both epidemic phases, thereby validating our modeling approach.

In consistence with the data, the epidemic trend of the COVID-19 in Nepal predicted by our model shows that the recorded COVID-19 cases increased slowly until the mid of May, attained the peak of the controlled phase during the mid-June, and then decreased until the end of the first phase (the controlled 215 phase), when the policies were altered. After the controlled phase, the cases again started to rise with a higher rate until the end of the study, giving the outgrown phase following the controlled phase. It's worth noting that the first peak observed during the controlled phase is around the same time when the maximum number of returned migrants were border-screened and quarantined 220 (Fig. 2) . 

From the epidemic trend it can be clearly seen that the major policies implemented by the government of Nepal, namely border screening and quarantine, 245 lockdown, and detection and isolation, were significantly effective because the disease spread was well-controlled while the policies were in place and became out of controls once the policies were lifted. We can use our model parameters φ, ξ, and ψ to quantify the effectiveness of these policies, border screen and quarantine, lockdown, and detection and isolation, respectively, on controlling 

In this section, we present our model prediction of epidemic outcome, espe- (Fig. 6b) .

At the current situation of the absence of pharmaceutical prevention, applying public health measures, including the ones the government of Nepal implemented during the controlled phase, are the most promising control mearues [21] [56]. We now assess the impact of these control measures on curbing COVID-19 epidemics from September 2020 to December 2021. Since the current trend (the 285 outgrown phase) shows that the imported COVID-19 cases are not important compared to the local transmission, we particularly focus on two control strategies, the lockdown and the detection and isolation. Note that the current value of infection rates is β = β o and the detection rate is θ = θ o (Table 1 ). In our model, the level of lockdown and detection and isolation can be incorporated 290 using the parameters ξ and ψ, respectively.

Our model predicts that both the lockdown (reduction on β) and the detection and isolation (increment in θ) are significantly impactful on curbing COVID-19 epidemic burden in Nepal (Fig. 7) . For example, 50% reduction of contact through lockdown (i.e., ξ = 0.5) can reduce the cumulative number thousand and the total death down to 13 thousand.

The basic reproduction number, R 0 , is an average number of secondary infections generated by a single infectious individual in a completely susceptible population. For infectious diseases, it is an important threshold, which helps determine whether outbreak occurs R 0 > 1 or is avoided (R 0 < 1) [34] . We used the Next Generation Matrix method [15, 52] to derive the expression of R 0 for 305 our model (see Appendix) and obtained

As expected, we are able to theoretically establish R 0 as the outbreak threshold for our model, as stated in the following theorem: R t . The value of R t allows us to track whether the epidemic at time t is in increasing (R t > 1) or decreasing (R t < 1) trend. For our model, the effective reproduction number is given by

.

Using the estimated parameters, we observed that the value of effective reproduction number R(t) remains about 0.21 until July 21 (Fig. 8b) in place, and upon lifting the policies the effective reproduction number rapidly rose to 1.80. As per our model evaluation, with these policies the government of Nepal was able to prevent more than 444 thousand cases and save more than 365 1,200 lives. Among these three policies, "lockdown" was found to be the most effective, followed by "border screen and quarantine" and then by "detection and isolation".

Consistent with the data based on the travel history of recorded infectious people (more than 80% came from aboard, especially from India) [17, 41] , our Nepal was key to avoid a potential early surge of cases from local transmission.

Our model predicts a high rate of local transmission, consistent with the data, during the outgrown phase (i.e., after the policies were lifted on July 21, 2020).

As a result, the contribution of the local transmission to epidemics became significantly high outcompeting the importation after July 21, 2020. We note March 04, 2021. However, we also acknowledge that there is a possibility for 390 the peak time to occur earlier, as projected by some studies [66] , if the government reduces the testing of asymptomatic cases (i.e., reduces the detection and isolation in our model) as mentioned in [38] . Because Nepal is in the high risk zone of COVID-19 due to its poor health system and porous borders with India, We acknowledge several limitations of our study. We used the limited data sets available publicly from the ministry of health and population of Nepal.

Because of poor policy at the border, the data related to border screen need 420 to be carefully considered. The detailed data with accurate border screen and quarantine will improve the predictions of our model. We first obtain the disease free equilibrium, E * , of the model system. Using the pre-pandemic condition λ = λ(0) and the disease-free conditions E = 0, I R = 0, I N = 0, we obtain ρ = 0. Then the model system provides the following disease free equilibrium: E * = (S * , Q * , 0, 0, 0, 0), where 720 S * = Λ µ + λ(0){τ + γ + µ(1 − φ)} µ(τ + γ + µ) and Q * = λ(0)φ τ + γ + µ .

According to the next generation matrix method, we divide the compartments used in the model into two groups: infected x = (x i , i = 1, 2, 3) = (E, I R , I N ) and non-infected group y = (y j , j = 1, 2, 3) = (S, Q, R). Then the model system can be written as:

x ′ i = f i ( x, y) and y ′ j = g j ( x, y) for i, j = 1, 2, 3.

We now write the right hand side of the system of infected compartments as f i ( x, y) = F i ( x, y) − V i ( x, y), where F i ( x, y) contains the terms representing the new infections in compartment i and V i ( x, y) contains the terms containing the difference between the transfer of individuals out of and into the compartment

We now take the values β(t) = β c and θ(t) = θ c corresponding to the beginning of the epidemic, and construct the following two matrices using F = ∂Fi ∂xj E * and V = ∂Vi

These matrices allow use to compute the second generation matrix as follows: 

.

A.2. Proof of Theorem 3.1 Jacobian of the system of equations (2-6) evaluated at the disease free equi-

The eigenvalues of this Jacobian are given by λ 1 = −µ, λ 2 = −µ, λ 3 = −(η + k + µ), λ 4 = −(γ + τ + µ),

and 730 λ 6 = −(δ + η + 2µ + k) + (δ + η + 2µ + k) 2 − 4(δ + µ)(η + µ + k) (1 − R 0 ) 2 .

We can clearly observe that all the eigenvalues are negative if R 0 < 1. Therefore, the disease free equilibrium, E * , is asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

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