key: cord-0824654-4tqvxo41
authors: Singh, Anurag; Arquam, Md
title: Epidemiological modeling for COVID-19 spread in India with the effect of testing
date: 2021-12-24
journal: Physica A
DOI: 10.1016/j.physa.2021.126774
sha: 55660861a61f8d3ebb31cc2463e79734e0f49ef8
doc_id: 824654
cord_uid: 4tqvxo41

A novel coronavirus has resulted in an outbreak of viral pneumonia in China. Person-to-person transmission has been demonstrated, but, to our knowledge, the spreading of novel coronavirus takes place due to an asymptomatic carrier. Most models are not considering testing and underlying network topology that shows the spreading pattern. By failing to integrate testing into the epidemiological model, models missed a vital opportunity to better understand the role of asymptomatic infection in transmission. In this work, we propose a model considering testing as well as asymptomatic infection considering underlying network topology. We extract the transmission parameters from the data set of COVID-19 of India and apply those parameters in our proposed model. The simulation results support our theoretical derivations, which show the impact of testing and asymptomatic carrier in infection spreading.

The classical SIR model [4] is given as ,

where S (t), I(t), R(t) denotes the proportion of the population that is susceptible, infected, and recovered at time t 82 with spreading rate β and recovery rate µ. 83 There have been various works that investigated for the understanding of pandemic spreading. For instance, 84 Shi et al. [26] proposed a model considering the propagation vector and examined that the propagation vector de-85 creased the epidemic threshold but lead to the spreading. After that, many researchers work on delayed SIR model Figure 1 : A Block diagram of the proposed Model is created, in which the host population is split into 10 states: Susceptible S (k, t), Exposed E(k, t), Quarantine Q(k, t), Tested T (k, t), Untested UT (k, t), Infectious (symptomatic) I s (k, t), Infectious (untested asymptomatic) I us (k, t), Infectious (asymptomatic) I as (k, t), Isolation J(k, t) and recovered R(k, t). Individuals may switch between states in the direction indicated by the arrow, as well as the direction of transmission.

interact with other susceptible individuals. This compartment is referred as isolation compartment. 159 10. R(k, t): the fraction of individuals who have been recovered from infection goes to recovered compartment. 160 The following are our assumptions about individual transmission from one compartment to another: 161 1. A healthy (or susceptible) individual after exposing to either symptomatic or asymptomatic infected indi-162 vidual(s) moves from susceptible to the exposed compartment. The spreading rate of the infection due to 163 symptomatic and asymptomatic infected individuals are β s and β as respectively. 164 2. An exposed individual can either develop symptoms and goes to Testing with rate σ T . After testing it is 3. A quarantine may show some symptom and can be tested with rate τ and move to test compartment. 5. An individual can recover spontaneously at any time with the recovery rates µ Q , µ J , µ us and µ as from 174 compartment quarantine, isolation, untested symptomatic infected and asymptomatic infected respectively. 175 The recovery of an individual is independent of any other compartments' individuals. 176 6. Once the individual gets recovered, it is assumed that it will become immune to the disease and thus, will 177 not transmit the infection to individuals to the susceptible population.

178 7. Furthermore, the demography, or the birth or death of individuals, is not taken into account in this model.

To put it another way, the population remains constant in the process.

The following mean-field equations is defined for the dynamics of the pandemic, considering the discussed 181 compartments, interactions and parameters,

where,

J o u r n a l P r e -p r o o f Journal Pre-proof Infectivity (Ψ(k )) may be considered as 1. Please refer Table 1 for notations and their meaning. Fraction of susceptible individual of degree k at time t E(k, t) Fraction of exposed individual of degree k at time t I s (k, t)

Fraction of symptomatic infected individual of degree k at time t I as (k, t)

Fraction of asymptomatic infected individual of degree k at time t Q(k, t)

Fraction of quarantine individual of degree k at time t T (k, t)

Fraction of tested individual of degree k at time t J(k, t)

Fraction of isolated individual of degree k at time t R(k, t)

Fraction of recovered individual of degree k at time t β s Spreading rate of symptomatic infected individuals β as Spreading rate of asymptomatic infected individuals σ Q Rate of individuals get quarantine σ T Rate of individuals being tested σ I Rate of exposed individual being infected τ

Rate of quarantined population being tested after showing symptom α

Rate of tested population transferred to Infection(symptomatic) κ

Rate of transfer of Infected population to Isolation µ Q Recovery rate of quarantine individuals µ J Recovery rate of isolated individuals µ s Recovery rate of symptomatic infected individuals µ as Recovery rate of asymptomatic infected individuals f proportion of population to be asymptomatic

Infectivity of a node with degree k Θ s (t) Θ as (t)

The mean-field Eq. (4-13) represents nonlinear dynamical system of pandemic spreading at any time t. The 186 sum of the fraction of all individuals is one,

We can represent X(t) = k X(k, t); X ∈ {S , E, Q, I s , I as , T, J, R} is the total population of the respective compart-188 ment at time t of any degree, k 189 Infection spreading occurs when the rate of infected individuals increases, that is ,

Where, I(t) = I s (t) + I us (t) + I as (t). Therefore, by using Eqs. 9 and 11,

The effective reproduction number R c is derived considering test, quarantine and isolated compartments. The 193 effective reproduction number R c is also a time-dependent which continues to track the expected number of sec-194 ondary infections caused by each infective as the epidemic continues with control measures considering test, 195 quarantine and isolation in place. Therefore, R t = R c * S (t) allows time-dependent parameter.

R 0 , or the basic reproductive number, is traditionally defined as the mean number of secondary cases that arise 197 from a primary infection in a completely susceptible population [48, 49] . In a deterministic model, if R 0 < 1 an 198 epidemic cannot develop from a small influx of SARS-infected individuals but an epidemic will develop if R 0 > 1.

By contrast, an epidemic is not guaranteed in a stochastic model if R 0 > 1, but the probability of an epidemic 200 increases with R 0 .

It is considered that the basic reproduction number, R 0 , is defined in the absence of control measures and introduce 202 the control reproduction number, R c , to denote the reproduction number when control measures are in place. R c 203 is derived in the same way as R 0 is derived but using the full model with test, quarantined and isolated classes.

There is also a time-dependent effective reproduction number R t which continues to track the expected number of 

where, R c is called effective reproduction number which determines the spread of infection with control paramet-213 ers If we don't consider the quarantine and isolation compartment then parameter of these compartment then it 214 give the basic reproduction number R 0 as

where, R 0 is called basic reproduction number which determines the spread of infection. One can see the derivation 216 of R 0 , it looks like ratio of spreading rate and recovery rate coupled with network parameter. When R 0 > 1, the 217 propagation occurs at a fast rate. When R 0 = 1 , the propagation happens at a slow rate. When R 0 < 1, the 218 propagation finishes. In this work, total population is considered as constant, therefore, k {S (k, t)+E(k, t)+Q(k, t)+I s (k, t)+I as (k, t)+ dS (k, t) dR(k, t) = −(β s Θ s (t) + β as Θ as (t))kS (k, t) µ Q Q(k, t) + µ J J(k, t) + µ s I us (k, t) + µ as I as (k, t)

β > k ((κ + µ s + µ as )) k 2 J o u r n a l P r e -p r o o f Journal Pre-proof Therefore , the critical spreading rate under heterogeneous network can defined as ,

If we consider all transmission parameter as constant then critical threshold β c will depend on network parameter 226 as k k 2 , which is similar to [44] . The basic reproduction number is calculated using a method proposed by Bettencourt et al.[51] . The author used the Bayesian method to estimate the R t . They consider all the new cases reported daily. The number of new cases gives an idea about the current value of R t . Therefore, today's value of R t is related to yesterday's value of R t . Based on these assumptions, the authors used the Bayes' Theorem as follows,

Where x is the new cases, so authors considered that the distribution of R t is dependent on the likelihood of coming x new cases given R t times, the prior of the value of p(R t ) without the data and divided by the probability of seeing new cases. This is considered for any particular day. When it is considered for a brief period of time: For each day authors used previous day prior P(R t−1 ) to estimate current day prior P(R t ). Prior is previous cases. The author considered the distribution of R t to be a Gaussian centered around R t−1 . Initially, (on the very first day, where no previous day data available), the probability of R t will be,

Further on next day, 240 P(R t 2 |x 1 , x 2 ) ∝ P(R t 2 ) · L(R t 2 |x 2 ) = R t 1 P(R t 1 |x 1 ) · P(R t 2 |R t 1 ) · L(R t 2 |x 2 ) J o u r n a l P r e -p r o o f

Where L is the likely-hood function. A likelihood function describes how likely x new cases are coming for a given a value of R t ).

The R t of India and some other Indian states are estimated and plotted in Figure (graph with no parallel edges and self-loops) by randomly assigning edges to fit the given degree sequence. Table 3 256 provides the statistics of the synthetic network. Based on the configuration model definition and various properties 257 of the synthetic network, this network reflects the real-world contact network [58] .

The aim of the simulations is to answer the following questions,

• How the rate β as affects the X(t)?

• How the rate σ T ) affects the the X(t)? The simulations are performed considering various transmission parameters in order to assess the impact of 316 spreading. Epidemic spreading in the human population increases with the increment in time at a certain point.

After that, it starts decreasing and finally vanishes. The susceptible population decreases to 0.1 at 100-time steps 318 and decreases slowly as compared to the initial trend (Figure 4(a) ). The exposed population are those that are in 319 close contact with infectious agents. The infection spread only to those population that is exposed. As time passes, 320 people start exposing to an infectious agent, and it reaches the maximum as 0.04 at 63-time steps (Figure 4(b) ).

The certain exposed population is quarantined till symptom arises. If a symptom arises, it is tested and moved to 322 the tested symptomatic infection compartment, and after that, it is sent to isolation (Figure 4(c) ). In this course, 323 the quarantined population has reached the peak of 0.125 in 72-time steps. In the meantime, the testing process 324 is also started. The testing process is done when the exposed population showing symptom of infection ( Figure   325 4(d)).

When the testing process is started, some infectious agent is detected while the infection is also increased 327 due to the exposed population which is untested or undetected. Untested Asymptomatic infection and Untested

Symptomatic infection plays a major role in the infection spreading as compared to tested symptomatic infection.

Due to testing, a number of symptomatic infection is checked and isolated. Therefore, the tested symptomatic infection. This is because the untested asymptomatic and symptomatic is directly dependent on the exposed The infection spreading vs. spreading rate is plotted in Figure 9 . When the spreading rate reaches 0.08, then β as and f from 0 to 0.6, changes the exposed population from 0.01 to 0.06. In same way, testing also increases 402 from 0 to 0.014 (Figure 11 (b) ). Infected tested symptomatic vary from 0 to 0.007 Figure 11 One can notice in Figure 12 that increasing the value of β as raises the peak of exposed (E k ), tested (T k ), In order to investigate the influence of the testing σ T in combination of fraction of exposed population f on the (Figure 14(a) ).

The exposed population are those that are in close contact with infectious agents. The infection spread only to 437 those populations that are exposed. As time passes, people start exposing and contacting with an infectious agent, 438 and it reaches maximum at 63-time steps. In the meantime, the testing process is also started from the exposed 439 J o u r n a l P r e -p r o o f Journal Pre-proof population. The testing process is done up to the 300-time steps (Figure14(b) ). The certain exposed population is quarantined till symptom arises. If a symptom arises, it is tested and moved to the tested symptomatic infection 441 compartment, and after that, it is sent to isolation . In this course, the quarantined population has reached the peak 442 of 0.125 in 72-time steps and vanishes after 200-time steps (Figure 14(c) ). In the meantime, the testing process 443 is also started. The testing process is done when the exposed population showing symptom of infection ( Figure   444 14(d)). As exposed population exists, tests are done. Both reaches to its peak at 63-time steps after that start 445 decreasing and vanishes after 175-time steps.

When the testing process is started, some infectious agent is detected while the infection is also increased 447 due to the exposed population which is untested or undetected. Untested Asymptomatic infection and Untested

Symptomatic infection plays a major role in the infection spreading as compared to tested symptomatic infection.

Due to testing, a number of symptomatic infection is checked and isolated. Therefore, the tested symptomatic 450 infection does not spread the infection to a large extent as compared to untested asymptomatic and symptomatic 451 infection. This is because the untested asymptomatic and symptomatic is directly dependent on the exposed dT (k, t) dt = 0 f σ T E(k, t) + τQ(k, t) − αT (k, t) = 0 Q(k, t) doesn't help to go infection out.

Similarly, 600 dE(k, t) dt = 0 (β s Θ s (t) + β as Θ as (t))kS (k, t) − σ Q E(k, t) − f σ T E(k, t)

−(1 − f )σ T E(k, t) = 0 E(k, t) = (β s Θ s (t) + β as Θ as (t))kS (k, t) αT (k, t) − κI s (k, t) + σ I UT (k, t) − µ s I us (k, t) − µ as I as (k, t) > 0

−κI s (k, t) − µ s I us (k, t) − µ as I as (k, t) > 0 σ T E(k, t) − κI s (k, t) − µ s I us (k, t) − µ as I as (k, t) > 0

putting the value of E(k, t) in Eq 27.

(σ T ) (β s Θ s (t) + β as Θ as (t))kS (k, t) σ Q + σ T − (µ s + κ + µ as )I(k, t) > 0 (σ T ) (β s Θ s (t) + β as Θ as (t))kS (k, t) σ Q + σ T > (µ s + κ + µ as )I(k, t) (σ T ) (β s I s (k, t) + β as I as (k, t)) k 2 k S (k, t) σ Q + σ T > (µ s + κ + µ as )I(k, t) dS (k, t) dR(k, t) = −(β s Θ s (t) + β as Θ as (t))kS (k, t) µ J J(k, t) + µ s I us (k, t) + µ as I as (k, t)

Population of J(k, t) is considered, when it is fixed, dS (k, t) dR(k, t) = −(β s Θ s (t) + β as Θ as (t))kS (k, t) κI s (k, t) + µ s I us (k, t) + µ as I as (k, t) dS (k, t) dR(k, t) = −(β s I s(k, t) + β as I as (k, t)) k max k=1 p(k |k)kS (k, t) κI s (k, t) + µ s I us (k, t) + µ as I as (k, t) dS (k, t) dR(k, t) = −(β s + β as ) k max k=1 p(k |k)kS (k, t) κ + µ s + µ as dS (k, t) S (k, t) = −(β s + β as )k k max k=1 p(k |k) (κ + µ s + µ as ) dR(k, t)

integrating both side S (k, t) = e −(β s + β as )k k max k=1 p(k |k)R(k, t) κ + µ s + µ as Let, Assume that f (Ω(k, ∞) is strictly increasing as a function of Ω(k, ∞). When R(k, ∞) is set to 0, the total 613 population recovers. It is also known as a disease-free state. As a result, we must find a solution that is between 0 614 and 1. However, since Eq. 34 is coupled with network parameters, we can't express it in terms of R(k, ∞). As a 615 result, we must find a solution 30,

Now deriving the f (Ω(k, ∞)) with respect to Ω(k, ∞), where, 

Agent-based modeling: Methods and techniques for simulating human systems

Dynamic models of segregation

Cognition and multi-agent interaction: From cognitive modeling to social simulation

A contribution to the mathematical theory of epidemics

London A: mathematical, physical and engineering sciences

The mathematics of infectious diseases

Modeling competitive marketing strategies in social networks

Modelling and analysis of delayed sir model on complex network

Modelling to contain pandemics

The basic si model, New Directions for Teaching and Learning

The quasi-stationary distribution of the closed endemic sis model

An sirs model with a nonlinear incidence rate

Mobility based sir model for pandemics-with case study of covid-19

Epidemic spreading in geometric network with mobile agents

The effect of 502 travel restrictions on the spread of the 2019 novel coronavirus

Modelling disease outbreaks in realistic urban 504 social networks

Reconstruction of the full transmission dynamics of covid-19 in wuhan

Coronavirus: India defiant as millions struggle under lockdown

Modi orders 3-week total lockdown for all 1.3 billion indians

How to protect yourself & others

Complex dynamics of an seir epidemic model with saturated incidence rate and treatment

Essai d'une nouvelle analyse de la mortalité causée par la petite vérole et des avantages de l'inoculation pour la prévenir

A contribution to the mathematical theory of epidemics

Contributions to the mathematical theory of epidemics. ii. &#x2014;the problem of 521 endemicity

Infectious diseases of humans: dynamics and control

An sis model with infective medium on complex networks

A delayed sir model with general nonlinear incidence rate

Analysis of a delayed sir epidemic model

A mathematical model for simulating the phase-based transmissibility of 532 a novel coronavirus

A conceptual model for the 534 coronavirus disease 2019 (covid-19) outbreak in wuhan, china with individual reaction and governmental action

Dynamics of the covid-19 pandemic in india

Modeling the dynamics of novel coronavirus (2019-ncov) with fractional derivative

Characteristics of and important lessons from the coronavirus disease 2019 (covid-19) outbreak in china: 540 summary of a report of 72 314 cases from the chinese center for disease control and prevention

Considerations for diagnostic covid-19 tests

1. kevin bacon, the small world, and why it all matters

Scale-free networks: a decade and beyond

Statistical mechanics of complex networks

The statistical physics of real-world networks

Nonlinear spread of rumor and inoculation strategies in the nodes with degree dependent tie strength in complex 549 networks

Epidemic spreading in scale-free networks

Epidemic processes in complex networks

Worm epidemics in wireless ad hoc networks

Theory of rumour spreading in complex social networks

Epidemic outbreaks in complex heterogeneous networks

Controlling the spreading in small-world evolving networks: stability, oscillation, and topology

Mathematical epidemiology of infectious diseases: model building, analysis and interpretation

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease 566 transmission

Real time bayesian estimation of the epidemic potential of emerging infectious diseases

Mathematical modelling and optimal cost-effective control of covid-19 571 transmission dynamics

Symptom duration and risk factors for delayed return to usual health among outpatients with covid-19 in a multistate 574 health care systems network-united states

The structure and function of complex networks

The origin of bursts and heavy tails in human dynamics

Graph theory and networks in biology

Exploring network structure, dynamics, and function using NetworkX

The configuration model for barabasi-albert networks

Coronavirus infection: How long do covid-19 patients remain infectious?

Coronavirus disease 2019 (covid-19) situation report -46

Icmr sero survey

Covid-19 pandemic in india: a mathematical model study

Early transmission dynamics in 589 wuhan, china, of novel coronavirus-infected pneumonia

Modelling and analysis of covid-19 epidemic in india

Report of the who-china joint mission on coronavirus disease

Estimation of the transmission risk of the 2019-ncov and its implication 594 for public health interventions

Mathematical modeling of covid-19 spreading with asymptomatic infected and interacting peoples