Enhancement and Numerical Assessment of Novel SARS-CoV-2 Virus Transmission Model
Authors: Tahir Khan
zhiqiang xu
Venue: Submitted to ICLR 2023 (2023)

Abstract: Recent pandemic of the coronavirus started in December 2019, which has affected almost all groups of humankind. In this regard, accurate epidemic models are not only crucial for demonstrating the mitigation of the current pandemic but also helpful for forecasting their future dynamics. In this work, we propose a model for SARS-CoV-2 virus transmission to forecast the temporal dynamics of the novel coronavirus disease by considering the characteristics of the disease and the recent literature. Due to the nondeterministic and stochastic nature of the novel-coronavirus disease, we present the model with the aid of stochastic differential equations by considering two infectious phases: pre-symptomatic and symptomatic, because both are significant in the spread of SARS-CoV-2 virus transmission. We ensure that the model is well-posed and identify the necessary conditions for disease eradication by proving the existence, uniqueness, and extinction analysis. The efficacy of the model and the importance of the current study are demonstrated using the actual data. Finally, the model will be simulated using Euler-Maruyama and Milstein’s numerical schemes to support the theoretical findings and show the significance of the results obtained.

Under review as a conference paper at ICLR 2023
ENHANCEMENT AND NUMERICAL ASSESSMENT OF NOVEL SARS-COV-2 VIRUS TRANSMISSION MODEL
Anonymous authors Paper under double-blind review
ABSTRACT
Recent pandemic of the coronavirus started in December 2019, which has af- fected almost all groups of humankind. In this regard, accurate epidemic mod- els are not only crucial for demonstrating the mitigation of the current pandemic but also helpful for forecasting their future dynamics. In this work, we propose model for SARS-CoV-2 virus transmission to forecast the temporal dynamics of the novel coronavirus disease by considering the characteristics of the disease and the recent literature. Due to the nondeterministic and stochastic nature of the novel-coronavirus disease, we present the model with the aid of stochastic dif- ferential equations by considering two infectious phases: pre-symptonatic and symptomatic, because both are significant in the spread of SARS-CoVa virus transmission. We ensure that the model is well-posed and identify the necessary conditions for disease eradication by proving the existence, uniqueness, and ex- tinction analysis. The efficacy of the model and the importance of the current study are demonstrated using the actual data. Finally, the model will be simulated using Maruuaaa and Milstein's 's numerical schemes to support the theoreti- cal findings and show the significance of the results obtained.
INTRODUCTION
Diseases mainly categorized into two groups: infectious and non- non-infectious diseases. Infectious are those caused by viruses, fungi. parasites, and bacteria, and usually transferred in numerous ways while causing fifty thousand deaths approximately every day all over the world. Some infectious diseases can be directly communicated, while many transfer indirectly. Infectious discases like hepatitis B, seasonal influenza, HIV (human immunodeficiency virus), Middle East Respiratory Syndrome (MERS), and SARS-CoVa. are major health issues, affecting millions of populations around the globe (Altamimi et al. 2020 Holmdahl & Buckee] [2020) Mann & Roberts 2011 Reich et al. 2019; Park et al. 2021). SARS-CoV is a f'amily of viruses that usually cause illnesses such as MERS and severe respiratory syndrome coronavirus (Syed) 2020). Coronaviruses are zoonotic diseases, and the novel one is new strain known as the SARS-CoV-2 virus, which broke out in 2019 and spread throughout the world (Rabaan et al., 2020 Bats are the most plausible ecological reservoirs for SARS-CoV-2. but it's also possible that the virus infected humans via an intermediate animal host. This intermediate animal host could be an unidentified domesticated food animal, a wild animal, or domesticated wild animal. Millions of people are infected and face consequences due to the novel disease of coronavirus (Shereen et al., 2020). Novel coronavirus transmits from one person to another by direct contact with an infected individual and indirect with objects used by the infected person (Chan et al.) 2020). A novel coronavirus disease has multiple phases of infections, pre-symptomatic, asymptomatic, environmental, and symptomatic (He et al. 2020). Especially, the pre-symptomatic and symptomatic phases are very significant because 47% and 38% of cases are reported respectively, by contact with these individuals (Ferretti al., 2020). Generally, pre- asymptomatic individuals have no symptoms while transmitting the disease to others. Therefore, the immigration pre-symptomatic and symptomatic patients from one place to another place leads to major source of novel coronavirus transmission. So, most countries around the globe restricted air traffic and announced a lockdown to use the precautionary measure to minimize human lives as much as possible. Also, every country tried reduce unnecessary traveling, which helped in the reduction of the newly reported cases. Many countries are badly affected by the pandemic of SARS- CoV-2. Moreover, the economy of different countries has been influenced by the novel deadly virus

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badly. Many organizations and companies have stopped their production. As result, the ratio of unemployment as well as poverty has increased in various countries. Besides, the health systems of many developed and powerful countries collapsed due to the consequences of the SARS-CoV-2 virus.
Mathematical modeling is one of the best tools to show disease mitigation and design control mecha-
nism. The epidemiology of infectious disease has a rich literature, see for instance, |Mnnn & Roberts
(2011) Hattaf et al. (2012); Shi et al. (2015); Alaa & van der Schaar (2019); | Kamarthi et al. (2021); }); Yin et al. (2021). Various models have been used extensively by researchers to study the temporal dynamics of different infectious diseases (Wang et al. 2014 Khan et al. 2018 Moreover, many mathematicians and biologists also studied the dynamical behaviors of SARS-CoV-2 transmission.
More precisely, a model study has been reported to forecast the spread of novel coronavirus out-
break in Wuhan (Wu et al., 2020). Guo et al. (2020) studied the prediction of host and infectiv- ity of novel diseases using deep learning algorithm. [Selvam et al.] (202T) analyzed the spread of corona virus diseases using mathematical modeling and performed stability analysis. Further, spatial-temporal model has been formulated to study the dynamic risk assessment of SARS-CoV-2
accounting for community virus exposure (Chen et al. 2021). Similarly, many other studies have been reported by various authors to forecast the pandemic trend and control of novel disease (SARS- CoV-2) transmissions (see for more details, Arii et al. (2020); Kucharski al. (2020); WWang et al.. 2 Flaxman et al. (2020); Chen et al. (2021); ZZang et al. (2021). Very recently, a stochas- tic epidemiological model has been analyzed for the dynamics of novel disease of corona virus by Khan et al. (202I). Models with appropriate structure and accurate dynamics are not only impor- lant show disease mitigation but also forecast the future dynamies of the discase, and thus are helpful for public health planning. Indeed, the above-reporte studies yielded some interesting out- puts. However, their main objectives are forecast disease dynamics using deterministic differential equations. While the model related to the stochastic analysis of SARS-CoV-2 virus transmission is reported in Khan et al. (202T), many factors are to improved. For example, they considered the random fluctuation only in the disease transmission rate, but the other parameters such mobility, vaccination, the occurrence of death, etc. also have stochastic nature. Moreover, the classification of various infection phases of SARS-CoV-2 virus disease has not been considered, such that important roles played by pre-symptomatic and symptomatic individuals were neglected in the pandemic trend of novel corona virus disease. In fact, small number of pre-symptomatic individuals will lead to major outbreak, because they have no symptoms while transmitting the disease to others. Further, the pandemic of SARS-CoV-2 rises due to human interaction, but initial sources of transmission were a reservoir that has been ignored.
Our goal is to enhance the model reported in Khan et al. (2021) by incorporating the missing parame- ters and characteristics of SARS-CoV-2 virus disease that can influence the disease transmission. To this end, we use various sources of randomness using different Brownian motions in each population group to include the stochastic effect in every parameter as well as in every group of population. We divide the total infected population into two sub-classes, namely pre-symptomatic and symptomatic, according to the characteristics of the disease. We will also assume that both the pre-sympiomatic and symptomatic individuals will contribute to producing the reservoirs. To do this, first, we formu- late the model and discuss its biological and mathematical fcasibility to show the well-posedness of the problem. For this purpose, we will use the combination of stochastic Lyapunov function theory and the Itô formula. Further, the disease extinction of the model will be discussed find the condi- tions for disease elimination. We then develop the algorithms for the proposed epidemic problem to discuss the numerical assessment and verify our analytical findings by using Gurer-Maruyama and Milstein's 's methods. To show the effectiveness and justify the proposed epidemic problem, we fit the model to real data of the SARS-CoVa virus reported in the United Arab Emirates the period of March, 2 2022 to Jun, 21st 2022. Finally, we compare our results to show the significance of the model solution.
PRELIMINARIES
In this section, we introduce some of the fundamental concepts and notations that will be helpful in getting our main results, including a multidimensional Itô formula, the strong law of large numbers, and some other results.
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Let R be the set of real numbers, R" the space of n-tuple, and R the set of n-tuple with non- negative entries. Fr, (Fi) P) represents a filtered probability space Fr,FP,P) with a right-continuous and complete filtration F = (Fe)ee(o.r) i.e., IF+ = F and each J. E contains all P-null spaces (Klenke 2007). If f(x)) is a function over [a, b), then the mean value of f(x) is defined as
(f(z)) = - - [- f(z)da.
In addition, (2)) denotes the set of all real-valued and twice continuously differentiable func- tions.
Lemma Kuo 2006) Let a = (a1....,aa) and (31...., B,,) represent the n-dimensional square-integrable adapted processes. We assume that X (Ni....: where Xr for k (1.....n) n) is driven by the stochastic differential equation (SDE) with standard Brownian motion B(I), ie.,
dX&(t)) = ag.t + BrdB(t), &x((0) E
Let H E C2 (R"), then
1 (0)((()) aM(((())) = = (H( (t.(t)) + Σ k 2 dr d (xx(
ark
where (.) represents the inner product between two functions C² (RR).
Lemma Birkel 1988) Let a be the intensity of environmental fluctuation and B(t) the standard Brownian motion, and assume that N {Ne)ezo continuous real-valued martingale defined by (t) = =o (s) and (N, N); =oo o2 dt, vanishes at t O. then
N
lim (N, NN,N), 00, implies lim
0.
N),
Also
lim sup (N, 0, implies
Ne 0.
lim
1-000 t
MODEL FORMULATION AND ANALYSIS
In this section, will discuss the enhancement of the model reported in Khan et al. (2021) For this, let us assume that FT, (FF))ejo.T)). P), on this space lives 5-D Brownian motion W
(W(t)=
l E [)} with W(()= == [Wi(t) assumed to generated by the Brownian motion W. Further, it could be noted that the spread of SARS-CoV-2 disease is uncertain due to its novel nature. Thus, it transmits from human to human as well as from reservoir to human while not uniform everywhere. We. therefore, assume that the various compartments of the population have stochastic nature driven by different randomness sources B, (t), i 1,....5.. 5. The proposed epidemiological model assumes that the variation in all groups of population is connected to various sources of information represented by (Fi) where F, == (B(()) symbolizes the alalgebra. We then assume the random fluctuation in every group of the population and formulate the model based available information about the biological process through which the SARS-CoV-2 virus spreads. The multiple infection phases and reservoirs play an important role in the novel disease transmission. The pre-symplomatic and symptomatic populations are especially very significant. We divide the entire host population into various classes: pre-symptomatic,
1, .... 5}. Moreover, the natural filtration (FF))eju,2) is
s = susceptible,
symptomatic, r = recovered, and m reservoir. The notion defines the un-infected population at time f, but has a chance to get infected at time small increment in time. Similarly, the pre-symptomatic are those who are
t + At, where At infected while having no symptoms and are very significant in SARS-CoV-2 transmission because of its novel nature, while the infected individuals showing symptoms are known as the symptomatic population. The individuals who become healthy after the infection assume that all newborn babies, as well as immigrants from other populations, are susceptible. Since still there is not enough evidence on the vertical transmission of the virus from the pregnant
the recovered ones. We
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woman to the newborn baby, however, immigrants may may not be infected. In the present case, for the sake of mathematical simplicity, we assume that the immigrants are virus-free. Since the population is assumed to be homogeneously mixed, the successful transmission of SARS-CoV-2 leads the susceptible individuals & into the pre-symptomatic or symptomatic with probability p and
(1 -), respectively. Both the infected population ppr-syymtooattc and symptomatic) and the reservoir are responsible for disease transmission. We specify two modes of recovery based on the characteristic of the discase. First, a fraction, denoted by p. of the pre-symptomatie population may recover directly with probability and enter the recovered class in case the individual has a strong immune system, while those who develop symptoms will lead to the symptomatic compartment with (1 - v) probability. Second, the individual leaves the symptomatic class only after they have fully recovered by taking proper treatment and move to the recovered class r or die. Based on the effectiveness of the vaccine, the w-th portion of the susceptible population will move to the recovered class directly. The ratio of the reservoirs contributing from infected (both the pre-symptomatic and symptomatic) population are respectively denoted by a1 and ag. while the removal rate for the reservoir is 17. Further, the geometry of the proposed problem is depicted in Figure T and so the
Co 0 0.Joy
Myyan
S(K)
a(t))
spa
EP 18))
(1 of ppa
Renerver
Figure 1: The schematic process of the SARS-CoVa virus transmission model I
enhanced stochastic epidemiological model looks like this:
ds(() { Φ (is((t)a(t)
()2b(t)s(t) sgss(t)m(t) (u (v- dt m mis(tdBBi())
m(()
nll)
n(l)
da(t) (ha(t)s(() (2b(())(()) + (sm(()sst)) (0 10+00 11+ + } dt (2()))dBu(t.
n(t) n(t)
n(t)
(1 (1) (11(())((t)
(2b(())(t) + (ss(t)m(tt) + ppa(l)
db(1)
n(t)
n(t)
nlt)
- (5 + 0 + 02)b(t) 02)0(t) dt mob(t)
dr(1) = (us(() + (1 - ()) + <b(t) (r({)} dt + nar(t)dBa dm(t)= = (a2b(t) + ana(t) - (1) (())d dt + mm ( (t)dBs(t).
(I)
where is the birth rate, th, i = 1,2,3, 3. are the disease transmission rates. The natural death is assumed be & while the moving rate of pre-symptomatic individuals to the symptomatic popula- tion is denoted by p. We also assume that 0,, while Ç represents the recovery rate of the symptomatic population. Moreover, a; 1, are the contributing rates for reservoir, and M), i = 1.... 5. are the intensities of white noise.
1, 2, are the death rates that occur from disease
3.1 WELL-POSEDNESS
This section is devoted to discussing the well-posedness of the proposed epidemic problem. We show that the epidemic model (I) is mathematically as well as biologically feasible by proving that the model has a unique and positive solution. We use the Lyapunov theory and Itô formula to
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perform the existence analysis of Eq. ① which describes by the following theorem. Due to space limit, all the proofs are deferred to appendix.
Theorem For any initial population sizes (s(0), a()), ((0),r(()) m(0) E R. there exists unique global solution of the model 1 remains in R.f, almost surely (a.s).
Theorem For any (0)) ((), (()) m(()) E R the solution of model is positive when- ever exists.
3.2 EXTINCTION
We discuss the novel disease extinction and find enough conditions for disease extinction form of expression containing the white noise intensities and epidemic parameters. Let us assume that Ro is the stochastic reproductive parameter defined as Rg = R2 + R2. where
( + t + 11)
4 (6) + t1 + (v + 2) 0
RS
and RS
(U+ + 0) ( 572)
Since the reproductive parameter is defined to be the average number of the secondary infectious produced by single infective individual whenever introduced into susceptible population, the infection dies out if Ro < 1, and spreads whenever Ro > 1. This shows that the threshold parameter Ro is important quantity, of which the sensitivity analysis against some chosen parameters described in the following figures.
15
35
(a) Variation of Ro verses v and ths
(b) Variation of Ro verses t/i and 8/2-
Figure 2: The graphical results show the variation of threshold parameter Ry against some chosen parameters (wh, 1h3, v). while the other parametric values are: +b 0.9, 1/1 0.2, 1/2 0.3,
0.53, 0.44, p 0.55, 01 0.45, 0₂ = 0.65, p = 0.41, 7/2 = 0.51 and 1/ 0.65.
(a) Variation of Ro verse 01 and Og.
(b) Variation of Ro verses Ç and 0.
Figure 3: This visualizes the variation of threshold parameter Ra versus some sensitive epidemic parameters {0, 02 {). while the other parametric values are: & 0.9, t 0.2, 1//2 0.3, 0.53, 0.44, p = 0.55, 0 = 0.45, 0₂ = 0.65, 0.41, f/2 = 0.51 and n = 0.65.
Further, regarding disease extinction, the following result is given.
Theorem The novel disease of coronavirus dies out, if the stochastic reproductive number is less
than one (Ro < 1) and then
In an)
1 S (p+0+ 0 0 (RF < 0,
lim sup 1-000
t
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and
Inb(t)
( ( 02 511 (R² 1) < O.
1
lim sup 20000
Also
+ -+
v$ 0(0 + v)
lim st) E-000
lim r(t) 10
lim b(t) lim a(t) 0, a.s. 1000 1-000 2-000
lim m(t)
4 MODEL DISCRETIZATION
To perform the numerical assessment of the model that is under consideration, we use Eu- ler-Maruyama (EM) method and Milstein's Higher Order Method. We assume that the de- sired time interval is T). The time-step At f. where L is positive integer. A point of the discretized interval is T3 = jåt. For the sake of simplicity, a solution of the prob- lem (s(7,), a(TT),())))))))))) b(), m(TT)) can be written as (sy. bj, r, ry,))) and (Ty)) Biy, i i= 11,,..... 5. The Fuler- Maruyama (EM) method for the proposed model will take the form:
V10j-188-1
8j = 85
(0 + 000))8-11 A
11-1
Ni-1
+ + (B1, - -B1,-1);
17 110j-189-1 + (220)-185- 118-1
1b3mj- 1
8j 0 + {
- (0 + + ++++)))--11 At
Ri-1
n1
nj-1
+ + (B2, - -B2
bj = b {11 p)
(4ha)-188-1 + 1/22--18 181-1
+
433 - + -paa-11
(3)
11-11
R1-11
n
- -S 0 + 02)b (dl + 113b-11 - -Ba, 1),
U8--1 + (1 - p)paa-11 + (bp - Or jatt + 1BT3-1(BA, - B,,_,
5j = ---1++ +
) |dl + 1
my m 011j-1 + 1
(1 -Bs, 1).
can be noticed that in the deterministic case (1); = 0), Eq. (3) reduces to Euler's 's method. The next step compute discretized Brownian paths that can be used to find (, (;(T)-1) needed in Eq. ① For this, let us assume that the step size A/ is an integral multiple of f 2 1 of the increment öt for the Brownian path. This ensures that the set of points t/ss (the discretized Brownian path) also contains Ty (used the EM solution). The increments B, (Ty)) (T, in the EM method (3) are given by
jR
BT(T,)- -B00)BB,Ty-)) = B.GRR()) (. - 1)Rot)
dBg.
J
R+1
If one replaces these Brownian increments by v A (V, takes the values +1 and -1 with equal probability), it will give the weak Euler Maruyama (WEM) procedure. The above steps can be concluded in Algorithm 1. Next, we truncate the Itô-Taylor expansion at a suitable point, which gives Milstein's 's procedure for SDE ① as follows:
#105-188-1
12bb-188-11 113 - -(u+0)s--1 )
8, = 83-1 T
71-1
n1-1
11j-1
1
+ 11 + (BI; - -B1, + + 8 ( - -B1, -B1,-1}² -
2
(
V V1a3-189-1 + d22b--18-11 + 233 ) - (p 1 0 -pp+0+0))0 } )
0j aj-1 +
{
nj-1
n
nÿ-1
+ (B22 - B2 + 1
- - 22y_))=AA11

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by = b-- (1 (1- 12b3-188-1
(120)-189-1
43 + 9pas-1
Rj-1
n4-1
1
- (5 + 8 + 02)bb-11 dt 1 -Bag-1) 1 + 17bb)-1(1BB) - -Bs At),
Tj = /ン= 18j-1 + (1 p)paa-1 + 5b,-1 - -Br,- At 17 11 -1,-1))
2 1 ((4s; Ba, 1)² -11,
m, = m,
01 3 + azbj-1 11 -mmm- At + +mmm--1 BB)-1)
+ 2 mg - ((15, Bs, 10-11)2- At).
(4)
The main algorithm of Milstein's Higher Order Method for the model is given in Algorithm 2.
5 EXPERIMENTAL RESULTS
In this section, visualize the large-scale numerical simulations of our model as described by Eq. (.. In the case of epidemiological models, it is very important to validate and justify the model with the help of available data. We fit our model the data of EARS-CoV-2 reported in the United Arab Emirates (UAE) between March 2 2022 Jun 21st, 2022, which clearly shows the effectiveness of the model and the importance of the current study as shown in Figure 4 To fit the model, parameters need be estimated. First, we import the data and accordingly use the theory of the Runge-Kutte Kutta method for the numerical solution of the proposed system. We then use the nonlinear fitting function 'Iqqcurvffit in matlab to estimate the epidemic parameter presented in table [ The error analysis of the model predictions with real data is also performed by calculating the етror and absolute error presented in Figure 4 Both the error and absolute error describe that within the first 60 days the eтrоr curve fluctuates in the neighborhood of O. That the model best fitting the data. Within this duration, one say that the model covers above 95% of the data points. Between 65 and 85 days, there is gap between the observed data and the curve predicted by the model. This is because,
have sudden increase in the reported cases and mathematically. such a drastic change in the dynamics of infection is very difficult to capture. Afterward. the error tends decrease, and the model again best fits the data which shows the effectiveness of the present work. We use the schemes as reported in Algorithms 1-2 with an application of model parameters
Aidel
5000
Mep
Figure 4: Parameter estimation and error analysis for the proposed model based on the data from United Arab Emirates (UAE). The model fitted with the real data of daily new confirmed cases for the time period March 21, 2022 to June 21, 2022. Moreover, calculated the error analysis of the values via simulation and real data.
in biologically feasible way, and initial sizes of populations, i.e., (20, 20, 32, 50). We also assume that the unit of the time interval is 0 to 10. Thus, the execution of the schemes as reported above with the biologically feasible parametric value and initial compartmental population sizes lead to the results as depicted in Figures 5 6, and 7 where the solution curves of the proposed epidemic problem 11 are represented by Figure 5 Particularly, the numerical solution retrieved by

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120
5(0
100
att
bitl
80
att
bit
r(t)
miti
mitt
60
x
X
60
40
40
10
(a) Solution curves via Euler- Galer-Maruyane Maruyama (EM) (b) Solution curves via Milstein's 's method. method.
Figure 5: The plots visualizes the solution curves of the model
| against the parameter values:
t 0.1, 1/1 = 0.2, 1/3 0.01, 1/2 0.3, 0.43, 0 0.44, !/1 0.41, 7/2 0.51, f/3 0..55,
7/ = 0.731, 175 = 0.64, 17 = 0.4, p 0.41, p = 0.55, 01 = 0.45, = 0.22, Ç 0.01, 02 = 0.65,
a1 0.43 and Q2 0.87.
Method - MHO Method
MHO Method
40
15
30
s(t)
r(U)
20
5-
10
2
6
10
t
(a) Susceptible Population
(b) Recovered Population
Figure 6: The graphs represent the temporal dynamics of the compartmental population as well as the comparative analysis of the results against parametric values; & 0.9, 1/1 0.2, 3/y 0.01, 2/2 = 0.3, // = 0.43, = 0.44, 7/1 0.41, 7/2 = 0.51, 173 = 0.65, 7/1 = 0.731, 7/5 = 0.64, 77 = 0.6,
p 0..1,, p 0.65, 01 0.45, q 0.022, Ç = 0.1, 02 0.65, an 0.43 and (t2 0.87.
Mathod
Mathod
Mathod
NHO Method
-HH Mehhod
MHO Mehodd
40
a/1
bi
m/t1
20
30
10
20:
5-
2
2
2
(a) Asymptomatic Population
(b) Symptomatic Population
(c) Reservoir
Figure 7: The graphs represent the dynamies of compartmental population as well as the com-
parative analysis of the results against parametric values: & = 0.9, 0.2, 3/3 = 0.01, t/2 0.3,
y 0.43, 0 = 0.44, 771 = 0.41, 72 = 0.51, 7/3 = 0.65, 77 = 0.731, n75 = 0.64, = 0.6, = 0.41, = 0.65, 0 = 0.45, g = 0.022, Ç 0.1, 02 0.65, on 0.43 and 02 0.87.
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Table 1: The values of epidemic parameters of model (I
estimated from real data of SARS-CoV-2
Parameters
Description
Value
Source
Disease transmission due pre-symptomatic
0.1387
Fitted
2/2
Disease transmission co-efficient due to symptomatic Disease transmission rate from reservoir Vaccination
0.6174
Fitted
4/3
0,001024
Fitted
t
0,00001
Fitted
p
Probability which the susceptible become vresyymptomatic Probability at which pre-symptomatic become recover
0.0003
Fitted
0.0102
Fitted
P
Recovery rate of pre-symptomatic Death rate due to disease Death rate due to disease Contributing rate for reservoir Contributing rate for reservoir Recovery rate of symptomatic individuals Removal rate of reservoir
0.09
Fitted
01
0.0054
Fitted
82
0.5245
Fitted
a1
0.3436
Fitted
Q2
0.0083
Fitted
C
0.001
Fitted
0.01
Fitted
Euler-Maruyama (EM) is visualized in Figure 5a while the graphs depicted in Figure 5b show the solution trajectories carried with the help of Milstein's method.
Further, to show the comparison of solution for the compartmental populations of model we present the temporal dynamics of susceptible, asymptomatic, symptomatic, recovered and reservoir in Figure bal 6b 7a 7b and [7c] respectively. More precisely, the solution curves of the model (I) for the susceptible population are demonstrated in Figure 6aa The two trajectories represented by the red and blue dashed lines, respectively, show the solution s(t) applying Fuler-Maruyama (FM) and Milstein's 's methods as in Figure [6a] Similarly. the long-run numerical simulation for the pre- asymptomatic, symptomatic, recovered, and reservoir visualized by the blue dashed line and red dashed line, respectively, represents the solution retrieved by Euler-Maruyama (EM) and Milstein's methods, see Figure [b] 7c It could be observed that the numerical solutions of the proposed prob- lem via Euler-Mand Maruyama (EM) and Milstein's 's methods agree with each other the long run, which shows the significance of the developed algorithm. Likewise, the numerical verification of the the- oretical results as stated by Theorem is shown in Figure 6 and 7 We obtained this by calculating the value of threshold parameter R4 0.2033 analytical findings that whenever Ro < the infection of novel disease dies out (infection popula- tion vanishes) while there will be only susceptible and recovered population as shown Figure Thus, it could be concluded that the elimination of the disease continuously depends on the threshold parameter, and it is very necessary to keep its value less than unity as much as possible.
in this case. The numerical results verify the
CONCLUSION
This paper formulated stochastic epidemiological model for SARS-CoV-2 virus mitigation cording to the characteristics of newly reported coronavirus transmission. The impact of pre- symptomatic, symptomatic, and reservoir has been incorporated to present accurate and well- structured dynamics behavior of the disease. For this purpose, we divided the total compartmental population into the various classes of susceptible, pre-symptomatic, symptomatic, recovered, and reservoir. We showed the proper disease mitigation with the help of the stochastic differential equa- tions model. We studied the existence and uniqueness analysis of the model and showed that the model is biologically model and calculated some sufficient conditions for the novel disease extinction. To perform the long-run simulation of the proposed model Euler-Maruyama and Milstein's schemes have been de- veloped and supported the analytical findings with graphical representation. We also used real data of the SARS-CoVa virus and fitted the model to the data and proved the effectiveness of the current study. In the future, we will work on the dynamics and control of novel corona virus disease using multiple strained model.
well as mathematically feasible. We also discussed the extinction of the

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Thomas Birkel. note on the strong law of large numbers for positively dependent random vari- ables. Statistics & Probability Letters, 7(1):17-20, 1988.
Jasper Fuk-Woo Chan, Shuofeng Yuan, Kin-Hang Kok, Kelvin Kai-Wang To, Hin Chu, Jin Yang. Fanfan Xing, Jieling Liu, Cyril Chik- Yan Yip, Rosana Wing-Shan Poon, et al. A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating person-to-pp to-person transmission: study of a family cluster. The lancet. 395(10223::51452222 2020.
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Luca Ferretti, Chris Wymant, Michelle Kendall, Lele Zhao, Anel Nurtay, Lucie Abeler-Dörner, Michael Parker, David Bonsall, and Christophe Fraser. Quantifying sars-cov-2 transmission sug- gests epidemic control with digital contact tracing. Science, 368(6491)ccab66933) 2020.
Seth Flaxman, Swapnil Mishra, Axel Gandy, Juliette T Unwin, Thomas A Mellan, Helen Cou- pland, Charles Whittaker, Harrison Zhu, Tresnia Berah, Jeffrey W Eaton, al. Estimating the effects of non-pharmaceutical interventions on covid-19 19 in europe. Nature, 584(7820)257-261, 2020.
Qian Guo, Mo Li, Chunhui Wang, Peihong Wang, Zhencheng Fang, Shufang Wu, Yonghong Xiao, Huaiqiu Zhu, et al. Host and infectivity prediction of wuhan 2019 novel coronavirus using deep learning algorithm. BioRxiv, 2020.
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Tahir Khan, Amir Khan, and Gul Zaman. The extinction and persistence of the stochastic hepatitis epidemic model. Chaos, Solitons & Fractals, 108:123-122 2018
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Achim Klenke. Probability Theory: Comprehensive Course. Springer, 2007.
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Adam J Kucharski, Timothy W Russell, Charlie Diamond, Yang Liu, John Edmunds. Sebastian Funk, Rosalind M Eggo, Fiona Sun, Mark Jit, James D Munday, et al. Early dynamics of trans- mission and control of covid- 19: mathematical modelling study. The lancet infectious diseases, 20(5):553-558, 2020.
Kuo. Introduction to stochastic integration springer. Berlin Heidelberg, 2006.
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Sangjoo Park, Gwanghyun Kim, Jeongsol Kim, Boah Kim, and Jong Chul Ye. Federated split vision transformer for covid- 19cxr diagnosis using task-agnostic training. arXiv preprint arXiv:2111.01338, 2021.
Ali A Rabaan, Shamsah H Al-Ahmed, Shafiul Haque, Ranjit Sah, Ruchi Tiwari, Yashpal Singh Ma- lik, Kuldeep Dhama, M Iqbal Yatoo, D Katterine Bonilla-Aldana, Alfonso J Rodriguez-Morales, et al. Sars-cov-2, sars-cov, and mers-cov: a comparative overview. Infez Med, 28(2):174-184, 2020.
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Xingjian Shi, Zhourong Chen, Hao Wang, Dit- Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional Istm network: A machine learning approach for precipitation nowcasting. Ad- vances neural information processing systems, 28, 2015.
Ashikujaman Syed. Coronavirus: mini-review. Int J Curr Res Med Sci, 6(1):8-10, 2020.
Jinliang Wang, Jingmei Pang, and Xianning Liu. Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model. Journal of biological dynamics, 8(1): 99-116, 2014.
Qinxia Wang, Shanghong Xie, Yuanjia Wang, and Donglin Zeng. Survival-convolttion models for predicting covid-19 cases and assessing effects of mitigation strategies. Frontiers in public health, pp. 325, 2020.
Joseph T Wu, Kathy Leung, and Gabriel M Leung. Nowcasting and forecasting the potential domes- tic and international spread of the 2019-ncov outbreak originating in wuhan, china: modelling study. The Lancet, 395(10225):689-693
2020.
Yuan Yin, Vincent Le Guen, Jérémie Dona, Emmanuel de Bézenac, Ibrahim Ayed, Nicolas Thome, and Patrick Gallinari. Augmenting physical models with deep networks for complex dynamics forecasting, Journal of Statistical Mechanics: Theory and Experiment, 2021(12):120122 2021.
Xu-Sheng Zhang, Emilia Vynnycky, Andre Charlett, Daniela De Angelis, Zhengji Chen, and Wei Liu. Transmission dynamics and control measures of covid-19 outbreak in china: modelling study. Scientific reports, 11(1):1-12, 2021.

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Under review as a conference paper at ICLR 2023
APPENDIX
A.1 PROOF OF THEOREM
the methodology adopted in Lei & Yang (017) which gives that for any initial size of population (s(0), a(0), 6(0), r(0), m()) E R2. the proposed epidemic system (I) admits unique local solution (s(t), att), b(t), m(t) on & [0, Te) (where Te is the explosion time). We investigate that
Proof. To begin,
oo to prove that the solution is global a.s. To do, we assume that No 2 0 is sufficiently large such that stopping time, т for any 2 Ko by
ND < (s(0), b(0), m(0) < Rg. We define the
1 T = inf { 1 [0, Te) minn(((), b(1) (0), m()) - k
or
max a(l), ((), m(() 2xj
(l),
Let us assume that & is the null set and inf o -= xc. Obviously, l. increasing whenever ls increasing unboundedly. We also set lim Tx Too whenever k oc, then т 2 Too- If we can show that Too oc then Te = oo will also hold. To complete the proof, only we need to show that Te = So, we are going to prove it on contrary basis that this statement is false, there exists a pair of constants T 0 and (0, 1), such that
P(T S TT} 2
(5)
for each R 2 Kg. Now define a function H E C2 by R2 - R.. by
H(ss, a, b, r, m) = s - 1 In + a Clearly, H is nen-negative, and thus by the virtue of the Itô formula, we obtain
In a + b 1- Imb r - Inr + m -
In m.
dH 1 ds + (1- 1 da (1
db (1-) 1 dr + 1
1
dm
a
m
1
(6)
+ 2
(1ī +...
+ +12) dt + + (1 - 8) dB, (t) + 7/2 (1 a) d12
+ 7s (1 - b) (1) + 1/4 (1 - r) (1) + 1/5 (1 (1-m) dBs
Since s s+a+btt+mm
then using Eq in Eq. 6 with little simplification and re-arrangement
implies
dH < (Ø + + +0
an + + 11) dt
+ 1
(ni +... dt (s- 73 (b dB)()))))))))) 174 1) (B) - dB, (mm
d12(())
For the shake of simplicity. let us assume that K. + + 2(1h1 + 1/2 + v/s) + e + 2 + + Ç + 40 01 - 02 + 17 + a1 + 02 + + _/2 (vī +..+++++0 +zE). the above inequality looks like
dHl < Kdt
17 (s 1) cB, (t) 122 (a dt)
- 173 (b - 1) dt, - 74 1) (t. 7s (m - 1) (t).
For any t E [0, T and
Ko, the application of integration with limits 0 to T. A t gives
At
TuAt
1. (s- 1)mdB (s-1)) - AhAt (a L)m2dB2
ThAt
dH <
Kdtt
0
Tahe
(raM(-- (r- 1) (1
( Tane
fram (m - 1)nddBs.
Tate
(b- 1)madB3
By taking the expectation we may arrive
&l of (nis(T At), At), a(Tu (t), A rTT. A m)tx AB))
s (()) + kdv
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Under review as a conference paper at ICLR 2023
which implies
ε H(s(Ta A at).aaa))))((()tttt) rT) At), ( At)) I H(s((), a()) b(0), r(0), (() + TK.
Let DR {T 2 THJ for every 2 No. then P(A.) > Note that some compo- nents of 5(TR A T), a(Tk A), b(TH A T), r(Ts A T), m(r T) equal either 1/x or R, and H(s(TT A At), a(T tt, b(T (t), A At), m(x A t)) for all w E Axs thus we have
is not less than +ln 1 or -n=+x--
lo-log (-1g A ( (
H(s, (.s.a,b,c.m)
1+ logx)
So from Eq. and 7 may conclude that
H(((s()) a()) 6(0)
+KC a(T 2 alt T), A AT), A m(To AT))
3 |an(=) logn - logg= b A (- log x
11 log 1+ m. ^ (-llog 1 n²) E (()an()))
The last inequality gives
+ TK 2 e In 1 - log 1o 1),
where 1 Am((w) is the indicator function for A, (w)) Thus for k approaches may derive that
Do ((() KT co, which is contradiction, so T 30 a.s.
A.2 PROOF OF THEOREM 2
Proof. Let L [0, oo) and we suppose that the solution of the model (I exists in C, then for every
E L, we have
(waa+ rizb (wia+vvb++ 3mm
s(t) = exp - (u -(++0) 0)t
du
171
d
2
(0) + + /' exp (ep(eett))tt 1-"
D2b + vimm
r
du + m 1-" dB,(o0) du
2
Next, the solution of the second equation of the proposed model looks like
1/ |pbis
1/
[" dB2(u)) x a(0))
a(t) = exp
(pp -pp+ 1 1p++++00 01)t+
du
+ 112
n
S /2bbs + thymes
(p + 2 01) 1" (pins
1/
du + 1/2 I dB2(m) ds
exp
2
Clearly, the above equation implies that s > 0 and a 2 0. In a similar fashion it can shown that b, and non-negative.
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A.3 PROOF OF THEOREM 3
Proof. To begin, first we integrate the proposed model ①1 which yields
1 ds(y)
Øt L. wias
wb2bs
Pams
(v +))ss dy I/ ms(x))dB1 (v).
+
n
n
da(y)) / v1ss
Vams
(0+ 2 + 01) 01)a dy + [ m2u(y)dBa(u),
+
1
n
n
S. db(y)
1/
V2bs
yyms
(C + + 02) + qpa jay du
(1 p)
+
72a(y)dB2ly)
S dr(y) I/" (us(x) (1-a)pa- ςb Or(y)) dy + ( (ar(y)dBa(y).
S dm(() S'" (122(x) + ana(x) -(m(n)))))))))))))))))))))) dy + 1- (sm(()ddBs())
implies that
stt) - s0)
0.
12268
Vams
- (v + 0) (s) + 1
7 / (B)
+ 1/
+
+
a(t) a(0)
0.
bs
ms
(p+ 0 1 01) -()+ } + { f m2a(y)dB2()).
1
1
+
4/22
+
n
71
b(1) - ((0)
bs
(1 p} 1/1
4/2
+ 1/3
0 (a) + {a)
TO
1
L'/ 12a(y)dB2(y).
+ 7
r()) - r(()) = v (8) + 4) (a) (b) - 1 (r) + S/ mar(y)dBa(),
t
m(t) m0)
(b) + a1 (a) - (m) + /'' (om(y)dBs()))
Adding the first three equations we obtain
s(t) + a(t) b(t) - s() - a()) b(0)
= + - (v+ + 0) (8) - (o 0 01) (C + + 02) (b)
t
1 + t 5 (ms(yyd (v) 12a(y)dB2(1) nzb(y)dB
Using some algebraic manipulation and little re-arrangement arrive
+ p(1 - g)
(s(t)) v +u 2 2 0 (a) (0 + +0) 02) f(t),
(8)
where
1
a(t) + b(t) + s(t) - b(0) + aa) + +0)
f(t)
+ 0
1 J. (ms(s)dBB(g)) + ma(y)dd2(v) + ngb())dBs(v))
Clearly, f(t) tend to 0 if increasing unboundedly
lim f(() = a.s. (+00
Now by Itô formula the second equation of model leads to the following assertion
dn a(t) = {
v2gbs
tyms
1 2/ + 12 (1).
(p + 2 + 0,)
an
an
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Under review as a conference paper at ICLR 2023
Applying integration we get
1 ? Ina(()) - p pf s1 (=:) + 1/2
bs
m.
}
(p+ - + 7 (t).
1
1
am
Since s +a a b+ r + m = n. so the above equation may takes the following form
1
ina(t)) S ( + 2 (1)) (s)) (p+
2 2
Bt)
(9)
Plugging the value of (s) by following Eq.
8 in 9 we reach to the following assertion
p 9)
++0
ina(t)) 1 (1 + + thy)
v+ 0 (a)
(a) - +2)+)22)00 (6)
1 1
v+
+ +5(1) - (p+ + 01) - 2
(t)dB2(t)
t
1
S p(1r + thy)
Nt v + +(t) (0 + 1 01)
2 2 + (t)B
Following strong law of large number Birkel 1988) as well as using Eq (2. the last inequality may takes the form
In att))
(R 1) a.s., { {p+ 0 01 S
1
lim sup 2-000
22
implies that lim a(t) = 0 whenever R < 1. In a similar the third equation of model that
I implies
d() -1-1) (1-0))
1ias n
422
4yms an
qpa
(S + 0 02) 2 + (t)BB(().
n
Taking the integral of both sides, we derive
1 In bnt(t) = (1 - p)
{ as
8
ms8 bm
+ qp ( (7) - - +
+ 2/2
+ tha
bn
1
2 73
+ (t)Ba(t).
Following the similar procedure as above we arrive at In 1n)(()) s (1 - p) (wh + t/ v/a)
+
p(1-4)
1 0+0
(a)
(a) - +p+ (b)
1+ 0
++
++
2 1
+ f(t) - (S 0 +0)0))
-
TaddB2 (t),
1
S (1-p) (1h + 1/2
v/a)
luto + +10) (C + 0 02)
-
Substituting Eq. 2p. and following again the strong law of large number, so the final inequality leads to
In b(t)
{ 02
lim sup to
2 13
2 (R 1) <.
which implies that lim b(t) for f tend 2o whenever R
Moreover, the last equation Thus, (q. gives that lim #4グ Therefore,
of model (1) yields that m(t) = if f conclude that the extinction of the novel disease depend continuously on Ro. and so vanishes whenever the value of R, is less then unity.
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Under review as a conference paper at ICLR 2023
A.4 ALGOITHTM
Algorithm Euler Maruyama Method (EMM)
1: Descretization of [o, T into L equal subintervals of width At = 1 > 0: 0 < ...
TL T.
2: Setting the initial data (so, 40, bo, ro, mo)-
Recursive definition of (sj, aj.bo.... Dj. T.. m,) for 1 S L as
(3)
4: Discretization of the Brownian paths that can be used in finding (:(T,) - BB((T--)) using the
constant R. 2 and the increment St for the Brownian path.
5: Finding of B, (T₃)) (T)))-1) = ((RRt) Bi(0 - 1)R8t) = - R+1 dB
Algorithm Milstein's 's Higher Order Method (MHOM)
1: Descretization of [0, T into L equal subintervals of width At 1 > 0: - <0
TT. with T. nntt.
2: Setting the initial data (so, do bo, ro, mo).
3: Recursive definition of (sj, aj bj. Ty. mj) for 1 1 L as
(4)
4: Discretization of the Brownian paths that can be used in finding B,(T,) - BB((T-1)) using the
constant R 2 1 and the increment 5t for the Brownian path.
5: Finding of (T;) Bi(T,-1) B.(iR()) - B.((i - 1)R8() = Ōk=jR-RA R++ dBy.
16


Reviews:
[{"review_id": "ToqLYKKEwr", "reviewer": "Reviewer_Xoxa", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this work, the authors propose a differential equation approach to model the temporal dynamics of SARS-Cov-2 in a single location. The model incorporates stochasticity in modeling the transitions among five partitions of the population. They also use standard analytic techniques to show that the proposed model is well-posed. Finally, a limited set of experiments suggests the model can model some of the dynamics involved in one small dataset.", "summary_of_the_review": "Overall, I believe the limited methodological contributions and lack of experimental evidence of generalizability will limit the impact of this work. Also, considering its strong emphasis on epidemiological modeling, it may interest a broader audience at a venue catered to that topic.", "strength_and_weaknesses": "The main strength of the paper lies in adding stochasticity to transitions among the population partitions in an existing model.\n\nAnother strength of the paper is the description and intuition given for the various parameters introduced in the equations throughout the text.\n\nAs described in more detail below, though, both the novelty and technical quality of the work is limited. The technical quality would be significantly improved by demonstrating the generalizability of  the model using a data-driven approach. Such an evaluation may also help improve the novelty, in the sense that it could help show the benefit of the stochasticity.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "6vxmCCaYOc", "reviewer": "Reviewer_qnjX", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper presents a new compartment based epidemic modelling for COVID which considers pre-symptomatic and symptomatic phases. The paper improves a prior model by Khan et al, 2021, and show empirically the new model better fits to a data from UAE. ", "summary_of_the_review": "Paper provides an epidemic modelling of COVID which is not ML related. ", "strength_and_weaknesses": "It is a mathematical modelling paper, there is no representation or learning. I don not think that this is within the scope of ICLR, and the audience will have limited interest. As a reviewer I also don't think that I can properly review this paper, given it is from an adjacent field. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "1ZR1Iy6fjOG", "reviewer": "Reviewer_wtTR", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work considers the SARS-CoV-2 transmission and forecast issue. To address this problem, a new model based on stochastic differential equations is proposed. For the proposed model, the necessary conditions for disease eradication are developed and also verified theoretically. To further validate the proposed model, numerical simulations based on real data are also provided.", "summary_of_the_review": "In my opinion, this work may not be suitable to be published in ICLR. I think it can only bring limited insights to both ML and optimization community.", "strength_and_weaknesses": "This work has a valid motivation, and it is well-organized. Also, it is technically sound with both theoretical support and empirical validations. However, this work can be further improved in the following aspects:\n\n- Fig. 1 is supposed to be a vector graph.\n- I have concerns about the impact of this paper. This work studies the method to track and control COVID, which does not use any ML or DL methods. Therefore, it may only bring limited insight or impact to the ML community. On the other hand, though a new compartmental model has been proposed, similar epidemic models have also been developed in recent studies. Meanwhile, this work does not provide any observations or suggestions which are not obvious or counterintuitive, which may also bring limited insights to the policymakers.\n- The form of Table 1 should be revised to meet the requirement of the conference.\n- It would be better to have some discussions after each theorem.\n- In equation 3, commas are missing in the first two equations.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "-Rj697w67W", "reviewer": "Reviewer_mAbY", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Authors present an extension to an already existing model for modelling dynamics of SARS-CoV-2 spread by accounting for pre-symptomatic and symptomatic populations. They present a real-world case study example to support their proposed model.", "summary_of_the_review": "Please look above. ", "strength_and_weaknesses": "Strengths: -\n * Idea of having compartments for pre-symptomatic and symptomatic populations is necessary if you want to model SARS-CoV-2 \n * dataset used is a real-world data of the epidemic \n * Model shows th established properties like having R0 < 1 for controlling the epidemic\n\nWeakness: -\n* The paper is not self-contained it is very cumbersome to actually get the whole idea behind the proposed model\n* Experiments section is written in a way that it is not obvious to know how experiments are conducted and how can you really get that model is well calibrated. For example, it is hard to know what Figures 4-7 are actually showing and how is that any validation of the model. \n* the claim of using pre-symptomatic and symptomatic as two new compartments as novel is incorrect. Almost all established models already have these. For example see [1]. \n* Paper has wrongfully claimed that they are first ones with stochastic model for SARS-CoV-2. Again, many of the models cited in paper are stochastic and the model in [1] is much more powerful than proposed by authors. \n* Fitting to only 3 months of data that too only on cases is not that much useful\n\n[1] Sonabend, R., Whittles, L. K., Imai, N., Perez-Guzman, P. N., Knock, E. S., Rawson, T., ...Cori, A. (2021). Non-pharmaceutical interventions, vaccination, and the SARS-CoV-2 delta variant in England: a mathematical modelling study. Lancet, 398(10313), 1825–1835. doi: 10.1016/S0140-6736(21)02276-5", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject