Transmission Dynamics of Hepatitis B: Analysis and Control
Authors: Tahir Khan
zhiqiang xu
Xin Cao
Venue: Submitted to ICLR 2023 (2023)

Abstract: The infection of hepatitis B attacks the liver and can produce acute and chronic diseases, while it is a major health problem and life-threatening around the globe. The control of this infection is a difficult task due to several reasons such as variation of human behavior, proper medication, vaccination, and existence of a large number of carries, etc., but understanding the dynamics of the infection helps to design appropriate control strategies. Thus, a proper complex dynamical system is needed to find the stability conditions and propose intervention strategies for forecasting the control of hepatitis B virus transmission. We formulate a model that will be helpful to investigate the temporal dynamics and suggest control strategies for hepatitis B infection. The well-posedness of the proposed model will be shown, and used to find the threshold parameter to analyze the model equilibria and its stability. We also perform the sensitive analysis of the threshold quantity to quantify the most sensitive epidemic parameters. Based on the temporal dynamics and sensitivity, we investigate effective methods to minimize the infection of hepatitis B, and develop the algorithms to support the theoretical results with the help of numerical simulations.

Under review as a conference paper at ICLR 2023
TRANSMISSION DYNAMICS OF HEPATITIS B: ANALY- SIS AND CONTROL
Anonymous authors Paper under double-blind review
ABSTRACT
The infection of hepatitis B attacks the liver and can produce acute and chronic diseases, while it is major health problem and life-threatening around the globe. The control of this infection is a difficult task due to several reasons such as vari- ation of human behavior, proper medication, vaceination, and existence of a large number of carries, etc., but understanding the dynamics of the infection helps to design appropriate control strategies. Thus, a proper complex dynamical system is needed to find the stability conditions and propose intervention strategies for fore- casting the control of hepatitis B virus transmission. We formulate a model that will be helpful to investigate the temporal dynamics and suggest control strate- gies for hepatitis B infection. The well-posedness of the proposed model will be shown, and used to find the threshold parameter to analyze the model equilibria and its stability. We also perform the sensitive analysis of the threshold quantity to quantify the most sensitive epidemic parameters. Based on the temporal dynam- ics and sensitivity, we investigate effective methods to minimize the infection of hepatitis B, and develop the algorithms to support the theoretical results with the help of numerical simulations.
INTRODUCTION
Hepatitis means liver inflammation produced by the virus's bacterial infections and continuous expo- sure alcohol or drugs (Ganem & Prince 2004). Hepatitis B infection is caused by non-cytopathic virus called the hepatitis B virus (HBV). The viruses are in vaginal fluids, blood, and semen, and are transferred in multiple ways from one individual to other. Major routes of the transmissions are blood, sharing razors, toothbrushes, and unprotected sexual contacts (Chang) 2007). Another source of HBV transmission maternal transmission (i.e., from an infected mother to her child). However, this virus can not be transmitted from causal contact (McMahon) 2005). The infectious hepatitis B has multiple infection phases: acute and chronic carries. The first phase refers to the initial three months after someone becomes infected with hepatitis B. The immune system can recover in this stage without taking any treatment/hospital care usually, however, may also lead to a long-term in- fection indicating the chronic phase for somcone. The individual with chronic HBV stage often has no acute history of infection. The chronic hepatitis B phase is a severe stage and causes many complications, for example, liver scarring, failure of the liver, and liver cancer (Ringehan et al.) 2017). In the chronic hepatitis B phase, there is a need for treatment with medicines, containing oral antiviral agents. The proper treatment and hospital care will be necessary for the rest of their lives, because this can slow cirrhosis progression and reduce the liver cancer incidence with an im- provement in long-term survival. In 2021, WHO recommended oral treatments for the hepatitis B virus as the most potent drugs. The symptom of hepatitis B includes skin yellowing, abdominal pain, urine darkness, fever, and loss of appetite, etc. But for every individual, this is not common to suffer from hepatitis B because 40% of the acute individuals have no symptoms in the acute stage of the infection. Although the vaccine for hepatitis B is available, new cases still have been reported and the infection of hepatitis B is one of the main public health issues that produce high mortality rate around the globe (Shepard et al., 2006). Worldwide two billion of the population are infected with hepatitis B, among which the number of chronic hepatitis B is 360 million (Libbus & Phillips, 2009). Viral hepatitis B is a leading source of death among other diseases while producing 820000 deaths in 2019, with 1.5 million newly infected.

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Mathematical modeling is useful tool and has been extensively used to test various theories related to the dynamics of infectious diseases. Various mathematicians, researchers, and biologists used mathematical models study the communication of transmittible diseases (Rao & Kumar 2015 Lessler et al. 2016 Ndeffo-Mbah et al. 2018: Li 2018). The temporal dynamics of hepatitis also have rich literature, and variety of epidemic models have been used frequently to know the dy- namics of the discase. For example, a model has been reported to analyze the dynamics of hepatítis in Ghana by Dontwi et al. (2014). The dynamics of the hepatitis B virus with fractional derivatives have been studied by Ullah et al. (2018a (al8alp b). Similarly, many more studies have been reported to dis- cuss the dynamics of hepatitis (Koonprasert et al., 2016 Khan al. 2019). However, the recent models are aimed to analyze the temporal dynamics of the infection using simple disease transmis- sion rate rather than probability-based transmission, which not realistic especially for disease having multiple infection phases. Since the contagious infection of hepatitis B has multiple phases and it would be better to use probability- transmission in the inflow and outflow of individuals from one compartment to another compartment. We try to fill the gap by formulating model using probability- baaeed rate of transmission according to the characteristics of hepatitis B virus transmis- sion. The model will be developed using the idea of classical model and probability- solution using the Cauchy abstract equation and calculate the threshold parameter of the model to analyze the sensitivity of important parameters. The threshold parameter and its sensitivity are dis- cussed with the help of the next-generation matrix method and sensitivity index, respectively. We also investigate the qualitative behaviors of the model to derive the stability conditions using linear stability analysis. Based on sensitivity and stability conditions, we use optimal control theories to formulate effective methods for hepatitis B eradication. Finally, algorithms for the proposed model will be developed to present the numerical simulations and verify the theoretical analysis.
(SIR) transmission of individuals. We then discuss the existence of the model
PRELIMINARIES
In this section, we recall some essential concepts, preliminaries, definitions, and methods for the analysis of proposed model that will be helpful for getting our results.
Let RD and R respectively represent the space of n-tuple and the space of n-tuple with negative entries. The system of non-lineat autonomous differential equation then looks like
du di
((((t), ū(to) To.
where य E R" and the function v implicitly depends only on the dependent variable i and not on the independent variable t, for example, du (a- bu(t)) u(t). Let J be the variational matrix of Eq.(T. then
3us
Fu.
J
(2)
8yl Bug
Theorem
Yassen 2005
Let assume that P(x)) is polynomial of degree such that P(x) = x² + -aja"11 + +...+ Ons
where O1,, ..,..... Ons are constants. The n-Hurwitz matrices for the above polynomial are defined
by
01
0
Hi 01,
H2
01
H.
03
0g
01
0'3
012
04
03
a1
0
0
0
03
Q2
Q1
Hr
0
2

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It should be noted thatfor m n, 0m 0. If the determinants of the Hurwitz matrices positive i.e. dett))................ det(()) > O, the roots of P(x) will be negative or having negative real parts.
2.1 CLASSIC SIR MODEL
In the history of epidemiology of the infectious disease, a classical model for the infectious disease mitigation was presented by Kermack and McKendrick which looks like
dS dt -IS.
dI =
dR dt
=
dt
where the state variables, S, and describe the susceptible, infectious, and recovered/removed population, respectively, while the disease mitigation rate is B and the recovery/removal rate is 7
Definition
Let it is a fixed (equilibrium) point of the dynamical system | and & is real valued C³ CP-function defined in some neighbourhood of i such that
a. (1x)) 0. and ((ū) > 0 if b. (tū s O, and (1) i7
then the function (i)) is Lyapunov, while the fixed point i' is stable asymptotically.
Definition 2 (Samsuzzoha et al. 2013 Ngoteya & Gyekye 2015) The basic reproductive number a very important quantity the study of epidemiological models. Usually, the estimation of epidemic parameters and uncertainties affect this quantity. The sensitivity analysis describes the relation, and its relative impact between the threshold quantity and epidemic parameters is defined 1 ØRa, where yp is any epidemic parameter and Ro is the threshold quantity. as Ro Bi
2.2 PONTRYAGIN'S S MAXIMUM PRINCIPLE
4 = (12 (t), (22(t)....., Mn. (t)) and s = (s1 (t)...... .........,,, 8n (t)) denote the state and control measures, respectively. Then the optimal problem for the dynamical system is
ds
dt = gll, (()), s(0) = 80, 0 OSLST is min ((T, + $ adc...)
then the Lagrangian L and the Hamiltonian H take the form
L(s, 1) st,, H(s, L(s.)) *f(ss
where
[(. = (J1 pn), f2 fn).).,.sss.ppp (s) x= 221;;2.....
If (s", ,x") represents the optimal solution of the above problem, then non-trivial function, denoted
by X, exists and satisfies
ds
0H
0H
dA(t)
0H
ax
3p
dt Is
at (t) and the maximality and transversal conditions
H(u {), = max H(s" n, A), and A(7) =
(3)
mE|o.1]
hold.
3 MODEL FORMULATION
In this section, we present the model framework and its detailed derivations. We follow the clas- sic SIR model, keep in view the transfer mechanism of the disease, and have the total population, represented by (t),, divided into various sub-classes. Particularly, the various epidemiological sub-classes are susceptible, acute hepatitis B, chronic/carries, recovered, and the vaccinated com- partments, indicated by S, A, C, R and V, respectively. We assume for the model constraints as follows:
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The various parameters and groups of population, S, A, C, R, and V are assumed to be non-negative at time t 0.
There are two infectious phases of hepatitis B: acute and chronic, and both cause the spread- ing of infection while the disease transmission co-efficient for acute and chronic are respec- tively denoted by 3a and 3e. Also, if p the probability of those who leads the acute portion after some one infected then (1 p)-th portion will go to the chronic phase of the infection.
Successfully vaccinated individual will lead to the vaccinated compartment while unsuc- cessful will go to the susceptible group population.
The death induced from hepatitis B is only considered in chronically infected group of individuals.
The inflow of maternally infected individuals go the chronic carries.
Srowma
6 8:475
A(E)
(2
((())
S((K)
P1--MBu +AC
Er trh3C
(1
-37
((1)
v(()
RoC
W
Cradomy Posdation
A5
Figure 1:
The graph represents the schematic process of HBV virus transmission of the proposed epidemic problem
Moreover, the schematic process of the model is depicted by Figure and thus the governing equa- tions of the epidemic problem subject to above assumptions is demonstrated by dS(() dt (1 -C(()CC()))))) + (V(t) - ((,,(()) + (.C(t) + 5))
dA(t)) dt
(.C))))))))+++)))))))))))))))))) + m))
(4)
dC(() (BaA(t) + B,C(t)) (1- ( - (17 + m1 + C(t),
dt dR(() (1 A(-)))AA()) r() ((),(), dv(t) (1- As(() - (1 C)V(I),
dt.
dt
with initial population sizes
S(0). V(0) 0
(5) the description of model parameters are as: I' is the birth rate
A(O). C(O). R(0)) 00
In the above epidemic problem while the hepatitis B transmission rates are 8a and 3... Moreover, 17 assumed to be the natural death rate. r1 and r2 respectively show the natural recovery rate, and recovery due to treatment. ?n the death rate from disease while is the vaccination rate of susceptible population. Further, the maternally infection rate (vertical transmission) is denoted by a, and is the proportion of un- successful vaccination rate. The probability of those individuals which have no acute history while directly leading to the chronic stage is denoted by Similarly, q is the probability of those which recover directly in acute stage.
It is worthy to mention that the proposed epidemic model is population dynamical problem, and to show the well- polledossed we describe the following propositions (all the missing proofs are given in appendix).
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Proposition 1
The set R2_ is invariant positively for the system 4
Proposition 2 The solution of the proposed epidemic problem is positive subject to the initial conditions
5
4 STEADY STATES ANALYSIS
We perform a qualitative analysis of the epidemic problem. We find the model equilibria and thresh- old parameter analyze the temporal dynamics of the model. We calculate the basic reproductive number to derive some sufficient conditions for the local and global dynamics of the model using linearization, Hurwitz criteria, and Lyapunov function theory.
4.1 STABILITY ANALYSIS
To perform the dynamical analysis of the epidemic problem, we first find the disease-free equi- librium the system (4). Let D be the disease-free state of the proposed system described by
r Dj - (S By, Ry, Vy), where Sy 7+0 A, C R2 O, and V/- d) (1++500==))
Further, the reproductive parameter, known the threshold parameter and generally denoted by Ro- represents the average of secondary infectious produced by an infective whenever put into suscep- tible individuals. Since, this quantity is the average of secondary infectives, the infection dies out if Ro < 1, and spreads whenever Ro > 1. To calculate this quantity we follow (Perasso) 2018), and let us assume that y = (A, which implies that dy variational matrices around the disease-free state D, and becomes
dt DD, F V, where F and are the
F F=
S (1 - (1- Bas
p
).
V ++
0
(1 (1-
-m
7 m + - pēa
Calculating the spectral radius of FV-1, 1ic,,, oFF-1)) have that oFV-11) Ro and
Bc p){
BapSs
Ro R1 + R2 + R3: R1 (1 + 111 + r2 rEa)
R2
(1+
(6)
BeqripS,
R3 (n ri (1) 11 72 r
We now calculate the endemic state of the model. To make our calculation easier, we substitute Q1 17 + 9, G2 17 + rı. q3 = 111 + 12 IEa, and a1 = 17 + Ç. The disease endemic state of the model is denoted by D. = (S.. .e. C., Re. Ve), where the components are defined as
S. Bege (1 - p) + p (gri3 + 7333a)
9192
Ve {(1 [)T + AS.).
014293 (Ro
< Vep8.ar (V.B.92(1 p) + (VoP433.
A [B.42(1 - p) Tr1p3. + (sp (pl + 424s + l'gage(1 - p)}
C. A. B.S. + p3.S.
R. ()) r2C.).
Q3
It is clear from the second equation of the above system 7 that the endemic state of the proposed epidemic problem continuously depends on the basic reproductive number Rg, therefore, it con- cluded that the existence of the endemic state is subjected to the condition of R₂ > 1. In the case of Ro 1, the proposed epidemic problem has only disease-free state while the endemic state does not exist.
Lemma The existence of the endemic state for the model is subject condition Ro > 1, otherwise it does not exist.
The temporal dynamics of the proposed epidemic problem around the steady-states de- scribed by the following results.
Theorem If the value of the threshold parameter is less than unity. i.e Ro 1, then the dynami- cal system (4) is asymptotically stable around disease-free equilibrium. But if it is greater than unity then the system is stable the endemic equilibrium.
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Table I: Parameters and its associated sensitivity indices along with the relative percentage impact on the threshold quantity (Ro)
Parameters
Indices
%
Increase or Decrease
Impact on Ro
0.2896
10
2.8960 %
Be
0.7103
10
7.1030 %
V
-0.2321
10
2.3210 %
T2
-0.0587
10
0,5870 %
i
-0,9803
10
9.8030 %
a 0.0014
10
0.0140 %
4.2 SENSITIVITY ANALYSIS
To analyze the sensitivity of the model parameters and their relative impact on the threshold param- eter, Ro, we calculate the sensitivity indices of the model parameters. These indices do not allow us to find the relative impact on basic reproductive numbers only but also will quantify the most sensitive parameters to the disease spreading and control, which is very useful for formulating control mechanism. To calculate the indices, let us assume the parametric values are as: I = 0.343, 8a = 0.44, 3. = 0.4570, r2 0.0081, rı 0.0590, / 0.2600, g = 0.59, 171 = 0.08, = 0.01, = 0.02, 5 = 0.03, 0 = 0.5. The various epidemic parameters sensitivity indices are calculated and given Table [ The sensitivity indices having positive signs show a direct relation with the threshold parameter, Ro. and whenever their values increase or decrease, the value of the threshold parameter will also increase or decrease. on the other hand, the parameters with negative sensitivity indices inversely proportional to the threshold parameter, Rg. That is, if their values increase, then the value of the threshold parameter will decrease, while if the value of the epidemic parameters decrease then the value of the threshold parameter will increase. It is clear, that the parameters which have a direct relation are 3a. fe, and a with accumulative sensitive index 1.0013. This implies that increase or decrease in the value of these parameters by 10% will increase or decrease the value of the threshold parameter by 10.013%. Similarly, the parameter with negative sensitive indices are r2, and 1 having -0.2321 00.0587 and -0.9803 sensitive indices, respectively. So, increase in the value of r1, r2 and 9 by 10% would decrease the value of the basic reproductive number by 2.321 %, 0.587% and 9.803%, respectively. We observe that 3e and are the most sensitive parameters hav- ing highest sensitivity indices and significantly affecting the threshold parameter. Keeping in view sensitivity analysis and temporal dynamic of the model, it easy to suggest control measure and mechanism for the disease HBV.
5 OPTIMAL CONTROL STRATEGIES
Various mathematical techniques are used to characterize optimal control analysis for infectious dis- eases (Rohani et al. 2008). Optimal control one the useful mathematical tool optimizing time- varying control of dynamical systems which has been widely used (Lenhart & Workman] 2007). By analyzing set of equations illustrating the dynamics of a disease, optimal control theory can math- ematically characterize the optimal strategy for a given control method and present insight into the underlying dynamics, without the repeated simulation required to optimize simulation models. We formulate control mechanism programme for the eradication of HBV transmission. Based on the dynamics and sensitivity, it is quantified that 3r and are the most sensitive epidemic parameters that significantly affect the threshold parameter well as the disease propagation. To formulate the control problem, we introduce the following two control measures:
(1(() (1) is time dependent control measure physically describing the treatment of hepatitis B infected population.
(t) is the time dependent control measure representing the vaccination of susceptible population.
The clear goal of the control mechanism is to maximize the number of recovered and vaccinated individuals and meanwhile minimize the acute and chronic individuals using the above time

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dependent controls, Ht (t) and (z)(). Thus, the proposed control problem is the modification of system which takes the following form
[/" { ((LA(t)+haCt)
==min min ==
(Nrui(t) + )323((t) dt.
(8)
subject to
dS(t)
I'E aC(()) (V(1)) (B.A(t)) + (,C(t) + + u2) S(t),
dl
dA(() (B,A(t) B.C(t)) S(t)p (n (n+ (t)) A(t),
di
dC(1)
(1 P) (8,A()) 8.C(1)) 8(0) + +(1 (u) + mn + C(o) -(- (9)
dt
d'((t) (1 ((t) + (2(() (A(t) + C(t) (t) nT((t),
(v (1 -)-
(17 ())(
dt
with initial condition
((0) > 0, A(O) 0, C(O) 2 0, R(0) 0. V(O) O.
(10)
where and kg 2 are the weight constants of infected hepatitis B population and measuring the associated cost with treatment and vaccination control measure respectively. The goal of the objective functional (8) is to minimize the population, A(t) and C(t) by taking the cost of control parameters. Here, we need to find the optimal measures denoted by (#] p5). such that
J (pT. (G) min { (((l))2)le E u, = 1,2
(11)
subject to the model (9. Moreover, assuming that 2 is the control set defined as
u = u:={(0m
(1I, is Lebesgue measurable on [0, 1], 0 S (1( $ i= 1
(12)
For these control measures, first we prove their existence. So the control system ( can be re-written as
dw
Kw + M(w),
(13)
di
where
(1 + ut))
0
0
-I'a
C
0
(n + (t))
K
0
47l
- (un + +
+ r2 I(a)
0
u2(
7)
S(L)
-
A(t)
p(s,A(t)+ + (.,((t)
w
C(t)
M(w)
(1 Alt+
R(t))
0
((t)
(1 (1-
Setting
F(w) Kw + M(w).
(14)
The second term the right hand side of Eq. 14) satisfies
M(w) - M(w2)
< m, (t) - S2(t) +m2 Az(t) - A1(t) +ms (1()) C2(t)
+ m4 RR(()) Rt) +mms [V(()) V2(() J.
< m St(()) - -Sx(t) + ||),(() - Aa((t) (t))
+ R2(t) (R,(t) | + vi(1)-V((1) 1}.

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where m max m2, my, m4. ms} is positive constant and does not depend the state variable of the system (9. So, it can be we-written as
F(wi) - F(2)) 1s Q w1 - w2
(15)
where Q= max(m, [|]]]] So, it implies that is uniformly Lipschitz continuous and by definition of controls (/11,)2) H2) and the state variables S()). V(() > 0, C()), R(() 2 0, imply that the
solution of Eq. (3) exists (Birkhoff & Rota 1978). as stated below.
Theorem There exists an optimal solution μ¹¹ (1i 12) to the problem
12 - contained in U.
12 To this end, we define the sociated Lagrangian and Hamiltonian. Let us assume that is the state variable, = (S, .A, C, R. V) and μ (M1, U22 is the control variable. Then the Lagrangian and the Hamiltonian are respectively described by the following assertions
We characterize the optimal solution to the control problem 8
1 L(s, = h1A hgC +
2 (kipf(t) kapg(t), H, L(s, Ag(,,
(16)
where
A = ds,
= (..
(17)
and
91 (s, (1- (V0) (80
+ + (((())(()))))
0s(5,1) (044400000--00000000000001111110 A(t),
02(5, p)) =
(Mi(1) n mn + C())
11) = (1 - ((0m)((0)+++(()))((((((+(++++))0 (()
()( = (1 - +) + (0))(())))+)))+))
6 NUMERICAL COMPUTATION
This section is devoted to numerical investigation of the model to verify the theoretical results with the help of some graphical visualization using Euler and Runge-Kutte Kutta methods. To verify the stabil- ity results of the model we use Euler method, while for the optimal control analysis the Runge- method is used, We will present the model discretization and then discuss the obtained results.
6.1 NUMERICAL SIMULATION
The numerical simulations are performed to verify our analytical findings with the help of some com- putational analysis. To do so, we use the Euler scheme and the fourth-order Runge-Kutte scheme given (due to space limit) in Algorithms and the appendix, respectively. More precisely, the Fuler scheme is used to simulate the proposed model 4 and analyze the stability results. For this, we assume the biological feasible parameters and different non- non-negative initial population sizes for well as the time interval of O to 100 units. In case of discase-free state the value of biological parameters are assumed as: T = 0.943, E = 0.05, Sa = 0.0044, 3. = 0.00035, r2 0.081, rı 0.0590, p 0.02, q 0.059, m 0.012, 17 0.09, <= 0.4, and i = 0.5. We respectively calculate values of the disease-fred state, D, and the thresh- old quantity (Ro) D (4.011, 0,,,0,110000 0, 0, 0, 2.79) and Ro = 0.00157. We execute the proposed model using the Euler algorithm along with the above parametric values and obtain the simulations carried out as reported in Figure 2 This clearly verifies the analytical findings as stated in Theorem
every class of individuals,
0.001,
2 We use the approach of linear stability analysis, that is, perturbing the initial sizes of the compart- mental population from the disease-free state while the solution trajectories go to the disease-free equilibrium irrespective of its initial values. This interpretation states that whenever R, < 1, each solution curve of the susceptible population takes 60 units of time to reach its equilibrium position of 4.011 as shown in Figure 2a Similarly, the dynamics of infected (acute & chronic) and the re- covered population reveal that the solution trajectories reach zero (vanishes) by taking 35, 30, and 70 units of time shown in Figures 2b 2c and 2d respectively. The dynamics of the vaccinated individual are depicted in Figure 2c] which describes that the solution trajectories of the vaccinated population will be non-zero and lead its equilibrium position by taking 70 units of time, as shown
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This ensures the stability of the proposed epidemic problem around disease- state, while the biological interpretation shows that there will be always susceptible and vaccinated individuals and the infected population vanishes if Ro < 1. It could be noted that the elimination of hepatitis B is subjected to the condition of Ro < 1, and therefore it is very necessary to optimize and keep the value of the threshold parameter as low as possible.
in Figure 2e
However, if this is not the case, that is, Ro > 1, we assume another set of parameters values as: I 0.943, & 0.05, 3n = 0.0044, 9e 0.00035, 72 0.081, Ti 0.0590. 0 - 0.02, 0.059, ?/| = 0.012, = 0.09, a = 0.001, Ç = 0.4, and d = 0.5 to perform stability analysis of the endemic state. We calculate the value of the endemic state and the threshold parameter: De = (0.531, 3.967, 1.632,..2222 1.725, 1.513) and Rg = 1.1364. Similar to the previous case, the same initial population sizes are taken and the model simulations are performed. We obtain the results as given in Figure 3 This describes the time dynamics of the proposed model at the endemic state. very much clear that the susceptible population decreases sharply from the very beginning of infection and leads to its associated endemic position of 0.351 as shown in Figure Ba while the dynamics of acute, chronic, and recovered population reveals that they are increasing at the initial time of the infection but decreasing then and attains its endemic positions, 3.967, 1.652, and 1.725 as shown in the Figure Figure and Figure [3], respectively. Similarly, the dynamics of vaccinated individuals are represented by Figure 3e which shows that it will attain the endemic position of 1 1.513 after a unit of time. It is clear from this analysis that the infected (acute & chronic) population persist in the community whenever no proper control measures are implemented. Furthermore, the optimal control problem ( 9 is simulated using Algorithm 2 by considering the set of parameter values corresponds to the endemic state of the proposed model. Moreover, the value of the weight constants are assumed as: h1 = 0.6, h2 = 0.9, hag = 0.44 and h4 = 0.2, while the time interval is taken to be to 20 units. Here, to see the effect of the optimal measures we plotted the compartmental population of the model with and without control as shown in Figure 4 It is very easy to observe that the goal of applying the control measures: to minimize the infected population and maximize the recovered, and vaccinated population. More precisely, the dynamics of infected population with and without optimal control are visualized in Figure Tigure44b|4c We note that the number of acutely infected individuals with optimal control are always decreasing from the beginning and vanishes in the first few units of time, while without control it is increasing and always exists. Similarly, the number of chronically infected population however, increasing without control at the beginning and then decreasing with permanent existence as shown in Figure 4c The number of recovered and vaccinated population is increasing and then reaches to certain level of amount when applying the optimal control measures, but decreasing in case of no control as shown in Figure Add and [4d] We clearly observe the difference between two cases (with optimal control and without optimal control) and believe that the proposed control strategy will be helpful for the elimination of hepatitis B virus elimination.
decreasing with optimal control application and reaches to zero,
7 CONCLUSION
The formulation of a mathematical model representing the complex biological or physical situation involves some amount of simplifications because of the purpose to describe and predict the essen- tial pattern of the process. The current work demonstrate the detailed derivation and analysis of a model for hepatitis B virus transmission with optimal control. We formulated the model and used a variety of mathematical methods including linear stability approach, basic reproductive number, sensitivity analysis, optimal control theory, Hamiltonian and Lagrangian, Pontryagins principles, numerical simulations to analyze the epidemic problem. We discussed the existence of model so- lution and showed that the model is feasible biologically and mathematically. We investigated the conditions for stability and performed the sensitivity analysis of the threshold quantity to figure out the most sensitive epidemic parameters. On the basis of sensitivity and stability conditions an opti- mal control mechanism has been developed in the form of two time dependent control measures, i.e., treatment and vaccination, for the minimization of acute and chronic individuals while maximizing the recovered and vaccinated population. We then conducted the numerical simulations to verify the the theoretical results. It could be concluded that two control measures play an important role to eliminate the infection of hepatitis B virus transmission.

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IK Dontwi, William Obeng-Denteh, L Obiri-Apraku, and EA Andam. Modelling hepatitis in high prevalence district in ghana. British Journal of Mathematics & Computer Science, 4(7):969, 2014.
Don Ganem and Alfred M Prince. Hepatitis b virus infection- natural history and clinical conse- quences. New England Journal of Medicine, (50(11):1118-112) 2004.
Tahir Khan, Zakir Ullah, Nigar Ali, and Gul Zaman. Modeling and control of the hepatitis b virus spreading using an epidemic model. Chaos, Solitons Fractals, 124:1-9, 2019.
Sanoe Koonprasert, Elvin Moore, and Saowanee Banyatlersthaworn. Sensitivity and stability anal- ysis of hepatitis b virus model with non-cytolytic process and logistic hepatocyte growth. Global Journal of Pure and Applied Mathematics, 12(3):2297-2311 2016.
Suzanne Lenhart and John T Workman. Optimal control applied to biological models. Chapman and Hall/CRC, 2007.
Justin Lessler, Andrew S Azman, M Kate Grabowski, Henrik Salje, and Isabel Boodriguez-Barraquee Trends in the mechanistic and dynamic modeling of infectious diseases. Current Epidemiology Reports, 33)22222- 2016.
Michael Y Li. An introduction mathematical modeling of infectious diseases, volume Springer, 2018.
Martha Kay Libbus and Lynelle M Phillips. Public health management of perinatal hepatitis b virus. Public health nursing, 26(4):353-361, 2009.
Brian McMahon. Epidemiology and natural history of hepatitis In Seminars in liver disease, vol- ume 25, pp. 3-8. Published in 2005 by Thieme Medical Publishers, Inc., 333 Seventh Avenue 2005.
Martial L Ndeffo-Mbah, Vivian S Vigliotti, Laura A Skrip, Kate Dolan, and Alison Galvani. Dynamic models of infectious disease transmission in prisons and the general population. Epi- demiologic reviews, 40(1):40-57, 2018.
Frank Nathan Ngoteya and YN Gyekye. Sensitivity analysis of parameters in a competition model. Appl. Comput. Math, 4(5):363-36) 2015.
Antoine Perasso. An introduction to the basic reproduction number in mathematical epidemiology. ESAIM: Proceedings and Surveys, 12:1233138 2018.
P Raja Sekhara Rao and M Naresh Kumar. A dynamic model for infectious diseases: The role of vaccination and treatment. Chaos, Solitons & Fractals, 75:34 49, 2015.
Mare Ringehan, Jane A McKeating, and Ulrike Protzer. Viral hepatitis and liver cancer. Philosoph- ical Transactions of the Royal Society B: Biological Sciences, 372(1732):201602) 2017
Pejman Rohani, Helen Wearing, Daniel A Vasco, Yunxin Huang, et al. Understanding host- multipathogen systems: modeling the interaction between ecology and immunology. Infectious Disease Ecology: Effects of Ecosystems on Disease and of Disease on Ecosystems, pp. 48-70, 2008.
MD Samsuzzoha, Manmohan Singh, and David Lucy. Uncertainty and sensitivity analysis of the basic reproduction number of vaceinated epidemic model of influenza. Applied Mathematical Modelling, 37(3):903-91 2013.
Colin W Shepard, Edgar Simard, Lyn Finelli, Anthony E Fiore, and Beth P Bell. Hepatitis virus infection: epidemiology and vaccination. Epidemiologic reviews,
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10

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Under review as a conference paper at ICLR 2023
Saif Ullah, Muhammad Altaf Khan, and Muhammad Farooq. Modeling and analysis of the fractional hbv model with atangana- atangana-baleam baleanu derivative. The European Physical Journal Plus, 133(8)11-18, 2018a.
Saif Ullah, Muhammad Altaf Khan, and Muhammad Farooq. A new fractional model for the dynam- ics of the hepatitis virus using the caputo-fabrizio derivative. The European Physical Journal Plus, 133, 2018b.
MT Yassen. Controlling chaos and synchronization for new chaotic system using linear feedback control. Chaos, Solitons & Fracials, 26(3):9139920, 2005.
A PROOFS OF PROPOSITIONS AND THEOREMS
A. PROOF OF PROPOSITION
Proof. Since, w = A, B, R. T) and let us assume that
V11 3.A + 3,C + 0, V31 (1 p) (BaA + B,C),
y21 p B,C)) 1/2 1 71
(18)
V33 = 1j + 1n + + - aTg, yo5 = 5.
then the proposed system 4 takes the form
dw dt
Aw + C.
where
V111
0
ollg
0
re
1/1
Y/2
0
V31
grl
9/3
0
C
0
(
r1
(1
②)
12
17
0
0
0
1/55
T
<)
Clearly, C is non-negative, and A is Metzler matrix, which enough to show that the model invariant positively in R
A.2 PROOF OF PROPOSITION
Proof. Let [0, +o0) is the interval solution for system 4). then the solution S(t) for the first equation looks like
S(l) exp - (1) (1++01 0)L 01- ( ((aA(t)+
[/ (I -a((=))( + (V(x)) exp (n (1)+00x 0)x ( (Ba(s) Ba(w)) du dx
+ S(0) exp (nn 8)L- f ( >0
Like wise, the solution A(I) may take the form
A(t) = exp { (-( / pBaS(z)do / p3.c(z) exp (n S (la())(()dy) do
A(0) exp
(n +1)t (3.5(z)do 20.
which describe that S(t) > 0 and A(t) 2 1 (0. +00). a similar way it is easy to perform
that C > 0. R > 0 and V 0.
11

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Under review as a conference paper at ICLR 2023
A.3 PROOF OF THEOREM 2
Proof. Obviously, from the fourth equation of the model 44. it is clear that R(() appears only in the fourth equation. so we can discuss the dynamics of the reduced system. Thus, using the linear stability analysis (2) the variational matrix of system (4) around disease- free state becomes
- S3
- -B P'a
5
JIn,
S
92
p3.S
0
(19)
0
(1 p)BcS1 93
0
94
Let Px()) is the characteristic polynomial of the matrix, JlD,, then
P(z) agæx 01 + azz² + az + ag.
(20)
where
a0 = 1, = + 0 R1) + 92 (1 (1-R2)
02 +)) 9.) (1 R1) + 22 (11 + q1) - (1-R2) + 9293 (1 Ro)
+ +S33aB.p - p);
03 79243 (1 Ro) + 43 (1 + 0) R.) q2 (1 < ++00 0) Ro))
5.73 (1 - p) - 13,90 (1-p) + f.93910 + 8.qq1r1P+ 8.991r1p).
44 92923344 (1 Ro) + Sy3 (1 - - --5551 Sa (1 -) - BuBep
+ Sy8ag00D + Sy 92930 - S......0..
The co-efficient a;'s 's reveals that, whenever, Ro and az, and a4 are positive, then the Routh- Hurwitz criterion implies that the roots the (2. are negative or having negative real parts, and so the model is locally asymptotically stable under the condition of Ro < 1 and 03, 04 0.
In a similar fashion, it can be shown that the model is stable at the endemic equilibrium if the value of the threshold quantity is greater than unity.
A.4 PROOF OF THEOREM
Proof. Since, the state and control functions non-negative while the control set (12) is closed and convex. The system ( is bounded, and ensures the compactness. Also the integrand in the Eq.(8) convex with respect to the control measure #1 (1()) and (t)))) Thus, it proves the conclusion of optimal controls (μi: p5) existence.
A.5 PROOF THEOREM 4
We use Pontryagin's 's Maximum Principle to obtain an optimal solution and thus we have the assertion given below.
Theorem
Let St. A", C², R- and V be optimal states associated with the optimal controls (ui, uz) for 88 12). then (t(t) (adjoint variables), i 1,..., exist, and satisfy
(1) = (t) - (t) - (1- ())a (8.
+,0 + (1) (n -(()))((())A(()) xx()) - h1 (t)(t) - pd2(t) (1 p))a(t)) + +(
- - X(t) - h2 (t)TEE + (At(t) pd2(t) - (1 (1-
- (
+ (ui(t) + mn + - + (r2+
X(0)=A000 X((1) (A))) - (i)) < +
with terminal (transversality) conditions
(T O.
12

---

Under review as a conference paper at ICLR 2023
Moreover, the optimal value of controls are
(A2(1) (00)) +As()) (1000000000)
(1(1) = max
min
h1
(Ai(1) - (Ai(0) (6100)000))0000..11
(2(1) max
min
h2
Proof. The adjoint system (x) (1)))))(1)) ((()) has been derived by the direct use of the Pontryagin Principle, while terminal (transversal) conditions are obtained from the use of the transversal condition. Moreover to derive the optimal values, uf and uz the Hamiltonian has been partially differentiated with respect to the control functions u1 and uo respectively, and then equating öf to zero, and solving. We then use the maximality conditions (3) and obtain the optimal value of the control variables.
du:
B DISCRETIZATION OF THE MODEL AND ALGORIHTMM FOR NUMERICAL SIMULATIONS
We shall present the temporal dynamics of the proposed model by creating algorithms using the Euler and Runge-Kutaa methods. Notice that the proposed epidemic problem contain five popula- tion groups and we will apply the algorithm of Euler and Runge-Kutta method to every group of population order to present the dynamics every compartment. For the shake of simplicity, let us assume some notations
1 = -
+ 2 (BaA, 8,C,8,)
S..
(r1 (r1+1) 1)) As:
Φ = (PaA, + (1 ()) 1m -(1 + Iga) C;
4 (1 A) r2C; - R,,, 4 (1- US; - (1 <)²,
then the algorithm for the model 4 solution with the aid of Euler method as concluded in the Algorithm I. To present the algorithms for the optimal control problem, we use the forward Runge-
Algorithm Euler Method (EM)
1: Input: Endpoints lo- 'max. integer n, parametric values, initial conditions
2: Output: approximation S, A. C, R, (n + 1) values of
3: Parameters and Initial Conditions: Setting the values for I', E. Pa. Bc, rı, p.g. m a, C.
0, and for the initial sizes S(0), A(0). C(O), R(0). ((0))
4: for i = 1, do
5: Recursive Formula:
t, = to + ih, S S h A,, V.),
(S,,
C++1 C. + 1 (S,, Ci) Ri+1 R, h.4 (Ai, Vi+1 V + h&b (S, V.).
6: end for
7: Output (t,, S1+1, A141, C++1, Ri41, V,-1)
13

---

Under review as a conference paper at ICLR 2023
Kutta method and assume the following notations:
of = = - (1- + (v; - Ai + BcC, + 0 + +1) S..
414 (
frr + n As,
of {S,A, +
p) + oriA, {1 + 111 + reafCs Ci,
8f (1 A)r +120. R
6 = (1 [ + AS:- - {1 + C)V,
hb4
1099
3 = T - a C +
Ba
A: +
+ Be Ci +
+ + +
2
2
2
h
hoy
S
+
+ 5
V: +
2
2
h
Ci + h04
h
Ai + 1Bg
8a
A +
+ Be
S
P (r1 + 11)
2
2
hd1
(c+ (ectt
ho9
h
hD1
8u
A
+ Be
S
p) - qri
A
2
2
2
2
(n + m + - -TEa} {
+ C3
2
A + hD1
h09
hBR
0. (1 r)rl
+ +12
C, +
R4
+
2
2
2
h
{
h V
66 (1 I + 1
S +
(1 + ()
1, +
2
2 S
h
hop
hBg
1
C
Be A +
+ 3c
C++ +
+ + +
2
2
2
h
h
8 +
+ 5
V. +
2
2
( c++
h
h
h
45
Ba
A +
+ B
S +
p - (r1 + 11)
A +
2
2
2
2
h
h
h
h
05
Bn
A +
+ Be
c++
S
(1 - p) - -r1
Ai +
2
2
2
2
{
1
(1) mn r2 rEa)
C,
+
A + h
h4c
h$
03 (1 ()ll
+ 12
C, +
Ri +
2
2
2
S + 34y
hΦy /
D/- (1 8) r +
(1+ 5)
11 +
2
2
h4g
h4g
11g
3 = T
1
C
+
B.
A: +
+ Be
C+
+ +0+n
2
2
2
hDy
8 +
+ <
V
2
2
h
c++ (c+
hbg
P (r1 + 11) { Ait + hTg
41
8a
A +
+ Be
S +
2
2
2
A + 1
(1 p) - 1ri { AA +
1D4
+ Bo
C. +
S
+
2
2
2
2
h
(17 + m71 + T2
Tεa)
C,
+
2
h
h$c
h R
o. (1 g)rl
A
+ +
+++
3
R4
+
2
2
h
8 (1 [ + 1
S
(1
2
2

---

Under review as a conference paper at ICLR 2023
Similarly by backward Runge- Kuunge-Kuttann
Kutta method, we have
@ PA1 = ={215 - pd22 (1 ()a + B,C;) + \1 [62) - M22 A631
PA21 - -1h1 + {(1s - pd2z - (1 - (1- Su + A2i fn+
- -13/or1 -4j(1-9))r - Aastiis
PxI = -12 + A1sI'Ea + {A1s - pd2j - (1 - p)>a}} S.S
+ A33 {uis + + 7/1 + [Ea} + Mas fr2 + mii))
Maaj
PAG
= [15s- - -X1i} < + 11551
11p111 } -pp xx2- 2 {
h22211
- - - p) -11-
h
2x12
A11
2 S -p
Aas
2
BBa (A} + A]-1) + BBcCC + +c--1)3 + { { X13
h
2
2
{
1
(U2z + (12 (1) {
1
124 Aa1
17 +
(12s + (a(--1)
2
2
2
h
(
h
1
+ {
4 A222
A1
A22
- (1 - p)
A13
Ba
2
(S}
2
2
1
h
+
A22
17 + 11
+
(255 + -i(G-1)
Xaj
gri
2
2
hrP
{
hDA
Nas
(1
q)rl
(His
+ +Mirs-n)
h1
2
2
2
= hh2 + {
h1P211
hoP
4 Ass
A115
TEa +
AA1j
2
2
ho
Be (S] + +Sp11) + Xaj
hrtba
1
{ 12 + 2 (mis mio-n))
(1
p)
Ausj
2
2
h
1
+
X31
(uīs
-))--)) + 1 + 1 + -Tça
2
h
{(As) [A5)
h
1
S S+ { {33
1D Aol
17
Aaj
A1j
2
2
2
2
10
hoP
1113
A1j
1x2)
(1
p)
2
2
2
BBa (A3 + 1)-1) + BBlCī + +cr-1)
\15
hrD2a1 X12
1
{ 17
5
1
(1a5 + m
2
A
(155
( 1)) Xas
2
11
x22
- pp1 ( (Aas
hoDxa
1 (S) + s
S A23
A15
A2j
Ba
2
2
1
1D Asa
+
Aaj
17 + 1i +
(ui, + mi(
X33
gri
2
hrPaa
{
{
1
Nas
9)rl
(Hi;
+
h1
2
2
h2 + { '115
h
1022
4xga
rea
Ad1j
p
2
2
- (1 p) (2) (20)
Be
h Au2
(Sj + +S;-1
A1j
2
{ r2 + _ (1 + M 1))
1
h Asz
Aa3
(uis
wi(--1)
+
+ 1 + -TEa
2
2
$ 24 17 { Aas 2
{f
h A02
Asj
(115
hdPx12
)
} c
17 As3
h >
2
2
2
15

---

Under review as a conference paper at ICLR 2023
{N1)
h4P
h Ana
h Asa
2214 =
A1
- (221
- - (11 -pppp (335
2
2
2
{
+ 3oC
+
A115
h x13
1111 M2(5-1) 11}
{11
- 1225 1)
A5j
h Asa
2
hMPAIa
h A2a
1 Asa
2220
A15
A2j
- - p)
Sa5
2
2
h A22
h4P Ama
+
A25
+ r1 + 2i(s-1)
A35
q7
2
ho
Xaa
Pe
1
Aea
Aas
(1
9)r1
Aa5
Mi(j-1) - h1,
2
2
h&
hPAAI
h&P
h Aaa
h2 +
A1j
rea
+
A1j
A2j
2
2
2
(1 - p)
As3
| + Aaj
1
( 12 + 1((G-1))
2
2
+
Azj
h
Asn
{ { (2ic-1)
+ + 17 + - rea],
2
11
{Ass) 11
Asa
1
)
Pr =
A15
{²
=
A115
2
2
2
1
Asa
+
17
Asy
2
then the algorithm of optimal control problem concluded the table as given below.
100
105
58
Suscapatble Poputation (Sm1)
500-30
A03-4
... 0%-
500m0
Acurte Poputation (A07))
A03=53
Chronioo Pos (Cr0)
... -Ma4
500.00
500-00
A/3-70
500-00
520-100
60
19
60
100
20
40
100
50
500
Time 00
Time
Time ()
(a)
(b)
(c)
80
Recovered Population (R(0)
A10
Voceinated Populaition (Vre))
V10
A00)-30
V00
80
A00
A10)-30
60
ViDk-30
A10
A10)-50
50
V00::60
V00+-00
up
30
20
%
(d)
(e)
Figure 2: The graphs visualize the solution trajectories for S(t), A(t), C(t), R(t), and V(t) of the proposed epidemic problem around disease free state.
16

---

Under review as a conference paper at ICLR 2023
Algorithm 2
1: Input: Endpoints to, tmax. integer parametric values, initial conditions
2: Output: approximation S, A. C. R. (n + 1) values of
3: Parameters and Initial Conditions: Setting the values for model parameters, weights con- stants, and for the initial sizes of population S(0), A(0), C(O), R(0), V(0).
4: for i = ,,,,,,, do
5: Recursive Formulas for both control system and without control system:
f4 to ih,
h
S S + (bs -2222 + 50 + +sa) Ait1 A; + A + A2 + 24 + A4),,
6
C++1 C, +
(Pc1 + 24c2 + 24c3 + Pc1) R1+1 = R4 + 6
(2R1 + 2 + 24 + +RR))
h
Vi+1 Vi + 6
(vvl + 20 + 20 + Pv4),
6: end for
7: for 1,... do
8: Recursive Formulas for adjoint system:
h (Axi + 24 + 222xi + 2xxa) A2(j-1) = A222
h
A1(j-1) =X1_-
@1)m +22 + 24 + (roa),
6
h
h
=λaj A3
(Pxa
+4 +24xas + ; Aaj-
( + +24x,s + 222xs + Pru))
6
h
A5(--1) =AsØ
(Pxs
2
rea 24 xxca)
6
9: end for
10: 1,2...., n, = n + 2 1 do
Optimal Control Variable (f1)
if mi() <0 then mi(j) = 0 else, if << m7(1 <0 then mil)) = pi(i) else mi()
12: end for
for = 1,2,...., n = n + 2 do
14: Optimal Control Variable (112)
if (56) < 0 then p2(j) = 0 else if 0< m2(1) < then p2(s) mi clse pe(j) = 1
15: end for
16: Output: (t,, S141, A4t1s Ci+1, Ri41, Vo+1).
17

---

Under review as a conference paper at ICLR 2023
100
135
500-0
A0--0
538.00
Acute Popurartio (An1)
A0
Chremic Populattion (Crt)
153
03:44
500-70
A03-80
500-00
400-70
533.00
100
403-50
510-10
A00-50
10
(a)
(c)
Recovered Population (Rtt)
Vancinated Pop (V!!1)
60
V10)-20
A10
V10)=30
A000-00
80
V10)-00
A101-50
V/Dl=50
A00
40
V/Diced
20
20
40
60
80
100
Time (d)
Time
(c)
Figure 3: The graphs visualize the solution trajectories for S(t), A(t), C(t), R(t), and V(t) of the proposed epidemic problem around the disease endemic state.
Ohvonicatt infee Poppotation (C20)
Suxcapiteee Propuaation (Sr1))) 40
4p
-it control WWooo contO
WiTh CONTS! WETENT cOODOl
centrll
wffess castro
Popurtation
15
(a)
(c)
Recovered Population (P(())
with
Vanctnated population (V(1))
control
75x
10
15
Time (d)
Time
(c)
Figure 4: The graphs represent the dynamics of the compartmental population with and without control applications.
18


Reviews:
[{"review_id": "zrC-lb1NpP", "reviewer": "Reviewer_aznX", "summary": "", "questions": "", "limitations": "", "rating": 1, "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": "This work proposes a new dynamical system for modeling the transmission of Hepatitis B virus (HBV). Based on the classicial SIR (susceptible-infected-recovered) framework, the paper develops a new probability-based HBV transmission model, investigates the sensitivity of the model parameters, discuss the method to eradicate HBV using optimal control theories.", "summary_of_the_review": "Overall, this paper is proposes a purely mathematical model of HBV transmission without a real-world application, a topic outside the usual scope of ICLR. The paper should be submitted to a more relevant venue such as BMC Public Health or JMIR Health and Surveillance.", "strength_and_weaknesses": "Strengths:\n- The paper derives a complex HBV transmission model using well-developed mathematical tools.\n- The paper provide a deep analysis of the proposed model and shows its stability and the existence of a solution\n\nWeaknesses:\n- All the modeling and analysis, solution derivation is done for only one disease, HBV, which is far from the topic of interest for ICLR audience.\n- The entire analysis of the proposed model is purely mathematical, and the paper does not show the model generalizes well to the real-world data.\n- The paper is not self-contained, as the main analysis results are in the appendix.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "hM_Xpx11pc", "reviewer": "Reviewer_znhF", "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 provides a mathematical compartment based model for Hepatitis B which uses a probability based transmission to be more realistic and consider different disease phases.  It studies the steady state and used optimal control theories to find a control mechanism for the eradication of HBV transmissions, without needing simulations. ", "summary_of_the_review": "Not within the scope of ICLR.", "strength_and_weaknesses": "This is not a representation learning or ML paper. There will be limited interest in the paper for the ICLR audience. This work should be submitted to and be peer-reviewed in a mathematics journal/conference or a more related venue. ", "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": "OVqwXf_wZB", "reviewer": "Reviewer_EeVw", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "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 formulates a mathematical model governing the dynamics of hepatitis B infection. The authors have obtained the threshold parameter to analyze the model equilibria and its stability. They also analyzed a model for hepatitis B virus transmission with optimal control. Some developed algorithms and numerical simulations were carried out  to validate the theoretical results.", "summary_of_the_review": "Overall, I think the paper is not appropriate enough of ICLR, and there are some incorrect or unsupported claims.", "strength_and_weaknesses": "**Weaknesses**:\n\n1) First off, I wonder whether this paper would be suitable enough for ICLR? To me, it seems to be more appropriate for a journal in mathematical biology or ...\n\n2) On page 2, Theorem 1:\n- $H_3$ is not correct (its last row is incorrect)\n- $H_n$ should be written in a more general form, please see the following references:\n\n[1] Süli, E., Numerical solution of ordinary differential equations. Lecture note, University of Oxford, (2013).\n\n[2] F. R. Gantmacher, L. B. Joel, Applications of the Theory of Matrices, Courier Corporation, (2005).\n\n3) The graph presented in Figure 1 is not well explained, and so the modeling of the system (4)-(5) is unclear. \n\n4) In Section 4, the authors have referred to Lyapunov function theory to investigate the global dynamics/stability of the model. However, if I understood correctly, this method is not explicitly employed in this paper (and the stability analysis has been performed using\nlinearization and Hurwitz criteria). Hence, the authors should make explicit the use of Lyapunov function theory in their work.\n\n5) On page5, Theorem 2 is **incorrect**. The mentioned condition on $R_0$ is not sufficient for asymptotically stability. The proof of Theorem is not correct either. If  $R_0 < 1$  and $a_3,a_4 >0$ , then all the determinants of the Hurwitz matrices cannot necessarily be positive. For instance, under the mentioned conditions, $H_2$ does not always have a positive determinant.\n\n", "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": "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": "OAYk0E3LGv", "reviewer": "Reviewer_FwjM", "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 transmission and control problem of hepatitis B. To address this issue, a mathematical dynamical system is constructed. Based on the proposed model, the sensitivity analyses for epidemic parameters are implemented. Also, some control methods were studied and analyzed.", "summary_of_the_review": "In summary, I think this work is not ready to be published yet. I have concerns for both motivation and novelty of the proposed model and algorithms.  Also, the organization of this paper is messy, and there exists some format issues. ", "strength_and_weaknesses": "This paper is technically sound. To show the effectiveness of the proposed model and algorithm, both theoretical support and empirical evaluations are provided. However, this work can be further improved in the following aspects:\n\n• According to the authors, the motivation of this paper is to study the strategies to forecast and control hepatitis B virus transmission. However, based on the data from CDC and WHO, Hepatitis B is not a major threat to us now. Compared to COVID, its transmission rate is not that high, and it is a vaccine-preventable disease. I am not sure if the study for control hepatitis B virus transmission at this point is essential or necessary.\n\n[1] https://www.cdc.gov/hepatitis/statistics/2020surveillance/hepatitis-b.htm\n[2] https://www.who.int/health-topics/hepatitis#tab=tab_1\n\n• I have concerns for the novelty of the proposed model and algorithm. Compartmental epidemics models have been well developed. Besides classical SIS or SIR model, there already be some works which divided infectious stages to two separate stages, also the stability of such models have been studied [3] [4]. Compare to these work, I didn’t see much novelty or improvement form this paper. In my opinion, it may be able to bring limited insights to the optimization and epidemics community.\n\n[3] https://arxiv.org/pdf/2010.12923.pdf\n[4] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3590621\n\n• Figure 1 is not vector graph, it should be updated.\n• The writing of this work can be further polished. There exists a lot of grammar issues.\n• This paper is not well-organized. Some of the sections are not consistent with the other part. For example, in Section 2.2, it is not clear why this principle is needed in the whole framework. Also, this section is not stated clearly, where the dynamical from and why there will an optimal problem were not clear. Besides, there exist a lot of long paragraphs which are not easy to follow. For instance, the two paragraphs in introduction, and the two paragraphs in numerical simulation. \n• In the numerical simulations, how these parameter values are chosen is not clear.\n• The format of Table 1 should be updated to meet the requirement of the conference.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and 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": {}}]

Author rebuttal:

Decision: Reject