Monkeypox with Cross Infection Hypothesis via Epidemiological Mode
Authors: Tahir Khan
zhiqiang xu
Venue: Submitted to ICLR 2023 (2023)

Abstract: A new re-emerging infectious disease of monkeypox 2022 is structurally related to smallpox that is induced by the monkeypox viruses and has caused 59,606 active cases with 18 deaths up to September 15, 2022. To end this ongoing epidemic, there is a need for population-wide control policies like reducing social interaction by keeping social distance, treatment of infected individuals, and restriction on animals, etc. We forecast the progression of the epidemic and come up with an efficient control mechanism by formulating a mathematical model. The biological feasibility and dynamical behavior of the proposed model are then investigated together with sensitivity analysis to obtain the effect of various epidemic parameters mitigating the spread of the disease. Subsequently, by taking non-pharmaceutical and pharmaceutical intervention strategies as control measures, an optimal control theory is applied to mitigate the fatality of the disease to minimize the infectious population and reduce the cost of controls, we construct an objective functional and solve it by using Pontryagin’s maximum principle. Finally, extensive numerical simulations are performed to show the impact of the application of intervention mechanisms in controlling the transmission of the monkeypox epidemic.

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MONKEYPOX 2022 WITH CROSS INFECTION HYPOTH- ESIS VIA EPIDEMIOLOGICAL MODEL
Anonymous authors Paper under double-blind review
ABSTRACT
A new re-emerging infectious disease of monkeypox 2022 is structurally related smallpox that is induced by the monkeypox viruses and has caused 59 99,606 active cases with 18 deaths up to September 15, 2022. To end this ongoing epidemic. there is need for population-wice control policies like reducing social interac- tion by keeping social distance, treatment of infected individuals, and restriction animals, etc. We forecast the progression of the epidemic and come up with an efficient control mechanism by formulating a mathematical model. The biological feasibility and dynamical behavior of the proposed model are then investigated to- gether with sensitivity analysis to obtain the effect of various epidemic parameters mitigating the spread of the disease. Subsequently, by taking non-phammaceutiall and pharmaceutical intervention strategies as control measures, an optimal control theory is applied to mitigate the fatality of the disease to minimize the infectious population and reduce the cost of controls, we construct an objective functional and solve it by using Pontryagin's 's maximum principle. Finally, extensive numeri- cal simulations are performed to show the impact of the application of intervention mechanisms in controlling the transmission of the monkeypox epidemic.
INTRODUCTION
The infection of monkeypox is a contagious disease resulting from the orthopoxvirus. This infection is zoonotic and was initially transported to humans by wild rodents in central and western Africa. But human-to-humant spread (horizontal transmission) also possible, particularly within the family home or in the context of care (Farahat et al., (022). The monkeypox viruses can be diffused by immediate contact with lesions on the skin or mucous membranes of a sick person, as well as by droplets (sneezing, saliva, sputters, etc.) (Singh et al. (02T). Generally, an individual can become infected through contact with patient's environment. is, therefore, important that patients respect isolation measures throughout the illness. Humans can also become infected through active contact with animals (rodents and monkeys) (Oladoye 202T). Usually, the monkeypox infection starts from fever, headaches, body aches, weakness, etc. (Deresinski 2022). The symptoms may lead to the appearance of blistering rash consisting of fluid-filled blisters that progress to dryness and crusting, then scarring and itching after two days. The bubbles most concentrated on the face, the forehands, and the feet soles. The disease more severe in children as well as those who have weak immune systems. Historically, monkeypox was identified first in the 1970s, but recently the re- emerging of the disease, cases are reported in various countries around the globe (ASSESSMENT) 2022). Usually monkeypox virus transmits from human interaction, but there of cross-infection (animal-to-he -o-uman) spread (Petersen et al., 2019). Therefore, the hypothesis of cross-infection between human and animals play a significant role and can not be neglected.
significant risk
Modeling and forecasting with the aid of dynamical system is a challenging domain in various dis- cipline, e.g., infectious disease epidemiology (Brauer) 2017 Saravanakumar et al., 2020 Guo al., 2020), health sciences (Choi et al. 2016), and various other fields of applied science and technology (Rolnick et al. 2022), and therefore attracted the considerable attention of researchers, see for in- stance, (Das et al. 2020b Yin et al. 202 Saha et al. 202I). Similarly, various models demonstrate different outlooks regarding the dynamical behavior of an epidemic (Busenberg & Cooke, 2012, Khajanchi et al. 2018 Das et al.) (220a). With the aim of these mathematical models, researchers want to understand the dynamics of a disease and then suggest control strategies to control or com- pletely eradicate the infection (Chen & Guo! 2016 Kumar et al. 2019). Besides the rich literature on

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infectious disease epidemiology, there have been enough studies found represent the temporal dynamics of monkeypox 2022, to the best of our knowledge. We try to formulate a model which describes the transmission of monkeypox 2022 to understand the dynamics and suggest control mechanism with the aid of optimal control theory. We summarize our contributions in this work as follows:
The cross infection between humans and animals plays sigmificant role in the dynamics of monkeypox virus transmission. We, therefore, propose model based the hypothesis of cross-infection between humans and animals. The model has two blocks: humans and animals.
The first block describes the evolution of monkeypox in the human population, while the second block represents the evolution the monkeypox virus among animals.
Four time-dependent control measures are then introduced in the model to demonstrate the utilization of optimal control measures: to minimize the infectious individuals and maximize the recovered population. Particularly, reducing the risk of disease transmission by educating people to rise awareness of risk factors, treatment of infected individuals, and restrictions on animals.
+ 8,MMa
Sancitive Papusatioo
Infected Popuistion
Recovered Population
(BM, {M,,)
yM
ORg
Sx
Ra
Cross-Infection
8Sp
4
(x + uz kg)
S
M
Infocted
Populetion of Aniimals
Amiratle
Figure 1: The plot represents the schematic process of the proposed monkeypox virus transmission
model
RELATED WORK
The analysis of infectious diseases with the aid of dynamical systems is a fascinating outlook to predict the dynamics of an epidemic. In the history of infectious disease epidemiology, Ker- mack and McKendrick were the pioneers to develop the three-population-gronn group epidemiological model busseepible-iifecttou---rooovvv
(Kermack & McKendrick, 1927). where various popu- lation groups are employed to signify the infection, demonstrating their progression and inter- actions. The classical suceeppilleinffectios--eeee endrick has been modified by incorporating an exposed compartment known as the susceptible- sypoeeddinfecttius-reeeeee
model formulated by Kermack and McK-
model (Anderson & May 1979), which is also extensively used to delineate the transmission of distinct diseases. Data-driven modeling methods have been also used wor-
investigate the transmission dynamics of infectious diseases (Heesterbeek al. 2015). thy to mention that the idea of the classical susceptible- besceptibleinfecctiuss--os infectious-rementt removed model has been further investigated by various researchers observe the transmission dynamics of distinct epidemics (see for instance Flaxmman et al. (2020); (Samii et al. (2020); Britton et al. (2020)) Optimal control the- ory has been extensively used and is very common in infectious disease epidemiology (Rohith & Devika 2020 Khajanchi al.) 2021). The adjustment of epidemic parameters in a feasible way, by taking the limits on the system to optimize a given function, can be applied with the help of control theory. Both non- paarmaacccuttia and pharmaceutical control measures infection. Especially, the non- non-pham pharmaceutical intervention strategies play a key role. Usually, with the help of optimal control analysis, are able know how to eradicate the disease.
be used to control the
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A re-emerging infectious disease of monkeypox 2022, was reported in May 2022. Here, interested to formulate a model by taking an extended cesceptibleinfeette infeccee-rrcooveeedee recovered-type model with two compartmental blocks: humans and animals. We then discuss the qualitative analysis of the proposed two-strained model. Further, applying the theory of optimal control to understand the progression of the monkeypox virus transmission. Since it is not merely medical problem, regarding a public health concern, both the combination of non-pharmaccutical and pharmaceutical intervention can be taken into account to propose control mechanism for the control of monkeypox virus transmission.
3
THE MODEL
We propose an epidemiological model for the dynamics of monkeypox based on the cross infection hypothesis: animal to human and human to human. The various compartmental population of the model divided into two blocks; human and animal. The first block represents the evolution of the human population, consequently distributed into three cpidemiological groups: sensitive individuals, infected by monkeypox, and recovered individuals, while the second block represents the evolution of animals, divided into two classes: susceptible animals and infected animals. To symbolize the population groups, let us assume that (t) represents the sensitive individuals at time t, which are not infected but have a chance to be infected at time t + At (At is the small increment in time). Mr denotes the individual infected with monkeypox, and R,, is the recovered individuals. Similarly, the susceptible and infected animals are denoted by (t) and Ia (t), respectively. Due to the assumption of a homogeneously mixed population for the successful transmission of the monkeypox virus, the risky humans will enter the infected human compartment at rate B, as well as, the susceptible animal getting infected will move to the infected animal at rate Φb. The individual leaves the infected human class only after they fully recover or die. The recovered individuals will enter R3. Moreover, the complete geometry of the epidemic problem is described by Figure | and thus, the evolution of the disease is represented by the following deterministic system of differential equations:
(1) Tb. (t)..) - (M...)) (.), dt (l,. (3M,()) + (Ma(1).a() (0 + 01 + M()), d.a = rMa(() (R.()),
dt
dt
dS.(1) = - (...MM(())S.(() - as) dMa(1) (Ma(() (a (a+ 01) Ma(t),
dt
dt
with biologically feasible non-negative initial population sizes
S. (0) > 0, (0) 2 0, (0) 2 0, S,(0)>0, 0, Ma 2 0.
In the above epidemic problem (I-2)) the parameters are described as: the newborn of human and animal are denoted by 4, and o respectively, while the monkeypox virus transmission rates are and E. The natural death rate of a human is assumed be 8, while the same ratio for the animal population is denoted by a. The monkeypox virus transmission rate from one animal to another is assumed to be Φ) and ai is the death rate that arises from the infection of monkeypox virus in the animal population. Moreover, the disease-induced death rate of a human is represented by 01, and is the recovery rate of an infected human.
To proceed, first, we show the mathematical, as well as, the biological feasibility of the proposed epidemic problem. To this end, we show the following result.
Proposition 1
The solution of the model 7-2 is positive and bounded.
3.1 DYNAMICAL ANALYSIS
In this section, we discuss the temporal dynamics of the model find the stability conditions for the monkeybox epidemic model. We find the monkeypox-free equilibrium state for the developed model 11 and calculate the reproductive number. Let Wo is the monkeypox-free equilibrium of the model, then, Wo
( ( 2 0). Moreover, the reproductive parameter, denoted by R...
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Parameters
Indices
% Increase Decrease
Impact on Ro
3
0.3677
3.677
01
-0.0288
10
0.288 %
r
-0.3316
10
3,316
Φ
0.6322
10
6.322 %
01
-0.5099
10
5.099 %
-0.7546
10
7.546 %
Table 1: Indices of the proposed epidemic problem parameters related with the basic reproductive number, and its relative impact.
represents the average of the secondary infectious produced by an infective whenever put into sen- sitive//usceptibbe individual. To calculate this quantity for the model reported in Eq. by following Van den Driessche & Watmough, 2002), let us assume that X = (Ma, Ma) " then
dXX F-VV
dt
where, F and V are the by 2 variational matrices at the monkeypox-free equilibrium defined as
9
+ 01 +r
0
c.
V
0
The reproductive number is the spectral radius of the matrix FV -1 and takes the following form
3
Rs R4 Ra:
RA
Rg a (a + a1)
9 (
Since the reproductive number is the expected average number of secondary infections. It is con- cluded that whenever R, < the disease will die out, otherwise spread if R. 1.
3.2 BIOLOGICAL INTERPRETATION OF THE BASIC REPRODUCTIVE NUMBER
Since the initial spread of any epidemic is related to the reproductive number, we analyze the nor- malized sensitivity of the proposed system parameters. The sensitivity of the threshold quantity will enable us recognize the most sensitive and effective parameters for discase transmission, because a small perturbation in the most sensitive parameter can produce a great influence on the associated epidemic model. To present the prediction for the prevalence of the monkeypox disease, reduction, and persistence the transmission of infection, we perform sensitivity analysis of the basic repro- ductive number R. Let assume that any epidemic parameter, then the normalized forward sensitivity co-efficient (index) related to the basic reproductive number R, is defined by:
Y 0R. θ7
7 R.
clear from the formula normalized sensitivity coefficient that may be dependent or inde- pendent of the model parameters. We calculate the associated sensitivity indices accordingly listed in Table [ It can be observed that some of the indices of the model parameters are negative and some are positive. The negative and positive signs demonstrate that the perturbation to these param- eters can produce decrease or increase in the value of the basic reproductive number, respectively. For example, the forward sensitivity index of the parameter /3 is r 0.3677, which indicates that if we increase the value of 33 by 10%, as result, the value of the basic reproductive number R, would increase by 3.677%. Similarly, the forward sensitivity indices of 91 and are 0.0288 and 0.6322, respectively, which implies that if we perturb the value of 1) and Φ by 10% it would result increase or decrease the value of the basic reproductive number by 0.288% and 6.322%, respectively. On the other hand, the sensitivity indices of r,, and are negatively associated with the basic reproductive number, i.e., increasing their values would decrease the value of R. we increase the value of a, and by 10% it casts the decrease of 15.961 % in the value of R,, In this analysis, we observed that the most effective parameters of the proposed epidemic problem are 3, r, &, a1, and a, therefore, special attention is required for the parameters with highly sensitive indices to forecast the transmission of monkeypox disease. We now state the dynamics of the model by proving the following results.
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Theorem If R. < 1, then the dynamical system 11 is locally and globally asymptotically stable around the monkeypox-free equilibrium state of the model and unstable if is greater than unity.
4 OPTIMAL CONTROL
The application of optimal control theory one of the important theoretical analyses associated with infectious diseases. We use this tool to produce the proper control mechanism for eliminating infection of monkeypox virus transmission. Our analysis is not limited to the theoretical analy- sis, and also perform some numerical experiments to show the effect of the proposed control strategies on the dynamics of monkeypox virus transmission. The key goal is to reduce the infected humans and animals while maximizing the recovered human using the optimal control theory. The parameters with certain assumptions lead to the monkeypox model as described by [. which is coupled system with five state variables (t), (t).). , R2 (t), (t()), (t)). We introduce four control measures (t) (i = 1.2,... .,....,,))))))). 4) that control the number of risky and infected individuals externally over given time frame.
4.1 REDUCING THE RISK OF HUMAN TO HUMAN AND ZOONOTIC TRANSMISSION
The main prevention measure for monkeypox is educating people to rise awareness of risk factors about the control measures they can take for the reduction of the virus transmission. Due suffi- cient information, the population needs to maintain social distance, wear masks, follow the strategy of home isolation, etc. Thus, we introduce the control factors to control the interaction between susceptible/risky human (t) with infected human (t and animal (t),), which represent the depletion 3 and E. respectively, while 7/I and 7/2 compute the usefulness of the control measure μi (t) and p2(() (where μn (()() [0, 1)). respectively.
m) (t) and (1 (2))
The most successful framework is (1) = = (2z((), which indicates that when the interaction of susceptible humans with infected humans and animals is almost perfectly avoided it makes the
transmission of the disease to zero. Here, #1 (t) = 1 = (2(() means fully response by implementing the given control mechanism, while /n (t) = 0 = (tz(t) implies response. The intensities of the responses are associated with the behavior sities are represented by (t) and p2(() control measures. We maximize the responses using isolation so that they change their behavior and the cost will correspond to a nonlinear function of (t( and #2 (t). Thus, we wish find the optimal response for risky individuals with the help of isolation as control measure.
the human population, and so these response inten-
4.2 TREATMENT FOR INFECTED INDIVIDUALS
The treatment of infected individuals not only controls the number of the infected individuals but also influences its development. Although there is no proper treatment for the monkeypox virus infection, smallpox and monkeypox are genetically similar, and there are antiviral drugs against smallpox that can be used for treatment purposes. So in the present scenario, we assume the accessi- bility of treatment for the infected population. We introduce the term -133/a (f) (1) as a treatment in the proposed model, where ')3 represents the treatment rate associated with the intensity (a(t). There are various costs associated with given medication, so we assume that the intensity treat- ment control measure Pa(t) lies between 0 and unity. The control pa(t) will attempt to change the fraction of the infected population to the recovered population.
4.3 RESTRICTION ON ANIMALS
While it seems to be difficult that how to restrict animals from the transmission of the monkeypox infection, is possible. In the current situation, various countries have restricted the importation of animals (rodents) and non-human primates. Animals that are infected with monkeypox should be placed into quarantine immediately to be isolated from other animals. Moreover, an animal that has close contact with another infected animal should be also isolated and accordingly quarantined to observe the symptoms of monkeypox for 30 days. We introduce the control factor (1 - (t)) to control the interactions of susceptible and infected animals, where (t( [0, 1).
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In this section, the main objective is to obtain the optimal control strategy that minimizes the infected population with the aid of the above control measures and with the minimum associated cost. Thus, the admissible set of control measures (t) is defined by
U (,(((), ..2.....4. 4:
0 S (((l) 1 LE [oTT)
We, therefore, develop the control problem by keeping in view the above strategies with the objective functional (((,}) be minimized:
(t), 1,....,)) 4) 0= (t)dt + +
1
Σ 1
4
(3)
subject to
(t),
4 {1 (,()))))))())))s - 6 72/22(t)) Ma) Ma (2/2(1)) (t)..)
ISA(t),
dt
(t)
= B (1 (t) M. (t) - - - 10+00+re0 ()))(t)))))) (M)
dt
(t) {r -a)4a (t)} (t) (R) (t)
4a - (t) - -
dt
dt
dMa(1) = (1 (1)4(11) (a(1)Sa(() - {a 0n Ma(t),
dt
(4)
with the initial population sizes in Eq. 2 In Eq. 3 the integrand represents the value of cost at time while the function W shows the sum of the cost described by the integrand the total incurred cost. The parameters hy and Ki 's are non-negative parameters that are weight constants to balance the units of the integrand. The control measures μe (i 1, 3, 4) exist in the admissible control set U that minimize W. We now discuss the existence of optimal control for our proposed control problem 4 then use the well-known Pontryagin's 's maximum principle for characterization and getting the necessary conditions of the optimal controls. The following result will be presented ensure the existence of μæ that minimizes the function W.
Theorem There exist optimal controls μ (1) = (2)() ua((), (1) in U that minimize the objective function W associated with the control problem in Eqs.
Since the above result ensures the existence of the controls to minimize the objective functional (3) subject to the state system (4), then derive the necessary conditions for characterization of the optimal control problem using Pontryagin's 's maximum principle, see Theorem 3 in the appendix.
5 NUMERICAL EXPERIMENTS
We perform numerical experiments test the model predictions and verify the analytical findings. We utilize a well-known numerical procedure of the Kange-Kutta method of the 4th order. First, we discretize the model and develop the algorithm to perform the numerical simulations.
5.1 DISCRITIZATION
To discritize the model, we set
S
- - S,, SMaS US
Mr
BM, + M, - (0 01 +T) Mr
X
Ra
F
rM, OR
Sa
$ 0 M.S.. aSS
Ma
SM, (a + an) +(+)
Ma
$1
4 - m111 Mr - {1 122222 M, - USh
42
3 111 M Sh +
{2222 M.S,, (1 + 01 + nouar Mn
Y =
43
==
[r + + M, - -
44
T 01 74/44] M.S. aSa
993
Φ {1 m144 } M, - -a + a1 MA

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and
fir (2) 71/))))))))))
+ -, - tf00
(11 (22 (3 (1
822 fr + U1) 43 - h1,
H
0p3, (94 95) 95) (0 mppp) Ma 0444;
{ie (2) (5 (1 - (1-1))))) 8;)} + (a+ Ps 111--45 10 (1 mmpr) St).
then Eq.
1.
4 4 and Eq. can be re-casted
dX (t)
dX(t,, dt
dY H(),))
= F(1, (()),
(5)
dt
dt
The application of forward and backward Runge-Kurte Kutta method of order four gives
1 (k1 + 2k: - 2k3 + ka),
X = X, + 6
(h1 + 2hv + 2hs + ha), Y =Y,
(6)
where
h1 = F(tn; Xn); l22 = (th 2 xn+ + 1/1 (h1). h3 = F
2 Xn + th2
x:
In +
2
2
Xn 1ki
h4 = (In + h, Xn + lha),
kn = Flln, Xn);
lo2 = hn
2"
2
l12 2
83 = hn 2' Xn
ha = F (tn - hi Nn 1k3).
Thus the rest of algorithms can be concluded as:
Algorithm Runge- Kutta Method (RK4)
KRunge-Kutta
1: Input: Endpoints to, max. integer parametric values, initial conditions
2: Output: approximation Sn, Mn. RR, Sa. MM at (n + 1) values of
3: Parameters and Initial Conditions: Setting the values of epidemic parameters and initial sizes
for compartmental populations
4: for i 1,...,... do
5: Recursive Formula: X,+1 for both control and without control system as given in Eq.(5)
and Eq. 6
6: end for
7: for i L, n. + - do
8: Recursive Formula: Y;-1 as given in Eq, and (q.(6)
9: end for
10: Optimal Control: Plugging optimal control variables as given by Eq.
Output: (t Si+1 .M++1 Ri+1 si+1,
Mc+1)
5.2 DISCUSSION
We perform numerical simulations to discuss sensitivity analysis and the application of optimal control strategies. We conduct numerical experiments to present the validation of our theoretical findings for the model parameters and initial sizes of populations as specified in Table 2 It could be noted that some of the parameters are directly correlated the basic reproductive number, Rg, while some are negatively correlated, as shown in Figure 2 Figure B represents the contour plot that describes the dependency of the basic reproductive number, R.. on 3 (disease transmission co-efficient of humans) and ø (disease transmission co-efficient of animals); & and i (natural death rate); 3 and r (recovery rate of infected population); and & and ai (the death rate arises from monkeypox in animal population). It is very much clear from Table Figure that the epidemic parameters, namely, 3 and ø have positive indices, and are also associated with the susceptible population. If we increase the value of 9 and o, it will increase the value of basic reproductive number R, and cross the value R, = 1, which leads to a substantial outbreak of the monkeypox virus transmission. To maintain the value of the basic reproductive number and deduce favorable

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control method to eradicate the spread of monkeypox virus transmission, need to restrict the value of these parameters, which is possible with the help of contact tracing and maintaining social distancing, On the other hand, the parameters with negative indices are 1, r. a, and at shown in Figure while the relative effect reported in Figure 3 Moreover, whenever the value of those parameters having negative indices increases, the basic reproductive number Rs will decrease, and if its value becomes less than unity, the infection will no longer persist. So, treatment of infected humans through medication has been incorporated with the aid of optimal control eliminate the monkeypox virus. To observe the influence of intervention strategies on the transmission dynamics of monkeypox, we perform the numerical simulations of the proposed optimal control problem with the help of the Runge-Kutte Kutta (RK4) scheme, as concluded in Algorithm 1. Moreover, the time frame is taken to be 20 units, and the value of parameters are borrowed from Table 2 while investigating and implementing the optimal control mechanism. To investigate the effect of optimal control measures for the monkeypox virus transmission, we execute the proposed problem in two folds: without control (t) (t) (t) = (t O and with the combination of four controls (1r (f), (t((),, (t), (t)) which leads to the results as presented in Figure 4 These graphs respectively demonstrate the dynamics of human and animal compartments of the proposed model under no control and with controls. The black dashed and red dashed curves respectively represent the dynamics of each compartmental population with and without the utilization of control strategies to highlight the effect of optimal policies implementation, see Figure 4 A significant reduction in the infected population, as well as, an increase in the non-infected population can be seen with the application of optimal control measures, whenever, compared without intervention strategies. We conclude that the combination of suggested optimal control measures can achieve significant reduction in monkeypox virus transmission whenever applied in true sense.
Figure 2: The plot represents the sensitivity indices and most sensitive epidemic parameters related to basic reproductive number Rg.
CONCLUSION
To keep the current scenario of the monkeypox virus transmission in our mind, we investigated and proposed a model for the dynamics of monkeypox virus transmission. Both theoretical and numer- ical analyses of the proposed epidemic problem have been studied with the aid of stability theory. Positivity, as well as, the boundedness of the states of the epidemic problem guarantees that the considered model is well-defined dynamical system. We performed the local and global dynamics of the model and derived the stability conditions. The proposed model has several epidemic pa- rameters and so with the help of normalized sensitivity analysis. the most sigmificant parameters are quantified. It could be observed that disease transmission from both numan-to-humee to human play a significant role in disease transmission. Besides the disease transmission rate, the parameter is also very effective and has major effect on infected individuals and R.. Moreover, we modified the proposed model by incorporating optimal measures with the aid of control theory to mitigate the monkeypox disease burden and describe the effect of control policies implementa- tion, and as a result to stop the disease from spreading. The findings of our optimal control problem show that the maintenance of the social distancing, tracing contact, transportation of animals and treatment of infected individuals may support mitigating monkeypox virus spreading in the current scenario of the epidemic.
and animals
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08
0.5
04
04
04
02
02
04
0.6
0.2
04
0.6 0.8
0.08
0.00
0.65
0.00
0,
Figure 3: Contour plots of the basic reproductive number Ro to show the sensitivity analysis. The basic reproductive number is taken to be the function of B, o and 01 to show the relative effect of these parameters on the basic reproductive number.
Buscapaibre Human 15,00
Ind Plurmarr (05,00
26
Recovered Homann (P L000
52.
St
18
20
13
Thee 51
7hne 20
with control withos contro!
with castral Witheet sontrs
16
Intected Animal (MR (C0)
-20
Animat
12-
22
10
15
16
(0)
Figure 4: The plots represent the dynamics of the compartmental population with the implementa- tion and without the implementation of control strategies
Given the increased development of fractional calculus, many operators of fractional orders were introduced to capture more valuable information. In our future work, we will generalize the proposed model its associated fractional version to discuss the dynamics of monkeypox virus disease.

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Hans Heesterbeek, Roy M Anderson, Viggo Andreasen, Shweta Bansal, Daniela De Angelis, Chris Dye, Ken TD Eames, W John Edmunds, Simon DW Frost, Sebastian Funk, et al. Modeling infec- tious disease dynamics the complex landscape global health. Science, 34766227) 2015.
William Ogilvy Kermack and Anderson G McKendrick. A contribution to the mathematical the- ory of epidemics. Proceedings of the royal society of london, Series A, Containing papers of mathematical and physical character, 115(772):70-77211 1927.
Subhas Khajanchi, Dhiraj Kumar Das, and Tapan Kumar Kar. Dynamics of tuberculosis transmis- sion with exogenous reinfections and endogenous reactivation. Physica A: Statistical Mechanics and its Applications, 497:52-71. 2018.
Subhas Khajanchi, Kankan Sarkar, Jayanta Mondal, Kottakkaran Sooppy Nisar, and Sayed F Ab- delwahab. Mathematical modeling of the covid-19 pandemic with intervention strategies. Results in Physics, 25:10428: 2021.
Devendra Kumar, Jagdev Singh, Maysaa Al Qurashi, and Dumitru Baleanu. A new fractional sirs-si malaria disease model with application of vaccines, antimalarial drugs, and spraying. Advances in Difference Equations, 2019(1):1-19, 2019.
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Moyinoluwa Joshua Oladoye. Monkeypox: A neglected viral zoonotic disease. Electronic Journal of Medical and Educational Technologies, 14(2)=em2108, 2021.
Eskild Petersen, Anu Kantele, Marion Koopmans, Danny Asogun, Adesola Yinka-Ogunleye Chikwe Ihekweazu, and Alimuddin Zumla. Human monkeypox: epidemiologic and clinical char- acteristics, diagnosis, and prevention. Infectious Disease Clinics, 33(4):1027-1043, 2019.
G Rohith and KB Devika. Dynamics and control of covid-19 pandemic with nonlinear incidence rates. Nonlinear Dynamics, 101(3)2003-2222, 2020.
David Rolnick, Priya Donti, Lynn Kaack, Kelly Kochanski, Alexandre Lacoste, Kris Sankaran, Andrew Slavin Ross, Nikola Milojevic-Dupen Natasha Jaques, Anna Waldman-Brown et al. Tackling climate change with machine learning. ACM Computing Surveys (CSUR), 55(2):1-96, 2022.
Priyabrata Saha, Saurabh Dash, and Saibal Mukhopadhyay. Physics-incorporatee convolutional recurrent neural networks for source identification and forecasting of dynamical systems. Neural Networks, 144:359-371, 2021.
Piu Samui, Jayanta Mondal, and Subhas Khajanchi. A mathematical model for covid- 19 transmis- sion dynamics with study of india. Chaos, Solitons & Fractals, 140: 10173, 2020.
S Saravanakumar, A Eswari, L Rajendran, and Marwan Abukhaled. A mathematical model of risk factors in hiv/aids transmission dynamics: observational study of female sexual network in india. Applied Mathematics & Information Sciences, 14(6):967-976, 2020.
Neha P Singh, Shivani Sharma, Garvit Ghai, and Amandeep Singh. A systematic review on epi- dermology of human monkeypox virus. Annals of the Romanian Society for Cell Biology, 25(7): 602-610, 2021.
Pauline Van den Driessche and James Watmough. Reproduction numbers and sub-threshole endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180(1- 2):29-48, 2002.
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):124012 2021.
APPENDIX
A. PROOF OF PROPOSITION I1
Proof. The differentiability of the right-hand side of the proposed model equations implies the existence of unique maximal solution for any associated Cauchy problem. So, the solution of the first equation of the model looks like
Sa(() = exp { {-01- S. (BMa(s)+ di
x 2xt @ntexp exp { {ot
f[ (BMM(W)+ + +MM) dy dr
St- --f (SMM(E)+ + (Ma(z) dx) dx
+a((0) exp
0.
The solution of second equation of the model
( takes the form
(t) exp {
(x))x I (M.()))((z))
+ 01 + r)t +
x exp (0+ 01 +rxx 1 3Sh(w)dy dr
( 01 r) L' (SS(())de) 20.
(0) exp
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Clearly, that (t) 0 and (t) 2 0 for Sk 0 and M, 2 0. Following the same steps, we can easily show that (t, 2 0, (a(t) > and Ma (t) 2 0. Moreover, to show the boundedness, let assume that the total human and animal populations are denoted by NNa(t) and (a( respectively, then the total dynamics of humans and animal looks like
dN S 2n (N),
dNa(t) p. S. - <
is easy solve the above inequalities and then taking unbounded limit i.e., t approaches infinity, (a(()) of and (t) 2n, and thus the feasible region for the proposed model is given by
4n g
(Sh, Mh, Rh, Sa, Ma) R4,
NA S
No
S
a
A.2 PROOF OF THEOREM
Proof. We use the approach of linear stability analysis and linearize the proposed model around the Monkeypox free equilibrium, Wo, then the variational matrix may takes the form
0
0
20%
01+01 +
0
0
J(W))
r
0
-(
0
0
0
(a + a1)
Let the associated eigenvalues of J(W) are X; where, i = 1,2,...,5, then we obtain
d = - A2, = - 01 + r) (1 - Rx) A4 = =-a=- = -aa + a1).
Obviously, Å1 A3. da and A are negative while 22 is negative whenever R^ 1 holds. Since, Rg < 1 provided that Ro < 1, therefore we conclude that the proposed Monkeypox model is locally asymptotically stable at the disease-free state Wo subject to the condition Ro holds. To discuss the global dynamics, let define St. and S 2a, and real-valued function, (t) = Ho + H1. such that
Ho (Sa 51) + {S. - 52), = MA + RA + Ma.
Clearly, (t) is non-negative and its temporal differentiation with the application of some algebraic manipulation leads to the assertion
dH - (S) - 5R) - a S - (0+ M, - OR, - (a a1) Ma.
dt
implies that Also it could be noted that, whenever, Sh, Sy and S. So. then 4H 0. which prove that (t) is Lyapunov function and the proposed model is stable globally asymptoti- cally.
A.3 PROOF OF THEOREM
Proof. Since the solutions of model (1p are bounded and non-negative for the initial population sizes 2). The objective functional is also non- negative which ultimately implies that W is bounded. Thus to minimize the sequence of controls 4¹' (t) = (n4 (t), (t))), (t), U exists, such that
lim W
inf (((()). #EU
The controls in set U are bounded uniformly in d space which implies that they are bounded uniformly in (10 TJ) space. Since, 122 space is reflexive, so there exists subsequence μ² (1) U such that
i (t) (t), (t)) pt(t), 15(t) PG(t), Mi()) + PG())
weakly in 2 space approaches oc, then the sequence defined by 27 (ST. MT RT. ST. MM) is bounded uniformly corresponding to (t). It can observed from the right- hand side of model [ that they uniformly bounded which gives the equi-continuity of the state sequence and uniform
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Under review as a conference paper at ICLR 2023
bounded-ness of the derivatives for 27. The Arzel-Ascoli theorem then gives that there exist 27 (S,, M, Ri, S MM²) such that x"' + xn uniformly on [0, T].
Further, to complete the proof, let assume proper subsequence passing through the limit of the system (I) and corresponding to u (t), p5(t), pī(t), (t))), we can obtain the optimal solution z². Thus, the lower semi-continuity of I norm with respect I,² weak convergence gives that
[ hy M,dt
2
inf W(n") lim W((" 2 MEU
Σ Repldt W(n").
7-000
Therefore, 10" (t) = (1i()), (5((), P5(1), (G))) is an optimal control measure.
A.4 CHARACTERIZATION OF OPTIMAL CONTROL
Theorem 3 There exists optimal controls p² (Mn 12, p3, p44) in U and the corresponding so-
lutions of optimal state z" = (Sh, My 52 M²) of the Eq.44) that minimize W (3). then the adjoint variables (p1. (22, (p3. (p4, and (ps satisfying the equations:
(i()) = =(4] (2} {B (1 7111) MA + E(1 72H2) +
(t) {41 52) {B (1 m1/11) Sn} {0 + 0 + 13333 P2 fr+ (13) iss h1,
(t) = 043: (1) = (44 ) (6 (1 74/44) Ma + a.44
(t)()) = {4p1 - }22 {E (1 72222) } + {a aa + (pp - pot t napl) Sat.
with the transversal condition:
(T) = (2)(T) = (T) (x
= (T) =0
Further, the associated optimal measures μ² (t)
are:
1
1
11 {p2 11) 3m MrS.:
11 K2
{ir (2)) MoSh.
P = K3
f:o2 sas) Ma
1 μ4 (pp-ypa) MaS.
R4
Proof. In order to investigate the necessary conditions for the control measures using the Pon- tryagin' smaximum principle for the proposed model we define the Hamiltonian H for every t E 00 T] as:
Σ K. p(t))(2,))) H hy + 1 2-11
where (t) = (t), (t), (t),, (t)) (t)( g- (g1.92. 93. I4. 94. and gl, 92- 93- I4 and gs respectively represent the right hand side of the first, second, third, fourth and fifth equation of the system (4) We derive the minimized Hamiltonian with the help of Pontryagin's 's maximum principle which minimize the objective function W (3). The Pontryagin' maximum principle relates the state variables of the model and the objective functional with including adjoint variables. Therefore, by using the Pontryagin's 's principle there exists &i (t), P2((), (t))), (t()) and (t((t) that satisfies
(t) ÖH
(t)()) = 0H
OR, (1)()) = QH
OH :
as, (t) = aM,"
as. :P((() = aM,
with the transversal conditions 1 (T) = 22 (T) = (T) = (T(TT = os (T) = Now by using the characterization of controls we have
aH
0H @1 0, 01 Opa(1) 0.
0H
0H
0pr (1) O,
AH = 0 (i 1,2,3,44 4), and at (1( = (f()), p2(t) = pā(t), Hz(t) = HG(() and pa(t) = Op: (t()) we obtain
From
1
1 {5ı - (2 Mr 1z K3
{g22 f1) 3m Sr. p== K2
{12 s23 ns
Ki
1
p K4
{4p (a) M...
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Under review as a conference paper at ICLR 2023
Parameter
Value
Parameter
Value
Parameter
Value
Pn
0,9430
8
0.0003
E 0.0044
0,4570
01
0.0910
0.0590
0,6000
Ø
0.0790
a
0.1200
on
0.0900
7)1
0.0900
7/2
0,5000
133
0.0400
1]4
0.0400
Y]5
0.0500
h1 0,0200
0.6000
h2
0.9000
k1
0.4400
K2
k3
0.2000
kis
0.2000
kg 0,2000
Table 2: Parametric values used in the numerical simulation of the optimal control problem
By considering the lower and upper bound for (t).), p2(t), Ma(t), and (t(t), we leads the fol- lowing assertions:
(t) = min
max
0.
(p2(t) - (t)) Mr J
0. (s1(()) - (2((1) Mr (a)Sa .1)
(5(() = min
max
R2
1G(t) = min
max
0. (12(()) - (P((t)) Ma /
Ki
M. Mc(1),.))) 11
pG(() = min
max
0. (1s(()) (2(11) 14
It is also clear from the Hamiltonian H that
0²H
8
8
0,
n 0, 0p3
x
0p3
which implies that the optimal measures minimize (H).
We state the optimality system with the help of pi, p2z 13 and 14 Thus, the optimality system that minimize H at (S%, Mi, RY Sh M 17 12 pg. 14g, 11,, 42, P3,, (p4) becomes:
dSn = Dy 3 (1 m MAS, - ワ2k M ST 05%.
dt
dMr
3 {1 -miit S₁ + & m1n} Mr (0 + 01 + + m315} Mj:
dl
(r + [t)} M OR dt dR,
dS,
t. o [1 mapa(1)) MrS: -as
dt
dMa =
M - fa + a1 M"
dt
with initial population sizes (0) > 0, MM (0) 2 0. (0) 0. 02 2 0. M2 2 0, and the associated co-states (adjoint variables) are
i {sp (2) (8 (1 m MM + {(1 72 1233) 0
P2 = {ip (2) (B - mιィ) [ (0 + 01 + r + + f2 - fr + 13) 43 - h1
P3 Dips. P {44 ps} (Ø (1 7)44) M.} a4p4
Pb = (4] (2) (E (1 m2/46) {t + (a + ++) 33 + {مp - ups) 10 (1 )THLر {
where (7) = (2(()) (7) = (T) (T) = 0 are the transversal conditions, and the
four control variables are stated by (q. Hence, the proof.
14


Reviews:
[{"review_id": "uqymW2779i", "reviewer": "Reviewer_8Yf3", "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 develop a differential equation-based model of the dynamics for monkeypox. The model explicitly includes both human and animal populations, as well as the cross-transmission from animal to human. The authors also propose several control measures affecting various transition rates in the model; a control theory approach is then used to derive optimal policies for the control measures. Finally, a limited set of numerical simulations confirm the derived analytic results concerning the plausibility of the model.", "summary_of_the_review": "Overall, while the paper proposes some interesting approaches to modeling monkeypox and some associated control strategies, the limited methodological contribution and complete lack of data-driven analysis likely limits the interest at a venue like ICLR. Improved discussion of the resulting policy for control measures, as well as a general discussion of the use of the approach beyond just monkeypox, may improve the impact.", "strength_and_weaknesses": "The main strength of the paper lies in the novelty of explicitly modeling the cross-transmission between human and animal populations.\n\nThe main weaknesses of the work are in the limited methodological contributions as well as lack of data-driven evaluation of the model. As described in more detail below, the evaluation could be improved by either including some type of real-world data for validating the model or at least providing a more thorough qualitative analysis (supported by synthetic/simulated data).", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "haOilLQGLJ", "reviewer": "Reviewer_8Pvu", "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": "In their manuscript entitled, \"Monkeypox 2022 with cross infection hypothesis via epidemiological model\", the authors present a two-population (human and animal) compartmental model for describing the evolution of the 2022 monkeypox outbreak.  Given various intervention options the authors present an investigation of optimal control policies for this system.\n", "summary_of_the_review": "An interesting paper for a mathematical disease modelling audience but not for ICLR.", "strength_and_weaknesses": "Considered across the spectrum of modelling approaches for all diseases, a two population compartmental model is at the lower level of complexity of those in use; its formal analysis by optimal control theory is less common and would be of interest to numerical modellers working in this space.  The investigation here is of a rather theoretical nature, more so than practical: that is, the problem is formulated in a general sense, rather than in collaboration with an actual disease control program or with a confrontation against real world data.  This would be suitable for a mathematical disease modelling journal; however, I do not see it as being suitable for ICLR as I cannot discern a novel contribution in the domain of deep learning (or an adjacent field).       \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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "cde1x3JdRf6", "reviewer": "Reviewer_zkQd", "summary": "", "questions": "", "limitations": "", "rating": 1, "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 introduces a compartmental model which allows for cross infections between animals and humans. The model is used to study the dynamics of the Monkeypox spread. \n\nI have major concerns for this paper to be almost a double submission of the paper http://www.aimspress.com/article/doi/10.3934/mbe.2022633  (Stochastic modeling of the Monkeypox 2022 epidemic with cross-infection hypothesis in a highly disturbed environment). Now it could be a case that it is not but then there are too many similarities between both papers\n* Abstract reads almost the same\n* Introduction is more or less same (in both points I am talking about literal word to word copies).\n* Proposed models are same if you account for description in the appendix \n* Even the major listed contributions are same for both papers\n* Figure 1 is more or less identical. \n\nNow a case can be made that first paper is essentially the model and this second paper is mainly about the optimal control. However, I doubt that as even in first paper authors talk about scenarios which are akin to the optimal control setting here and importantly in words of authors itself, they list their main contributions as:-\n```\n\" \n   * The cross infection between humans and animals plays a significant role in the dynamics\n     of monkeypox virus transmission. We, therefore, propose a model based on the hypothesis\n     of cross-infection between humans and animals. The model has two blocks: humans and\n      animals.\n   • The first block describes the evolution of monkeypox in the human population, while the\n      second block represents the evolution of the monkeypox virus among animals.\n   • Four time-dependent control measures are then introduced in the model to demonstrate\n      the utilization of optimal control measures: to minimize the infectious individuals and\n      maximize the recovered population. Particularly, reducing the risk of disease transmission\n       by educating people to rise awareness of risk factors, treatment of infected individuals, and\n       restrictions on animals.\n\"\n```\n\nNow first two can't be contributions of this paper in presence of the first paper I am talking about. Also quite mysteriously the first paper is nowhere cited in the main paper which raises even more concern. \n\nI will finish my review here and hope that authors prove me wrong and their rebuttal makes me look at paper differently to evaluate other points. ", "summary_of_the_review": "look above", "strength_and_weaknesses": "look above", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}]

Author rebuttal:

Decision: Reject