MONKEYPOX MATHEMATICAL MODEL WITH SURVEILLANCE AS CONTROL

. A compartmental mathematical model of the transmission dynamics of the monkeypox virus (MPXV) was developed and analyzed. The model incorporates proper surveillance and contact tracing as effective controls. The equilibrium states of the model were obtained and analyzed both locally and globally. The effective reproduction number, R m was obtained and the sensitivity of the model parameters were studied using R m as the threshold of transmission. When the infection becomes endemic, R m (cid:117) 1 , the model exhibits a backward bifurcation but R m < 1 which means that the interventions tend to MPXV containment. Numerical simulations to bespeak our ﬁndings and discussions are provided. Our result shows that surveillance and contact tracing are effective for the containment of MPXV in the absence of a perfect vaccine.


INTRODUCTION
Among the epidemic rampaging the world lately, monkeypox is the one with disturbing reemergence. Since the first recorded case of monkeypox virus (MPXV) on human in 1970 in the Democratic Republic of Congo (DRC), regardless that it was initially discovered in Denmark in 1958 [16,3,13], it has been trending in Central and West African nations [10,15]. The about ten days [21]. The rashes affect the face, palms of the hands, soles of the feet, oral mucous membranes, and sometimes the genitalia, conjunctivae, or cornea [15,27,12]. Like any other viral infection, monkeypox is a self-limited viral infection with symptoms lasting from 2 -4 weeks. The severity of the infection is determined by the health status of the patient before the invasion, virus exposure, and the nature of the virus strand.
The present therapeutic method for monkeypox is clinical care, which is targeted squarely at symptom alleviation, management of possible complications, that includes encephalitis, pneumonia, etc, and prevention of long-term sequelae like keloid scars [31]. A healthy diet and fluid are used to maintain adequate nutritional status and regulation and optimality of internal organs.
Vaccination against defeated smallpox has shown through a series of observational studies that it is 85% effective in the prevention of monkeypox disease except that it is no more accessible after the eradication of smallpox in 1980. There are other variants of vaccines going through trials and some countries have started administering modified attenuated vaccinia virus -Ankara strain which is a two dose vaccine but the availability remains limited; which means that most of the countries cannot access it at present [27,12]. As a result of this gap, the most effective prevention strategy currently for monkeypox is constructed awareness of the risk factors and education on effective measures to curtail the exposure to the virus [7]. To contain the monkeypox outbreak, swift surveillance and rapid identification of the new cases are key; unprotected contact with wild animals especially sick or dead ones should be avoided and animal trading should also be monitored.
Mathematical models have been instrumental in understanding the dynamics of different diseases: HIV [23], Malaria [24], conjunctivitis [28] and tuberculosis [25] etc and other areas of endeavors like heat transfer [2], psychiatric [11], bio-medical engineering [1] and other countless fields of life. Considerable researchers have applied mathematical modeling in the study of transmission dynamics of monkeypox. The study [6] showed that with treatment intervention monkeypox will be eradicated from human and rodents population, [14] studied the transmission dynamics of monkeypox virus which they concluded that isolating infected individuals in the human population helps to reduce the disease transmission. Treatment and vaccination interventions were considered by [8] as an effective measure for the containment of monkeypox, [16] incorporates impact vaccine on the human population as a means of boosting the immune system of the people, [26] modeled human to human transmission, [29] used game theory to model monkeypox vaccination assess strategies and likewise [4]. There is other priced work on monkeypox transmission but to the best of our knowledge, none has considered infection cautioned by infected individuals who are out of radar and surveillance subpopulation. United kingdom as of 12 July 2022 has a total of 1735 confirmed cases just between 6 May to 11 July of which most of the patients are traced to men without documented history of travel to endemic countries [30]. We aim to study the effect of effective surveillance in containing the transmission dynamics of monkeypox whether the suspected or probable case is symptomatic or asymptomatic; to buttress the importance of public health awareness on how the disease is transmitted, its symptoms, preventive measures, and action points when the infection is suspected or confirmed.
The remaining part of the paper is structured as follows: model formulation in section 2 and model analysis in section 3 which includes stability of the equilibrium points and the bifurcation analysis. Section 4 consists of the numerical results which include sensitivity analysis of the model parameters and numerical simulation and result of the model. Finally, in section 5, the discussion and conclusion are presented.

MODEL FORMULATION
A deterministic compartmental model is used to represent the transmission dynamics of monkeypox virus infection in two populations; human and host animals. The human population is partitioned into the susceptible human population, S h , which are humans that have the potency of contacting MPXV when they come in contact with infected individual; exposed human population, E h , are persons who have been in contact with at least an infected person either directly or indirectly; population of humans under surveillance, H h , are individuals who have been exposed to MPXV and through contact tracing are under watch of health worker(s); Isolated infected human population, I h , are individuals that have been proved to be infected by MPXV after medical diagnosis and have been isolated using standard procedure. Unidentified infected human population, I u , are individuals who have been infected by MPXV with symptoms that are outside radar; recovered human population, R h , are individuals that were infected previously with Monkeypox virus but have recovered. The animal host population is sub-divided into Susceptible animals, S a , infected animals, I a and recovered animal population, R a . The influx of humans into the population which is either by birth or immigration is at the rate Λ h .
The level of health awareness carried out within the population is represented with θ , while the probability of an individual being infected with the virus per contact with unidentified infected human and isolated infected human are β and β 1 respectively. β 2 is the product of effective contact rate and the probability of a human being infected by an infectious animal or its product.
The proportion of the exposed individuals that the virus overcomes their immunity and hence making such a person infectious is ω, and the level of contact tracing implored in a community to isolate all infected humans is ρ. Human inherent recovery rate without any help from a health worker is k and γ when a health worker guides the infected human to recovery. The disease-induced death for unidentified infected human population is d 1 and the induced death rate for the isolated infected human which is predominantly children with a history of terminal disease is d 2 . The proportion of the exposed that are under surveillance which is either passively, actively, or directly is represented as δ , under this condition, collection and dispatch of specimens for monkeypox laboratory examination is carried out. When the laboratory results are out, the rate of humans under surveillance that is negative to MPXV is τ, and they go back to the susceptible population. The rate of individuals whose diagnosis comes out positive is α, and these are isolated completely. The natural death rates of humans and non-human primates are µ and µ a respectively. The recruitment of the animal host population is at the rate Λ, the rate of disease-induced death in the animal is d 3 , the natural recovery rate of animals is b a and a a is the effective contact rate with the probability of an animal being infected per contact with an infected animal. Standard incidence function is used as the force of infection because monkeypox is a frequency dependent infection, then λ = (1 − θ ) β I u +β 1 I h +β 2 I a N where N = S h + E h + H h + I h + I u + R h , N a = S a + I a + R a and all the state variables are function of time, t is in months/years. The schematic representation of the above assumptions and illustrations is shown in Figure 1. Taking into consideration the above assumptions, the model is governed by the following system of differential equations: subject to the following initial conditions and N a (0) ≥ S a (0) + I a (0) + R a (0). The epidemiological interpretation of all the parameters found in model (2.1) is given in Table 1; all the parameters are assumed to have non -negative numerical value(s). µ Natural death rate of human 0.01794 [5] Λ a influx of animal 0.2 [10] a a Animal to animal contact rate 0.027 [6] b Recovery rate of animal 0.6 [14] d 3 Disease induced death of animal 0.4 [6] µ a Natural death rate of animal 0.006 [14] 3. MODEL ANALYSIS 3.1. Model Properties.
As t → ∞, we have that lim sup Hence Ω is a feasible solution set for the model (2.1), which implies that all solution sets of model (2.1) are bounded for t ∈ [0, ∞). is satisfied.

Positivity of Solution.
Clearly W > 0 and also W < ∞. From (2.1) we have that By method of integrating factor, we have that Using the same approach on ther state variables in (2.1), we can easily show that Similarly, we can easily show that Therefore any solution of model (2.1) when (2.2) holds is non -negative for t ∈ [0, ∞) and buttress the epidemiological meaningfulness.

Model Equilibrium Points. At equilibrium, dS
; λ a = a a I a N a . From this steady state of the model we can observe the following types of equilibrium point: (a) Monkeypox Free Equilibrium (MFE), E 0 : This is a state where there are no infected persons or non-human primates in the community. In this regard I h (t) = I u (t) = I a (0) = 0 and also λ = λ a = 0 which means eradication of monkeypox pathogen that leads to reduction of the worldwide incidence of MPXV to zero by deliberate efforts, obviating the necessity for further control. The state of MFE is made possible by the following eradication strategies: an effective intervention that can truncate the transmission of monkeypox, practical and sensitive diagnostic kits that will be available with the ability to determine the level of infection that is transmissible, and public health awareness. The MFE of model (2.1) is obtain when λ = λ a = 0 in the above steady-state to obtain (b) Monkeypox Endemic Equilibrium (MEE), E 1 : At this steady state, the force of infection λ > 0, λ a > 0 which entails that the infection will persist in the community hence, there will be at least one of the infected human or infected reservoir that must pass it on to one other human or non -human pirate on average. The MEE of model (2.1) is given as where the state variables are defined as in (3.1). The MEE exists when k 1 k 2 (λ + µ) > τδ λ , I * h ≥ 1, I * u ≥ 1 and I * a ≥ 1 at any given time.
3.3. Effective Reproduction Number, R m . Reproduction number is a threshold used to study the prevalence and long-term behaviour of a given disease. It is the number of secondary cases that emanates from a single infected person in the course of its infectious stage. The next generation matrix as defined in [9] is used to compute the reproduction index which we term as an effective reproduction number because of the involvement of awareness and surveillance parameters used as controls in the model. Using this method, f i is defined as the influx of the infection into the compartments and v i is the transfer of infection into other compartments as the disease pathogen progresses. Hence Therefore, the transmission matrix F and transition V evaluated at E 0 is given as respectively. From [9], R m = ρ(FV −1 ), where ρ is the spectral radius. Then After further simplification, we obtain that 3.4. Stability of Monkeypox -Free Equilibrium.
Theorem 3.3. Local stability of MFE: The monkeypox -free equilibrium is locally asymptotically stable when R m < 1 and unstable when R m > 1.
Proof. Using the next generation matrix to obtain R m and the linearization of (2.1) at E 0 taking (2.2) into consideration satisfies the five conditions in theorem 2 of [9]. Hence, the MFE is locally asymptotically stable whenever R m < 1 but unstable if R m > 1. The epidemiological interpretation is that monkeypox will be gradually eliminated from the population whenever R m < 1 given that the initial size of the subpopulations is within the manifold of attraction Ω.
By implication diminutive influx of monkeypox infectious individuals into the population will not loom large the possibility of a monkeypox outbreak, and the disease dies out in due time. Proof. Rewriting model (2.1) in the form R 5 is the latent/infected compartments. The following conditions must be satisfied for E 0 to be globally asymptotically stable: At MFE, we have which shows global convergence of (3.2) in Ω. From (2.1) we have Clearly S h N h ≥ 0 and S a N a ≥ 0 which clearly lie within the manifold Ω andF 2 (X 1 , X 2 ) ≥ 0. Observe that R is actually a M− matrix, hence conditions l 1 and l 2 are satisfied and is sufficient for the global asymptotic stability of E 0 when R m < 1.

Stability of the Monkeypox Endemic Equilibrium.
Theorem 3.5. The system (2.1) has an endemic equilibrium, E 1 that is globally asymptotically stable whenever R m > 1.
Proof. Since R m > 1, E 0 loses its stability and a unique endemic equilibrium emerges in Ω and is locally asymptotically stable. Using the method employed in [11] we show that model (2.1) has no limit cycles. Let the vector M = (S h , E h , H h , I h , I u , R h , S a , I a , R a ), we define a Dulac's < 0.
FM dt < 0 implies that E 1 will stay non-positive for all t ∈ [0, ∞) and confined in Ω. Hence by Bendixson -Dulac criterion, there will be no limit cycle in Ω and E 1 is globally asymptotically stable.

Bifurcation Analysis.
To study the stability of the equilibrium points, it is needful to check for simultaneous and consecutive occurrence of the two existing steady states. To investigate the coexistence of the monkeypox -free equilibrium and endemic equilibrium we implore theorem 4.1 in [22] and center manifold theory. Assuming that p and q represent the dynamic of the center manifold such that β * = β 1 is the bifurcation parameter with critical value at R m = 1, gives Observe that E 0 is a non -hyperbolic equilibrium, hence J E 0 has zero eigenvalue while others have negative real part. If w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 , w 9 ) T is the eigenvector corresponding to zero eigenvalues, we have

Let the associated left eigenvector that corresponds to the zero eigenvalue be
From theorem 4.1 in [22] we have that Since v 1 = v 6 = v 7 = v 9 = 0, the second order partial derivative of f 1 , f 6 , f 7 and f 9 will give zero, hence we obtain the second -order partial derivatives of f 2 , f 3 , f 4 , f 5 , and f 8 . We have that and all the other second -order derivatives gives a zero. Therefore, Observe that w 1 < 0 and µ Λ h −1 < 0, then p > 0 at β * = β 1 . From theorem 4.1 in [22], model (2.1) exhibits backward bifurcation at R m = 1 since p > 0 and q > 0. Epidemiologically, it means that factors that help in making R m < 1 are only necessary conditions but not sufficient for eradicating monkeypox in the community. Monkeypox occurs predominantly in remote villages of central and West Africa close to tropical rain forests which mounts a challenge of how to reduce the burden of the viral disease. As a result, the endemic equilibrium co-exists with the monkeypox-free equilibrium when R m < 1 which explains why the viral infection resurfaces in various countries when it appears that it has been eradicated.

NUMERICAL ANALYSIS
4.1. Sensitivity Analysis. Studying the transmission dynamics of monkeypox, it is observed that it has a reemergence tendency because it is a zoonotic viral disease that the main reservoir is yet to be identified and humans come in contact with animals daily either in the form of food, pet or game. We apply a normalized forward sensitivity index to the reproduction number to identify the parameter that is sensitive to the transmission and prevalence of the viral infection.
The elasticity sensitivity index of R m concerning the parameter p i is given by , The sensitivity index of our model is given below ( Table 2)     be taken to reduce exposure to Orthopoxvirus. It is observed in Figure 2 that when public awareness is intensified, coordinated, and carried out strictly; the transmission rate through unidentified infected persons will be reduced drastically because the community will be aware of all the flag-offs of MPXV which leads to proper guidance individually and collectively. Awareness also reduces the transmission through the isolated infected population as shown in Figure 3 because health workers and household members will apply standard infection control precautions in handling patients and the specimen that emanates from them. The awareness will also be extended to how humans come in contact with animals especially the sick or dead ones, meat that is not thoroughly cooked, blood, and other lesions because most human infections is a s a result of animals. As a control measure, the main objective of surveillance and case investigation is to swiftly identify cases and clusters to provide optimal clinical care; to isolate cases to avert further transmission; to identify, manage and follow up contacts to be able to recognize early signs of infection; to protect front-line health workers; to identify risk groups; to tailor effective control and prevention measures [15]. Figure 4 shows that increase of contact tracing, ρ; reduces the transmission rate cautioned by infected persons outside the radar and hence reduces the reproduction number. Case investigation will also help in containing the transmission of MPXV by increasing the number of patients that will be isolated and given clinical care as shown in Figure 5.  As case investigation and contact tracing are amplified, the human-to-human transmission will be reduced, and hence exposed individuals will be under surveillance regardless of associated symptoms of their absence, in other to categorize pre-symptomatic, pauci -symptomatic or asymptomatic infection as shown in Figure 6 and 7. In Figure 8 and 9 it is observed that when a greater population of the exposed individuals is under surveillance, the reproduction number reduces considerably which is a green -flag that the epidemic will be contained and eradicated in due time.
When these control measures are not in place, the transmission rate caused by unidentified infected individuals will make the containment of the infection difficult as shown in Figure 9.

Numerical Simulation.
In this section, the parameters of the model (2.1) are considered using secondary data and numerical simulation performed to support the analytical results. The values of parameters used in the simulation is shown in Table 1 with some estimated, assumed, calculated and obtained from literature, initial conditions used is follows: of specimens for monkeypox virus laboratory examination. This process when carried out dutifully will reduce the population of the unidentified infected individuals as shown in Figure 11, thereby localizing the transmission which can be easily contained; in addition, it will increase the population of the isolated infected patients' population as shown in Figure 12 where proper clinical care will be given to the patients. Coordinated surveillance makes sure that any contact with animal or human confirmed to have MPXV is under surveillance for a minimum of 21 days from the day of exposure, hence increase in the surveillance activities reduces the population of the exposed as shown in Figure   13; which means that the population of the isolated patients under clinical watch will increase as shown in Figure 14 and population of the unidentified infected patients will reduce as shown in Figure 15.    An increase in the level of surveillance will consequently lead to an upsurge of coordinated public awareness, case investigation, and contact tracing which will accidentally increase the population under surveillance, and hence the eradication of monkeypox in any community will be possible as shown in Figure 16. Since the monkeypox vaccine is still in the developmental stage in most countries, therefore proper surveillance and case investigation which will lead to early diagnosis and isolation are the most effective means of containing the outbreak of monkeypox at present. In addition to this control, strategic caution should be taken on how humans come in contact with animals without protection, and awareness should be directed to abattoirs, slaughters, herdsmen, and any other person that deals directly with animals on the outbreak and the need to be always vigilant of symptoms of monkeypox. Figure 17 shows that actions should be taken to reduce the number of infected animals in the community because they are the primary reservoir of which the main host is yet to be discovered. We also showed that when monkeypox is endemic, the system exhibits backward bifurcation of which we showed the condition for this existence.

CONCLUSION
Furthermore, a numerical analysis was conducted in which we used the standard sensitivity index to study the sensitivity of model parameters using the effective reproduction number as a threshold. We obtained that contact with infected humans who are out of the radar is what sustains the burden of MPXV but with coordinated contact tracing and awareness it can be contained because of the absence of a perfect vaccine at present. Numerical simulation was carried out to underscore the role of monkeypox awareness, surveillance, and contact tracing in the containment of the transmission dynamics of MPXV. Conclusively, proper surveillance and case investigation comprising of early diagnosis, isolation, clinical care, and contact tracing are strategic for effective control of the monkeypox outbreak at present.

AUTHOR CONTRIBUTIONS
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.