Optimal control model of human-to-human transmission of monkeypox virus

Joseph Ackora-Prah; Samuel Okyere; Ebenezer Bonyah; Atinuke Olusola Adebanji; Yaw Boateng

doi:10.12688/f1000research.130276.2

Home Browse Optimal control model of human-to-human transmission of monkeypox...

ALL Metrics

Views

Downloads

Get PDF

Get XML

Export

▬

✚

Research Article

Revised

Optimal control model of human-to-human transmission of monkeypox virus

[version 2; peer review: 1 approved, 1 approved with reservations, 1 not approved]

Joseph Ackora-Prah¹, Samuel Okyere ¹, Ebenezer Bonyah², Atinuke Olusola Adebanji³, Yaw Boateng¹

Joseph Ackora-Prah¹, Samuel Okyere ¹, [...] Ebenezer Bonyah², Atinuke Olusola Adebanji³, Yaw Boateng¹

PUBLISHED 29 Aug 2023

Author details Author details

¹ Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
² Mathematics Education, Akenten Appiah-Menka University of Skills Training and Enterpreneurial Development, Kumasi, Ghana
³ Statistics and Actuarial Science, Kwame Nkrumah University of Science and Technology, kumasi, Ghana

Joseph Ackora-Prah
Roles: Conceptualization, Methodology, Resources, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Samuel Okyere
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

Ebenezer Bonyah
Roles: Formal Analysis, Methodology, Resources, Supervision

Atinuke Olusola Adebanji
Roles: Conceptualization, Formal Analysis, Methodology, Supervision

Yaw Boateng
Roles: Conceptualization, Formal Analysis, Methodology, Writing – Original Draft Preparation

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Emerging Diseases and Outbreaks gateway.

This article is included in the Trends and Advances in Counteracting Mpox: A Global Public Health Emergency collection.

Abstract

Background: The number of monkeypox cases is rising globally, but it’s unclear how many instances there will be in the near future. The disease has been one of the major problems for sub-Saharan Africans in the past few years. This study seeks to suggest optimal strategies for curbing the disease in Ghana and preventing future occurrences.
Methods: A deterministic mathematical model incorporating optimal controls has been developed in this research to investigate the transmission of the monkeypox virus. The model’s fundamental properties such as positivity and boundedness of solution, and basic reproduction number have been examined. In order to assess the efficacy of two preventative control strategies—public education and vaccination—optimal controls were included in the model and Pontragyin’s maximum principle was used to characterize the model.
Results: The disease was observed to be not endemic with infection going extinct after ten days. The Monkeypox-related deaths were insignificant. The optimal strategies revealed that public education had less of an effect on those who were vulnerable than the vaccine control strategy. However, both approaches were successful in reducing the number of people who were exposed to the illness and reducing the number of fatalities. Vaccination reduced the number by 32.35% and public education by 50% at the peak of the exposed phase. Additionally, vaccination increases a person’s immunity, which speeds up their recovery.
Conclusions: A deterministic classical model incorporating optimal controls was proposed to study the monkeypox virus dynamics in a population. The disease is not endemic, which is explained by the model’s basic reproduction number, which was less than unity. Based on the findings of this study, we advise the use of both strategies in controlling the disease.

Keywords

Monkeypox, Equilibrium points, Sensitivity analysis, basic reproduction number, optimal control.

Corresponding author: Samuel Okyere

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2023 Ackora-Prah J et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Ackora-Prah J, Okyere S, Bonyah E et al. Optimal control model of human-to-human transmission of monkeypox virus [version 2; peer review: 1 approved, 1 approved with reservations, 1 not approved]. F1000Research 2023, 12:326 (https://doi.org/10.12688/f1000research.130276.2) First published: 23 Mar 2023, 12:326 (https://doi.org/10.12688/f1000research.130276.1) Latest published: 29 Aug 2023, 12:326 (https://doi.org/10.12688/f1000research.130276.2)

Revised Amendments from Version 1

A few typographical errors in version 1 have been corrected and the introduction of six new figures (Figures 2-7).

See the authors' detailed response to the review by Olumuyiwa James Peter
See the authors' detailed response to the review by Stephen Lasong

Introduction

The rare disease monkeypox is brought on by the monkeypox virus according to the Centers for Disease Control and Prevention (CDC). The variola virus, which causes smallpox, is related to the virus that causes monkeypox CDC. Although less severe, monkeypox symptoms are similar to those of smallpox, but it seldom causes fatalities as detailed by the World Health Organization (WHO). Despite being known as “monkeypox,” the disease’s genesis remains a mystery according to the CDC. However, the virus might spread from non-human animals like monkeys to humans.

A human case of monkeypox was reported by the WHO for the first time in 1970. In 1970, while smallpox eradication initiatives were intensifying, a nine-month-old infant in the Democratic Republic of the Congo became the first human to be identified as having monkeypox. In a number of central and western African countries, incidences of monkeypox were known from 1970 before the outbreak in 2022 as reported by WHO. Until recently, almost all cases of monkeypox in people living outside of Africa were attributed to either imported animals or international travel to places where the disease regularly occurs (CDC). These occurrences have been recorded over numerous continents (CDC). Direct contact with animals or other people who have the monkeypox virus can cause human infection.¹

The number of cases of monkeypox is rising globally as reported by (WHO), but it’s unclear how many instances there will be in the near future. At least 110 countries around the globe have already been identified to have the virus (CDC). Earlier in 2022, the disease started spreading in non-endemic regions at an unprecedented rate.² The Centers for Disease Control and Prevention (CDC) data reveals that more than 84,716 cases have been documented globally in the 110 countries, including 83,511 in 103 nations that have never before reported instances of monkeypox (CDC). The US, 29,980 cases is the greatest number of confirmed cases followed by Brazil with 10,625.³ As of January 16th, 2023, the Ghana Health Service (GHS) had identified 121 cases of the disease with 4 recorded deaths (CDC).

Historically, the disease has received little attention, which has made it difficult to understand how it spreads. Mathematical models has been used to study the dynamics of diseases in the past.⁴^–⁸ The transmission of the Cholera-COVID-19 co-infection in Yemen was studied using an ordinary differential equation model, according to Hezam et al.⁴ To reduce the number of cases in the community, they built the model with the best control strategies, including social isolation, lockdown, the number of tests, and the quantity of chlorine water tablets. To analyze the transmission of cholera, Tessema et al.⁵ devised a mathematical model that included drug resistance and immunization of newborns. The model included the best control strategies, including prevention, instruction, and treatment of sick people. In order to better understand the COVID-19 disease’s transmission dynamics in Senegal, Perkins and España⁷ proposed a mathematical model that included both of the disease’s primary therapeutic approaches: immunization of susceptible individuals and recovery/treatment of infected individuals. A model was proposed by Deressa and Duressa⁶ to investigate COVID-19 transmission in Ethiopia. The model was altered to include the best preventative methods, including treating hospitalized cases and promoting public health awareness. The authors in Ref. 8 proposed a model to study the co-dynamics of diabetes and COVID-19 in Ghana.

A few researchers including Okyere and Ackora-Prah⁹ have tried to analyze the monkeypox virus mathematically.⁹^–¹⁴ A mathematical model was developed by Somma et al.¹⁰ to investigate the spread of monkeypox throughout the human and rodent populations. According to the model proposed in Ref. 10, the human population is provided with a quarantine class and a public awareness campaign to prevent the spread of the disease. In order to restrict the spread of the monkeypox virus infection among humans and rodents, Usman and Adamu¹¹ devised a mathematical model that incorporates both vaccination and treatment interventions for the human population. Peter et al.¹² created and investigated a classical model for the spread of the monkeypox virus from people to rodents in Nigeria. Peter et al.¹³ developed both traditional and Caputo-Fabrizio fractional-order derivative models to investigate the disease’s spread in Nigeria. To examine the transmission of the monkeypox and smallpox viruses in the Democratic Republic of the Congo, Grant et al.¹⁴ devised a model. In our previous work in Ref. 9, a fractional-order model was formulated to study the disease dynamics in Ghana, taking into consideration individuals immune to the virus, however the model proposed in Ref. 9 fails to suggest preventive mechanisms to curtail the spread of the disease.

After looking over a number of academic publications, we discovered that very few research on the monkeypox virus and its mechanisms of transmission considered the virus’s capacity to propagate from person to person and that no previous research has taken into account the implications of individuals who have been deceased. Although there is evidence of virus transmission from human to human,⁹^,¹⁵^,¹⁶ which is the main mechanism of illness transmission in Ghana, current models of the disease concentrate on rodent-to-human transmission. This study extends our previous work done by Okyere and Ackora-Prah in Ref. 9 by introducing a compartment for the deceased individuals and incorporating optimal preventive methods in the proposed model.

The remainder of the paper is divided into the following sections: We extend a previous model proposed in Ref. 9 by two authors of this study and investigate the classical model in the methods section. We identify the model’s qualitative characteristics such as the positivity and boundedness of solutions, equilibrium points and basic reproduction number. We finally include optimal controls in the new model that has been developed in the optimal control section. We numerically solve the optimal control model.

Methods

We propose a deterministic model to study the viral transmittal of monkeypox by examining our previous fractional monkeypox model proposed in Ref. 9. Our previous model fails to suggest preventive mechanisms for curtailing the spread of the disease. In order to avoid the spread of the disease, this conventional model instead offers preventative control measures thereby excluding the immune compartment proposed in the previous model and introducing a compartment for individuals deceased of monkeypox virus. It was relevant to exclude the immune compartment as the recovered compartment serves it purposes in this new model. Only human-to-human virus transmission was taken into account in Ref. 9 and in this current model. The population is segmented into six compartments, including: Susceptible $(H_{S})$ , exposed $(H_{E})$ , infected $(H_{I})$ , hospitalised $(H_{Q})$ , recovered $(H_{R})$ , and individuals deceased from the disease $(D)$ . We let $N$ , denotes the total population. Therefore, $N (t) = H_{S} (t) + H_{E} (t) + H_{I} (t) + H_{Q} (t) + H_{R} (t) + D (t)$ .

The rate of new recruits into the vulnerable class is $ζ$ , whereas the rate of natural mortality is $ζ_{1}$ . Those who are vulnerable have a $ζ_{2}$ risk of contracting the sickness from those in $H_{I}$ . The recovery rates of a hospitalized person and infected patients are shown by the values $ζ_{6}$ and $ζ_{3}$ , respectively. The infectious rate and disease-induced death rate are denoted by $ε$ and $ζ_{5}$ , respectively. $ζ_{5}$ gives the rate of progression from the infected segment to the hospitalized segment. Figure 1 depicts the model’s flowchart.

Figure 1. The flow diagram of the extended monkeypox model.

Our modified model is described by the following ordinary differential equations.

(1)

\begin{array}{l} \frac{{dH}_{S}}{dt} = ζ - \frac{ζ_{2} H_{I} H_{S}}{N} - ζ_{1} H_{S}, \\ \frac{{dH}_{E}}{dt} = \frac{ζ_{2} H_{I} H_{S}}{N} - (ε + ζ_{1}) H_{E}, \\ \frac{{dH}_{I}}{dt} = ε H_{E} - (ζ_{3} + ζ_{4} + ζ_{1} + ζ_{5}) H_{I}, \\ \frac{{dH}_{Q}}{dt} = ζ_{4} H_{I} - (ζ_{1} + ζ_{5} + ζ_{6}) H_{Q} \\ \frac{{dH}_{R}}{dt} = ζ_{6} H_{Q} + ζ_{3} H_{I} - ζ_{1} H_{R}, \\ \frac{dD}{dt} = ζ_{5} (H_{Q} + H_{I}) . \end{array}

With initial conditions $H_{S} (0) \geq 0, H_{E} (0) \geq 0, H_{I} (0) \geq 0, H_{Q} (0) \geq 0, H_{R} (0) \geq 0, D (0) \geq 0 .$

Positivity and boundedness of solution

In order to prove that system (1) is mathematically and epidemiologically well-posed in a workable region $Φ$ , we present the results shown below.

(2)

Φ = \{(H_{S}, H_{E}, H_{I}, H_{Q}, H_{R}, D) \in R_{+}^{6} : 0 \leq N \leq \frac{ζ}{ζ_{1}}\}

Theorem 1

There is a domain 0 ≤ Φ < ∞ that contains and bounds the solution set (H_S, H_E, H_I, H_Q, H_R, D) of the system (1).⁸

Proof

Considering the solution set $(H_{S}, H_{E}, H_{I}, H_{Q}, H_{R}, D)$ with positive initial conditions

\{H_{S} (0), H_{E} (0), H_{I} (0), H_{Q} (0), H_{R} (0), D (0)\} \geq 0 .

We let, $N (t) = H_{S} (t) + H_{E} (t) + H_{I} (t) + H_{Q} (t) + H_{R} (t) + D (t)$ , then

N^{'} (t) = H_{S}^{'} (t) + H_{E}^{'} (t) + H_{I}^{'} (t) + H_{Q}^{'} (t) + H_{R}^{'} (t) + D^{'} (t) .

it follows that

N^{'} (t) < ζ - ζ_{1} N

The results of solving the differential inequalities are

N^{'} (t) \leq \frac{ζ}{ζ_{1}} (1 - e^{- ζ_{1} (t)}) + N (0) e^{- ζ_{1} (t)} .

Taking the limits as $t \to \infty$ gives $N^{'} (t) \leq \frac{ζ}{ζ_{1}} .$

To put it another way, every solution is constrained to the $Φ$ zone of feasibility. The system (1) solutions are now shown to be nonnegative in $Φ$ .

Theorem 2

For any $t > 0$ in the domain $Φ$ , the initial states must not be negative in order to remain such.⁸

Proof

We construct our argument based on the concepts from.¹³ Clearly, it is simple to understand that $H_{S} > 0$ , if not, for the condition that $t_{*} > 0$ such that $H_{S} (t_{*}) > 0$ and $H_{S}^{'} (t)$ $\leq 0$ for all $0 < t \leq t_{*}$

\frac{d}{dt} (H_{S} e^{(\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t}) = ζ e^{(\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t} .

By integrating from 0 to $t_{*}$ , we get the following:

H_{S} (t_{*}) e^{(\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t} - H_{S} (0) = \int_{0}^{t_{*}} ζ e^{(\frac{ζ_{2} H_{I}}{N} + ζ_{1}) τ} dτ .

The result is obtained by multiplying through by $e^{- (\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t}$ ,

$H_{S} (t_{*}) = H_{S} (0) e^{- (\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t} + e^{- (\frac{ζ_{2} H_{I}}{N} + ζ_{1}) t} [\int_{0}^{t_{*}} ζ e^{(\frac{ζ_{2} H_{I}}{N} + ζ_{1}) τ} dτ] > 0$ , which contradicts $H_{S} (t_{*}) = 0$ .

The following conclusions can be drawn from the remaining three (3) equations in system (1).

H_{E} (t_{*}) = e^{- (ε + ζ_{1}) t} (H_{E} (0) + [\int_{0}^{t_{*}} ζ_{2} \frac{H_{S} (τ) H_{I} (τ)}{N (τ)} e^{(ε + ζ_{1}) τ} d τ]) > 0,

H_{I} (t_{*}) = e^{- (ζ_{1} + ζ_{3} + ζ_{4} + ζ_{5}) t} (H_{I} (0) + [\int_{0}^{t_{*}} ε H_{E} (τ) e^{(ζ_{1} + ζ_{3} + ζ_{4} + ζ_{5}) τ} d τ]) > 0,

H_{Q} (t_{*}) = e^{- (ζ_{1} + ζ_{6} + ζ_{5}) t} (H_{Q} (0) + [\int_{0}^{t_{*}} ζ_{4} H_{I} (τ) e^{(ζ_{1} + ζ_{6} + ζ_{5}) τ} d τ]) > 0,

H_{R} (t_{*}) = H_{R} (0) e^{- ζ_{1} t} [\int_{0}^{t_{*}} [ζ_{3} H_{I} (τ) + ζ_{6} H_{Q} (τ)] e^{ζ_{1} τ} d τ] > 0,

which goes against the logic $H_{E} (t_{*}) = H_{I} (t_{*}) = H_{Q} (t_{*}) = H_{R} (t_{*}) = 0$ . This concludes the proof.

Steady states of the model

The monkeypox-free and endemic equilibrium point given by the model in Ref. 9 are

(3)

Monkeypox‐free = (H_{S}^{0}, H_{E}^{0}, H_{I}^{0}, H_{Q}^{0}, H_{R}^{0}) = (\frac{ζ}{ζ_{1}}, 0, 0, 0, 0)

The endemic equilibrium point, $ee = (H_{S}^{*}, H_{E}^{*}, H_{I}^{*}, H_{Q}^{*}, H_{R}^{*})$ where,

(4)

\begin{array}{l} H_{S}^{*} = \frac{ζN}{ζ_{2} H_{I}^{*} + ζ_{1} N}, H_{E}^{*} = ζ_{2} \frac{H_{S}^{*} H_{I}^{*}}{(ε + ζ_{1}) N}, H_{I}^{*} = \frac{ε H_{E}^{*}}{(ζ_{3} + ζ_{4} + ζ_{1} + ζ_{5})}, \\ H_{Q}^{*} = \frac{ζ_{4} H_{I}^{*}}{(ζ_{6} + ζ_{1} + ζ_{5})}, H_{R}^{*} = \frac{ζ_{6} H_{Q}^{*} + ζ_{3} H_{I}^{*}}{ζ_{1}} . \end{array}

The basic reproduction number $(R_{0})$ is the key element that determines whether or not a disease will proliferate throughout a community. The $R_{0}$ as computed in Ref. 9, is

(5)

R_{0} = \frac{ε ζ_{2}}{(ζ_{3} + ζ_{4} + ζ_{1} + ζ_{5}) (ε + ζ_{1})} .

Optimal control

In this section, we introduce the control $u (t)$ into the system (1), where control $u_{1} (t)$ stands for public education on monkeypox and control $u_{2} (t)$ stands for vaccination/immunization of those who are vulnerable. By incorporating time-dependent controls onto system (1), we are able to produce

(6)

\begin{array}{l} \frac{{dH}_{S}}{dt} = ζ - (1 - u_{1}) \frac{ζ_{2} H_{I} H_{S}}{N} - (ζ_{1} + u_{2}) H_{S}, \\ \frac{{dH}_{E}}{dt} = (1 - u_{1}) \frac{ζ_{2} H_{I} H_{S}}{N} - (ε + ζ_{1}) H_{E}, \\ \frac{{dH}_{I}}{dt} = ε H_{E} - (ζ_{3} + ζ_{4} + ζ_{1} + ζ_{5}) H_{I}, \\ \frac{{dH}_{Q}}{dt} = ζ_{4} H_{I} - (ζ_{1} + ζ_{5} + ζ_{6}) H_{Q} \\ \frac{{dH}_{R}}{dt} = ζ_{6} H_{Q} + ζ_{3} H_{I} - ζ_{1} H_{R} + u_{2} H_{S} \end{array}

Analysis of the optimal control model

Pontragyin’s maximal principles are used to examine the behaviour of the system (6). The objective function for a fixed time $t_{f}$ is given by

(7)

J (u_{1}, u_{2}) = \int_{0}^{t_{f}} [b_{1} H_{S} + b_{2} H_{E} + b_{3} H_{I} + b_{4} H_{Q} + c_{1} u_{1}^{2} + c_{2} u_{2}^{2}] dt,

where $t_{f}$ is the control’s final time and the parameters $b_{1}$ , $b_{2}$ , $b_{3}$ , $b_{4}$ , $c_{1}$ and $c_{2}$ indicate the balancing cost factors for the two controls, respectively. Therefore, we investigate to find the best controls $u_{1}^{*}$ and $u_{2}^{*}$ so that

(8)

J (u_{1}^{*}, u_{2}^{*}) = min_{u_{1}, u_{2} \in U} J (u_{1}, u_{2}),

The Pontryagin maximum concept, which is described in Ref. 18, was utilized to determine the prerequisite for the ideal control. This idea transforms the (6), (7) and (8) into a problem of minimizing a Hamiltonian, H, which is described by

(9)

\begin{array}{l} H = b_{1} H_{S} (t) + b_{2} H_{E} (t) + b_{3} H_{I} (t) + b_{4} H_{Q} (t) + \frac{1}{2} (c_{1} u_{1}^{2} + c_{2}^{2}) \\ + Λ_{H_{S}} \{ζ - (1 - u_{1}) ζ_{2} \frac{H_{I} H_{S}}{N} - ζ_{1} H_{S} - u_{2} H_{S}\} \\ + Λ_{H_{E}} \{(1 - u_{1}) ζ_{2} \frac{H_{I} H_{S}}{N} - (ε + ζ_{1}) H_{E}\} \\ + Λ_{H_{I}} \{ε H_{E} - (ζ_{1} + ζ_{3} + ζ_{4} + ζ_{5}) H_{I}\} \\ + Λ_{H_{Q}} \{ζ_{4} H_{I} - (ζ_{1} + ζ_{6} + ζ_{5}) H_{Q}\} \\ + Λ_{H_{R}} \{ζ_{3} H_{I} + ζ_{6} H_{Q} - ζ_{1} H_{R}\}, \end{array}

where the adjoint or costate variables are represented by $Λ_{H_{S}}, Λ_{H_{E}}, Λ_{H_{I}}, Λ_{H_{Q}}, Λ_{H_{R}}$ , represents the adjoint variables or costate variables. By considering the correct partial derivatives of system (9) with regard to the related state variables, the system of equations is generated.

Theorem 3

Given an optimal control $(u_{1}^{*}, u_{2}^{*})$ and corresponding solution $H_{S}^{*}, H_{E}^{*}, H_{I}^{*}, H_{Q}^{*}, H_{R}^{*}$ that minimizes $J (u_{1}, u_{2})$ over U, there exist adjoint variables $Λ_{H_{S}}, Λ_{H_{E}}, Λ_{H_{I}}, Λ_{H_{Q}}, Λ_{H_{R}}$ , satisfying

(10)

- \frac{d Λ_{i}}{dt} = \frac{\partial H}{\partial i},

where

i = H_{S}, H_{E}, H_{I}, H_{Q}, H_{R}

, with the transversality conditions

Λ_{H_{S}} (t_{f}) = Λ_{H_{E}} (t_{f}) = Λ_{H_{I}} (t_{f}) = Λ_{H_{Q}} (t_{f}) = Λ_{H_{R}} (t_{f}) = 0 .

Proof

The right hand side differentiation of the system (9) calculated at the optimal control is taken into consideration to produce the differential equation described by the adjoint variables. The obtained adjoint equations are presented as

(11)

\begin{array}{l} \frac{d Λ_{H_{S}}}{dt} = - b_{1} + \frac{ζ_{2} H_{I}}{N} (1 - u_{1}) [Λ_{H_{S}} - Λ_{H_{E}}] + (ζ_{1} + u_{2}) Λ_{H_{S}}, \\ \frac{d Λ_{H_{E}}}{dt} = - b_{2} + (ζ_{1} + ε) Λ_{H_{E}} - ε Λ_{H_{I}}, \\ \frac{d Λ_{H_{I}}}{dt} = - b_{3} + \frac{ζ_{2} H_{S}}{N} (1 - u_{1}) [Λ_{H_{S}} - Λ_{H_{E}}] + (ζ_{3} + ζ_{4} + ζ_{5} + ζ_{1}) Λ_{H_{I}} - ζ_{3} Λ_{H_{R}} - ζ_{4} λ_{H_{Q}}, \\ \frac{d Λ_{H_{Q}}}{dt} = - b_{4} + (ζ_{6} + ζ_{5} + ζ_{1}) Λ_{H_{Q}} - ζ_{6} Λ_{H_{R}}, \\ \frac{d Λ_{H_{R}}}{dt} = ζ_{1} Λ_{H_{R}} . \end{array}

Obtaining the solution for $u_{1}^{*}$ and $u_{2}^{*}$ subject to the constraints gives,

(12)

\begin{array}{l} 0 = \frac{\partial H}{\partial u_{1}} = - c_{1} u_{1} + \frac{ζ_{2} H_{I} H_{S}}{N} [Λ_{H_{E}} - Λ_{H_{S}}], \\ 0 = \frac{\partial H}{\partial u_{2}} = - c_{2} u_{2} + H_{S} [Λ_{H_{S}} - Λ_{H_{R}}] . \end{array}

This gives

(13)

\begin{array}{l} u_{1}^{*} = min (1, max (0, \frac{\frac{ζ_{2} H_{I} H_{S}}{N} [Λ_{H_{E}} - Λ_{H_{S}}]}{c_{1}})), \\ u_{2}^{*} = min (1, max (0, \frac{H_{S} [Λ_{H_{S}} - Λ_{H_{R}}]}{c_{2}})) . \end{array}

Numerical analysis of the model

In this part, we numerically analyze the behavior of the system (1) and (6) optimum control model and display the effects of varying the controls $u_{1}$ and $u_{2}$ . We consider the deceased compartment for the purposes of the numerical simulations. In the first instance, using the parameter values listed in Table 1. The outcome of using the parameter values from Table 1 to evaluate the $R_{0}$ gives $R_{0} = 0.1940$ , proving that the illness is not endemic to the country. We begin our simulation on May 24, 2022, the day the first five cases of monkeypox are confirmed by the Ghana Health Service. According to the Ghana Statistical Service, in the 2021 population and housing census, there are 30.8 million people living in Ghana. Our model is consistent and includes the entire population under study (N = 30.8 million). We begin with $H_{S} (0) = 30799995, H_{I} (0) = 5, H_{E} (0) = H_{Q} (0) = H_{R} (0) = D (0) = 0$ . The outcomes of the MATLAB (version R2016a) (RRID:SCR_001622)¹⁹^–²¹ (Python (RRID:SCR_008394) could potentially be used as an open alternative using similar methods described in this manuscript) ODE45 using the parameter values and the initial conditions are shown in Figures 2–25.

Table 1. Description of the parameters and values.

Parameters	Description	Estimate (per year)	Source
$ζ$	Recruitment of vulnerable people	$2908 \times 10^{- 2}$	⁸^,⁹
$ζ_{1}$	Natural mortality rate	$0.4252912 \times 10^{- 4}$	⁸^,⁹
$ζ_{2}$	Contact incidence between infected and vulnerable people	$0.22325 \times 10^{- 1}$	¹³
$ζ_{3}$	The frequency of natural immunity-induced recovery in sick people	$0.88366 \times 10^{- 1}$	⁹^,¹³
$ζ_{4}$	The probability at which infected people get very ill	$0.5$	⁹^,¹³
$ε$	The frequency at which an exposed person contracts an infection	$0.16744 \times 10^{- 1}$	⁹^,¹³
$ζ_{5}$	Deaths from the sickness caused by monkeypox	$0.3286 \times 10^{- 2}$	⁹^,¹³
$ζ_{6}$	The pace of recovery for people in critical condition	$0.36246 \times 10^{- 1}$	⁹^,¹³

The susceptible compartment as depicted by Figure 2, shows a decline in the number of susceptible individuals with time.

Figure 2. Dynamics of the susceptible compartment.

The exposed compartment as depicted by Figure 3, is seen to reach an early peak in the first ten days, then declines sharply with time.

Figure 3. Dynamics of individuals exposed to the monkeypox.

The number of infected individuals is seen to extinct in the first ten days (see Figure 4) and this is explained by the basic reproduction number computed to be 0.1940.

Figure 4. Dynamics of the infected compartment.

The hospitalized compartment depicted by Figure 5, is seen to show an early peak in the first ten days and decline sharply, however the individuals hospitalized does not extinct at the end of the period.

Figure 5. Dynamics of the hospitalised compartment.

The number of recoveries increases exponentially with time. This is depicted by Figure 6. The marginal number of recoveries is accounted for as a result of the low infected and hospitalized cases.

Figure 6. Dynamics of the recovered compartment.

The deceased dynamics is depicted by Figure 7. The model shows insignificant number of individuals deceased and this is not strange as the country recorded no Monkeypox-related deaths.

Figure 7. Dynamics of the deceased compartment.

Control strategy I (public education)

Setting control $u_{2} = 0$ and varying control $u_{1}$ , the results obtained are displayed in Figures 8–13. This strategy was ineffective on the susceptible, infected and hospitalised compartments (see Figures 8, 10 and 11 respectively). However, the strategy effectively reduced the number of individuals exposed to the virus (see Figure 9). There were a marginal reduction in the number of recoveries and individuals deceased at the end of the period (see Figures 12 and 13).

Figure 8. Behaviour of the susceptible with optimal control.

Figure 9. Behaviour of the exposed individuals with optimal control.

Figure 10. Behaviour of the infected with optimal control.

Figure 11. Behaviour of the hospitalised individuals with optimal control.

Figure 12. Behaviour of the recovered individuals with optimal control.

Figure 13. Behaviour of the deceased with optimal control.

Control strategy II (immunization)

The results obtained by setting control $u_{1} = 0$ and varying control $u_{2}$ , are shown in Figures 14-19. Figure 14 and Figure 15 depicts a decline in the number of susceptible and exposed individuals respectively. The vaccination control led to a marginal decrease in the number of infected, hospitalized and deceased individuals (see Figures 16, 17 and 19). Recoveries increases as people get permanent immunity through vaccination (see Figure 18).

Figure 14. Behaviour of the susceptible with optimal control.

Figure 15. Behaviour of the exposed individuals with optimal control.

Figure 16. Behaviour of the infected with optimal control.

Figure 17. Behaviour of the hospitalised individuals with optimal control.

Figure 18. Behaviour of the recovered individuals with optimal control.

Figure 19. Behaviour of the deceased with optimal control.

Strategy III (public education vs immunization)

We set $u_{1} \neq 0$ and $u_{2} \neq 0$ and compare the effect of the two control measures on the system (1). The results obtained are given as Figures 20–25. Vaccination strategy was the only effective control measure in curbing vulnerability of the population (see Figure 20). The two strategies were effective in bring down the number of exposed individuals (see Figure 21). However, they were ineffective in the infected, hospitalised and deceased classes (see Figures 22, 23 and 25). Greater number of recoveries with vaccination strategy unlike the public education (see Figure 24).

Figure 20. Behaviour of the susceptible with optimal control.

Figure 21. Behaviour of the exposed individuals with optimal control.

Figure 22. Behaviour of the infected individuals with optimal control.

Figure 23. Behaviour of the hospitalised individuals with optimal control.

Figure 24. Behaviour of the recovered individuals with optimal control.

Figure 25. Behaviour of the deceased compartment with optimal control.

Conclusion

To better understand how the monkeypox virus spreads, a deterministic model has been put forth. The disease is not endemic, which is explained by the model’s basic reproduction number, which was less than unity and an extinction of the disease in the first ten days and an insignificant monkey pox disease related deaths. In order to assess the efficacy of two preventative control strategies–public education and vaccination–optimal controls were included in the model. Public education was found to have less of an effect on those who were vulnerable than vaccine control. However, both approaches were successful in reducing the number of people who were exposed to the illness. Vaccination reduced number by $32.35 %$ and public education by $50 %$ at the peak of the exposed phase. Additionally, vaccination increases a person’s immunity, which speeds up their recovery. Both strategies had a minor impact on the number of fatalities during the course of the time period. Based on the findings of this study, we advise the use of both strategies in controlling the disease.

Data availability

Underlying data

OSF: Raw_data_monkeypox. https://doi.org/10.17605/OSF.IO/69RYP.¹⁹

This project contains the following underlying data:

Raw data.docx (data input into matlab simulations)

Raw data.pdf (data input into matlab simulations)

Extended data

OSF: Monkeypox_data. https://doi.org/10.17605/OSF.IO/ZQ65J.²⁰

This project contains the following extended data:

MATLAB Command Window.pdf (Matlab output data that accompanied the numerical simulation figures)

Data are available under the terms of the Creative Commons Zero “No rights reserved” data waiver (CC0 1.0 Public domain dedication).

Software availability

Source code available from: https://github.com/okyere2015/Matlab_codes/tree/v1.0.0

Archived source code at the time of publication: https://doi.org/10.5281/zenodo.7603824.²¹

License: Apache 2.0

References

1. Hammarlund E, Lewis MW, Carter SV, et al.: Multiple diagnostic techniques identify previously vaccinated individuals with protective immunity against monkeypox. Nat. Med. 2005; 11(9): 1005–1011. PubMed Abstract | Publisher Full Text
2. Kalyar F, Chen X, Kunasekaran M, et al.: An unprecedented global resurgence of monkeypox. Global Biosecurity. 2022; 4. Publisher Full Text
3. Alah MA, Abdeen S, Tayar E, et al.: The story behind the first few cases of monkeypox infection in non-endemic countries. J. Infect. Public Health. 2022; Volume 15(Issue 9): Pages 970–974. 1876-0341.
4. Hezam IM, Foul A, Alrasheedi A: A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen. Adv. Differ. Equ. 2021; 2021: 108. PubMed Abstract | Publisher Full Text | Free Full Text
5. Tessema FS, Koya PR, Bole BK: Optimal Control and Cost-Effectiveness Analysis of Cholera with Vaccination. J. Math. 2022; 2022: 1–16. Publisher Full Text
6. Deressa CT, Duressa GF: Modeling and optimal control analysis of transmission dynamics of COVID-19: the case of Ethiopia. Alex. Eng. J. 2021; 60(1): 719–732. Publisher Full Text
7. Perkins A, España G: Optimal control of the COVID-19 pandemic with nonpharmaceutical interventions. Bull. Math. Biol. 2020; 82(9): 118. PubMed Abstract | Publisher Full Text | Free Full Text
8. Okyere S, Ackora-Prah J: A mathematical model of transmission dynamics of SARS-CoV-2 (COVID-19) with an underlying condition of diabetes. Int. J. Math. Math. Sci. 2022; 2022: 1–15. Publisher Full Text
9. Okyere S, Ackora-Prah J: Modelling and analysis of monkeypox disease using fractional derivatives. Results Eng. 2023; Vol. 17: 100786. 2590–1230. PubMed Abstract | Publisher Full Text | Free Full Text
10. Somma SA, Akinwade NI, Chado UD: A mathematical model of monkey pox virus transmission dynamics. Ife Journal of Science, AJOA. 2019; 21(1). elSSN: 0794-4896.
11. Usman S, Adamu I: Modeling the Transmission Dynamics of the Monkeypox Virus Infection with Treatment and Vaccination Interventions. J. Appl. Math. Phys. 2017; 5(12). Reference Source
12. Peter OJ, Kumar S, Kumari N, et al.: Transmission dynamics of Monkeypox virus: a mathematical modelling approach. Model. Earth Syst. Environ. 2021; 8: 3423–3434. PubMed Abstract | Publisher Full Text | Free Full Text
13. Peter OJ, Oguntolu FA, Ojo MM, et al.: Fractional order mathematical model of monkeypox transmission dynamics. Phys. Scr. 2022; 97(8): 084005. Publisher Full Text
14. Grant R, Nguyen LL, Breban R: Modelling human-to-human transmission of monkeypox. Bull. World Health Organ. 2020; 98(9): 638–640. PubMed Abstract | Publisher Full Text | Free Full Text
15. Vaughan A, Aarons E, Astbury J, et al.: Human-to-Human Transmission of Monkeypox Virus, United Kingdom, October 2018. Emerg. Infect. Dis. 2020; 26(4): 782–785. PubMed Abstract | Publisher Full Text | Free Full Text
16. Thornhill JP, Barkati S, Walmsley S, et al.: Monkeypox virus infection in humans across 16 countries – April–June 2022. N. Engl. J. Med. 2022; 387: 679–691. PubMed Abstract | Publisher Full Text
17. Castillo-Chavez C, Blower S, van den Driessche P , et al.: Mathematical approach for emerging and reemerging infectious diseases. New York: Springer Verlag.
18. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, et al.: The mathematical theory of optimal processes. New York/London: Wiley and sons; 1963; vol. VIII +360S. . Publisher Full Text
19. Okyere S: Raw_data_monkeypox. [Data]. OSF. 2023 February 14. Publisher Full Text
20. Okyere S: Monkeypox_data. [Data]. OSF. 2023 February 14. Publisher Full Text
21. okyere2015: okyere2015/Matlab_codes: Matlab codes for monkeypox deterministic models (v1.0.0). [Code] Zenodo. 2023. Publisher Full Text

Comments on this article Comments (0)

Version 2

VERSION 2 PUBLISHED 23 Mar 2023

Author details Author details

Joseph Ackora-Prah
Roles: Conceptualization, Methodology, Resources, Supervision, Writing – Original Draft Preparation, Writing – Review & Editing

Samuel Okyere
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Validation, Writing – Original Draft Preparation, Writing – Review & Editing

Ebenezer Bonyah
Roles: Formal Analysis, Methodology, Resources, Supervision

Atinuke Olusola Adebanji
Roles: Conceptualization, Formal Analysis, Methodology, Supervision

Yaw Boateng
Roles: Conceptualization, Formal Analysis, Methodology, Writing – Original Draft Preparation

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (2)

version 2

Revised

Published: 29 Aug 2023, 12:326

https://doi.org/10.12688/f1000research.130276.2

version 1

Published: 23 Mar 2023, 12:326

https://doi.org/10.12688/f1000research.130276.1

© 2023 Ackora-Prah J et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Download

Export To

metrics

	Views	Downloads
F1000Research	-	-
PubMed Central Data from PMC are received and updated monthly.	-	-

Citations

SEE MORE DETAILS

CITE

how to cite this article

Ackora-Prah J, Okyere S, Bonyah E et al. Optimal control model of human-to-human transmission of monkeypox virus [version 2; peer review: 1 approved, 1 approved with reservations, 1 not approved]. F1000Research 2023, 12:326 (https://doi.org/10.12688/f1000research.130276.2)

NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article.

track

receive updates on this article

Track an article to receive email alerts on any updates to this article.

Open Peer Review

Current Reviewer Status: ?

Key to Reviewer Statuses VIEW HIDE

ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions

Version 2

VERSION 2

PUBLISHED 29 Aug 2023

Revised

Views

Reviewer Report 24 Nov 2023

Olumuyiwa James Peter, Department Mathematical and Computer Sciences, University of Medical Sciences, Ondo, Nigeria

Approved

https://doi.org/10.5256/f1000research.154608.r201250

The authors have not attended to all the issues raised; authors ... Continue reading

CITE

Report a concern

Respond or Comment

Views

Reviewer Report 23 Nov 2023

Kristan Alexander Schneider, Department of Applied Computer- and Biosciences, University of Applied Sciences Mittweida, Mittweida, Germany

Girma Mesfin Zelleke, Department of Mathematics, University of Buea, Buea, Cameroon

Martin Eichner, Institute for Clinical Epidemiology and Applied Biometrics, University of Tübingen, Tübingen, Germany

Not Approved

https://doi.org/10.5256/f1000research.154608.r203226

Please find our report attached here since it is necessary to show some formulae, as well as a Julia script (attached here) with an implementation of the model, showing that Theorem 1 of the article is incorrect.

References
1. Diekmann O, ... Continue reading

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

1. Diekmann O, Heesterbeek JA, Metz JA: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations.J Math Biol. 1990; 28 (4): 365-82 PubMed Abstract | Publisher Full Text
2. Schneider KA, Eichner M: Does it matter who is spreading monkeypox?. Lancet Infect Dis. 2022; 22 (9): 1266-1267 PubMed Abstract | Publisher Full Text
3. Mizushima D, Shintani Y, Takano M, Shiojiri D, et al.: Prevalence of Asymptomatic Mpox among Men Who Have Sex with Men, Japan, January-March 2023.Emerg Infect Dis. 2023; 29 (9): 1872-1876 PubMed Abstract | Publisher Full Text
4. Sanz-Muñoz I, Sánchez-dePrada L, Sánchez-Martínez J, Rojo-Rello S, et al.: Possible Mpox Protection from Smallpox Vaccine–Generated Antibodies among Older Adults. Emerging Infectious Diseases. 2023; 29 (3): 656-658 Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematical modelling, epidemiology, bio-statistics, global health

We confirm that we have read this submission and believe that we have an appropriate level of expertise to state that we do not consider it to be of an acceptable scientific standard, for reasons outlined above.

CITE

Report a concern

Respond or Comment

Version 1

VERSION 1

PUBLISHED 23 Mar 2023

Views

Reviewer Report 08 Aug 2023

Stephen Lasong, Kumasi Technical University, Kumasi, Ghana

Approved with Reservations

https://doi.org/10.5256/f1000research.143020.r193971

The manuscript is good. The findings are new, accurate, and comprehensive. The procedures are clearly laid out. The findings fully support the discussion and conclusions. The writing is decent. There are corrections that need to be done:

$N(t)\leq \frac{\zeta}{\zeta_{1}}(1-e^{-\zeta_{1}(t)})+N(0)e^{-\zeta_{1}(t)}$

not

$N'(t)\leq \frac{\zeta}{\zeta_{1}}(1-e^{-\zeta_{1}(t)})+N(0)e^{-\zeta_{1}(t)}$
The same applies to t approaches infinity.
Check equation 6: $\frac{dH_{I}}{dt}$ not $\frac{dH_{A}}{dt}$
Check equation 11:

$\frac{d\Lambda_{H_{Q}}}{dt} = -b_{4} + (\zeta_{6} + \zeta_{1} + \zeta_{5})\Lambda_{H_{Q}}-\zeta _{6} \Lambda _{H_{R}}$

not

$\frac{d\Lambda_{H_{Q}}}{dt} = -b_{4} + (\zeta_{6} + \zeta_{1} + \zeta_{5})\Lambda_{H_{Q}}-\zeta _{3}\Lambda _{H_{R}}$

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematical Modeling

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

CITE

Report a concern

Author Response 29 Aug 2023

Samuel Okyere, Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

29 Aug 2023

Author Response

Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Competing Interests: No competing interest
Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Competing Interests: No competing interest Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 29 Aug 2023

Samuel Okyere, Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

29 Aug 2023

Author Response

Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Competing Interests: No competing interest
Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Corrections have been rectified in the revised version, however, the first equation is still valid and not the one suggested by the Reviewer.
Competing Interests: No competing interest Close
Report a concern

Views

Reviewer Report 20 Apr 2023

Olumuyiwa James Peter, Department Mathematical and Computer Sciences, University of Medical Sciences, Ondo, Nigeria

Approved with Reservations

https://doi.org/10.5256/f1000research.143020.r168328

The article investigates an optimal control model of human-to-human transmission of monkeypox virus. This work has potential and my comments are as follows:

Please add the main findings and objective of the current study in the

The article investigates an optimal control model of human-to-human transmission of monkeypox virus. This work has potential and my comments are as follows:

Please add the main findings and objective of the current study in the abstract.
What are the benchmark cases in your study?
What are the special cases of your study?
Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations.
Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
For enhancing the introduction section please include the following recent studies on monkeypox models: Peter et al. (2023¹) and Peter et al. (2022²).

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Peter O, Abidemi A, Ojo M, Ayoola T: Mathematical model and analysis of monkeypox with control strategies. The European Physical Journal Plus. 2023; 138 (3). Publisher Full Text
2. Peter O, Madubueze C, Ojo M, Oguntolu F, et al.: Modeling and optimal control of monkeypox with cost-effective strategies. Modeling Earth Systems and Environment. 2022. Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Mathematical Biology, Infectious Disease Modelling

CITE

Report a concern

Author Response 29 Aug 2023

Samuel Okyere, Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

29 Aug 2023

Author Response

1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have ... Continue reading 1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have been included in the results of the abstract.

2. What are the benchmark cases in your study?
The study extends and analyses the monkeypox model proposed in Ref. 9, which also takes inspiration from the study in Ref. 12 and 13 and suggests intervention strategies in curbing the disease in Ghana and preventing future occurrences.

3. What are the special cases of your study?
The study reveals an insignificant number of monkeypox-related deaths in Ghana. It also reveals the disease getting extinct in the first two weeks of the outbreak. The study reveals the effectiveness of the two control measures with each strategy leading to a greater percentage reduction in the number of exposed individuals.

4. Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations. All equations and theorems have been referenced with the exception of the model formulated by the authors. Punctuations and corrections have been done.

5. Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
This has been done with the introduction of even new figures to give more insight into the model.
1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have been included in the results of the abstract.

2. What are the benchmark cases in your study?
The study extends and analyses the monkeypox model proposed in Ref. 9, which also takes inspiration from the study in Ref. 12 and 13 and suggests intervention strategies in curbing the disease in Ghana and preventing future occurrences.

3. What are the special cases of your study?
The study reveals an insignificant number of monkeypox-related deaths in Ghana. It also reveals the disease getting extinct in the first two weeks of the outbreak. The study reveals the effectiveness of the two control measures with each strategy leading to a greater percentage reduction in the number of exposed individuals.

4. Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations. All equations and theorems have been referenced with the exception of the model formulated by the authors. Punctuations and corrections have been done.

5. Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
This has been done with the introduction of even new figures to give more insight into the model.
Competing Interests: These two articles (Peter et al. (2023, 1) and Peter et al. (2022, 2)) cannot be included in the manuscript as they may constitute a potential conflict of interest. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 29 Aug 2023

Samuel Okyere, Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

29 Aug 2023

Author Response

1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have ... Continue reading 1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have been included in the results of the abstract.

2. What are the benchmark cases in your study?
The study extends and analyses the monkeypox model proposed in Ref. 9, which also takes inspiration from the study in Ref. 12 and 13 and suggests intervention strategies in curbing the disease in Ghana and preventing future occurrences.

3. What are the special cases of your study?
The study reveals an insignificant number of monkeypox-related deaths in Ghana. It also reveals the disease getting extinct in the first two weeks of the outbreak. The study reveals the effectiveness of the two control measures with each strategy leading to a greater percentage reduction in the number of exposed individuals.

4. Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations. All equations and theorems have been referenced with the exception of the model formulated by the authors. Punctuations and corrections have been done.

5. Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
This has been done with the introduction of even new figures to give more insight into the model.
1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have been included in the results of the abstract.

2. What are the benchmark cases in your study?
The study extends and analyses the monkeypox model proposed in Ref. 9, which also takes inspiration from the study in Ref. 12 and 13 and suggests intervention strategies in curbing the disease in Ghana and preventing future occurrences.

3. What are the special cases of your study?
The study reveals an insignificant number of monkeypox-related deaths in Ghana. It also reveals the disease getting extinct in the first two weeks of the outbreak. The study reveals the effectiveness of the two control measures with each strategy leading to a greater percentage reduction in the number of exposed individuals.

4. Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations. All equations and theorems have been referenced with the exception of the model formulated by the authors. Punctuations and corrections have been done.

5. Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
This has been done with the introduction of even new figures to give more insight into the model.
Competing Interests: These two articles (Peter et al. (2023, 1) and Peter et al. (2022, 2)) cannot be included in the manuscript as they may constitute a potential conflict of interest. Close
Report a concern

Comments on this article Comments (0)

Version 2

VERSION 2 PUBLISHED 23 Mar 2023

Open Peer Review

Reviewer Status

Reviewer Reports

	Invited Reviewers
	1	2	3
Version 2 (revision) 29 Aug 23	read		read
Version 1 23 Mar 23	read	read

Olumuyiwa James Peter, University of Medical Sciences, Ondo, Nigeria
Stephen Lasong, Kumasi Technical University, Kumasi, Ghana
Kristan Alexander Schneider, University of Applied Sciences Mittweida, Mittweida, Germany

Girma Mesfin Zelleke, University of Buea, Buea, Cameroon

Martin Eichner, University of Tübingen, Tübingen, Germany

Comments on this article

All Comments(0)

Add a comment

Browse by related subjects

Back to all reports

Reviewer Report

6 Views

24 Nov 2023 | for Version 2

Olumuyiwa James Peter, Department Mathematical and Computer Sciences, University of Medical Sciences, Ondo, Nigeria

6 Views Cite this report Responses(0)

Approved

The authors have not attended to all the issues raised; authors are advised to reference all the recent papers suggested on mpox models.

Competing Interests

No competing interests were disclosed.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

Responses (0)

Back to all reports

Reviewer Report

24 Views

23 Nov 2023 | for Version 2

Kristan Alexander Schneider, Department of Applied Computer- and Biosciences, University of Applied Sciences Mittweida, Mittweida, Germany

Girma Mesfin Zelleke, Department of Mathematics, University of Buea, Buea, Cameroon

Martin Eichner, Institute for Clinical Epidemiology and Applied Biometrics, University of Tübingen, Tübingen, Germany

24 Views Cite this report Responses(0)

Not Approved

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

No

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematical modelling, epidemiology, bio-statistics, global health

Respond to this report

Responses (0)

Back to all reports

Reviewer Report

14 Views

08 Aug 2023 | for Version 1

Stephen Lasong, Kumasi Technical University, Kumasi, Ghana

14 Views Cite this report Responses(1)

Approved With Reservations

$N(t)\leq \frac{\zeta}{\zeta_{1}}(1-e^{-\zeta_{1}(t)})+N(0)e^{-\zeta_{1}(t)}$

not

$N'(t)\leq \frac{\zeta}{\zeta_{1}}(1-e^{-\zeta_{1}(t)})+N(0)e^{-\zeta_{1}(t)}$
The same applies to t approaches infinity.
Check equation 6: $\frac{dH_{I}}{dt}$ not $\frac{dH_{A}}{dt}$
Check equation 11:

$\frac{d\Lambda_{H_{Q}}}{dt} = -b_{4} + (\zeta_{6} + \zeta_{1} + \zeta_{5})\Lambda_{H_{Q}}-\zeta _{6} \Lambda _{H_{R}}$

not

$\frac{d\Lambda_{H_{Q}}}{dt} = -b_{4} + (\zeta_{6} + \zeta_{1} + \zeta_{5})\Lambda_{H_{Q}}-\zeta _{3}\Lambda _{H_{R}}$

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematical Modeling

Respond to this report

Responses (1)

Back to all reports

Reviewer Report

25 Views

20 Apr 2023 | for Version 1

Olumuyiwa James Peter, Department Mathematical and Computer Sciences, University of Medical Sciences, Ondo, Nigeria

25 Views Cite this report Responses(1)

Approved With Reservations

The article investigates an optimal control model of human-to-human transmission of monkeypox virus. This work has potential and my comments are as follows:

Please add the main findings and objective of the current study in the abstract.
What are the benchmark cases in your study?
What are the special cases of your study?
Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations.
Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
For enhancing the introduction section please include the following recent studies on monkeypox models: Peter et al. (2023¹) and Peter et al. (2022²).

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Yes
Are all the source data underlying the results available to ensure full reproducibility?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

References

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Mathematical Biology, Infectious Disease Modelling

Respond to this report

Responses (1)

Author Response

29 Aug 2023

Samuel Okyere, Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

1. Please add the main findings and objective of the current study in the abstract
The objective has been specified in the background of the abstract. The main findings have been included in the results of the abstract.

2. What are the benchmark cases in your study?
The study extends and analyses the monkeypox model proposed in Ref. 9, which also takes inspiration from the study in Ref. 12 and 13 and suggests intervention strategies in curbing the disease in Ghana and preventing future occurrences.

3. What are the special cases of your study?
The study reveals an insignificant number of monkeypox-related deaths in Ghana. It also reveals the disease getting extinct in the first two weeks of the outbreak. The study reveals the effectiveness of the two control measures with each strategy leading to a greater percentage reduction in the number of exposed individuals.

4. Table needs to be referenced. Main equations and propositions need to be referenced. Punctuation is missing after some equations. All equations and theorems have been referenced with the exception of the model formulated by the authors. Punctuations and corrections have been done.

5. Try to show more of the physical situation in the results and discussion to reflect the proposed notion of the whole study.
This has been done with the introduction of even new figures to give more insight into the model.

View more View less

Competing Interests

These two articles (Peter et al. (2023, 1) and Peter et al. (2022, 2)) cannot be included in the manuscript as they may constitute a potential conflict of interest.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

[1] 1. Hammarlund E, Lewis MW, Carter SV, et al.: Multiple diagnostic techniques identify previously vaccinated individuals with protective immunity against monkeypox. Nat. Med. 2005; 11(9): 1005–1011. PubMed Abstract | Publisher Full Text

[2] 2. Kalyar F, Chen X, Kunasekaran M, et al.: An unprecedented global resurgence of monkeypox. Global Biosecurity. 2022; 4. Publisher Full Text

[3] 3. Alah MA, Abdeen S, Tayar E, et al.: The story behind the first few cases of monkeypox infection in non-endemic countries. J. Infect. Public Health. 2022; Volume 15(Issue 9): Pages 970–974. 1876-0341.

[4] 4. Hezam IM, Foul A, Alrasheedi A: A dynamic optimal control model for COVID-19 and cholera co-infection in Yemen. Adv. Differ. Equ. 2021; 2021: 108. PubMed Abstract | Publisher Full Text | Free Full Text

[5] 5. Tessema FS, Koya PR, Bole BK: Optimal Control and Cost-Effectiveness Analysis of Cholera with Vaccination. J. Math. 2022; 2022: 1–16. Publisher Full Text

[6] 6. Deressa CT, Duressa GF: Modeling and optimal control analysis of transmission dynamics of COVID-19: the case of Ethiopia. Alex. Eng. J. 2021; 60(1): 719–732. Publisher Full Text

[7] 7. Perkins A, España G: Optimal control of the COVID-19 pandemic with nonpharmaceutical interventions. Bull. Math. Biol. 2020; 82(9): 118. PubMed Abstract | Publisher Full Text | Free Full Text

[8] 8. Okyere S, Ackora-Prah J: A mathematical model of transmission dynamics of SARS-CoV-2 (COVID-19) with an underlying condition of diabetes. Int. J. Math. Math. Sci. 2022; 2022: 1–15. Publisher Full Text

[9] 9. Okyere S, Ackora-Prah J: Modelling and analysis of monkeypox disease using fractional derivatives. Results Eng. 2023; Vol. 17: 100786. 2590–1230. PubMed Abstract | Publisher Full Text | Free Full Text

[10] 10. Somma SA, Akinwade NI, Chado UD: A mathematical model of monkey pox virus transmission dynamics. Ife Journal of Science, AJOA. 2019; 21(1). elSSN: 0794-4896.

[11] 11. Usman S, Adamu I: Modeling the Transmission Dynamics of the Monkeypox Virus Infection with Treatment and Vaccination Interventions. J. Appl. Math. Phys. 2017; 5(12). Reference Source

[12] 12. Peter OJ, Kumar S, Kumari N, et al.: Transmission dynamics of Monkeypox virus: a mathematical modelling approach. Model. Earth Syst. Environ. 2021; 8: 3423–3434. PubMed Abstract | Publisher Full Text | Free Full Text

[13] 13. Peter OJ, Oguntolu FA, Ojo MM, et al.: Fractional order mathematical model of monkeypox transmission dynamics. Phys. Scr. 2022; 97(8): 084005. Publisher Full Text

[14] 14. Grant R, Nguyen LL, Breban R: Modelling human-to-human transmission of monkeypox. Bull. World Health Organ. 2020; 98(9): 638–640. PubMed Abstract | Publisher Full Text | Free Full Text

[15] 15. Vaughan A, Aarons E, Astbury J, et al.: Human-to-Human Transmission of Monkeypox Virus, United Kingdom, October 2018. Emerg. Infect. Dis. 2020; 26(4): 782–785. PubMed Abstract | Publisher Full Text | Free Full Text

[16] 16. Thornhill JP, Barkati S, Walmsley S, et al.: Monkeypox virus infection in humans across 16 countries – April–June 2022. N. Engl. J. Med. 2022; 387: 679–691. PubMed Abstract | Publisher Full Text

[17] 17. Castillo-Chavez C, Blower S, van den Driessche P , et al.: Mathematical approach for emerging and reemerging infectious diseases. New York: Springer Verlag.

[18] 18. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, et al.: The mathematical theory of optimal processes. New York/London: Wiley and sons; 1963; vol. VIII +360S. . Publisher Full Text

[19] 19. Okyere S: Raw_data_monkeypox. [Data]. OSF. 2023 February 14. Publisher Full Text

[20] 20. Okyere S: Monkeypox_data. [Data]. OSF. 2023 February 14. Publisher Full Text

[21] 21. okyere2015: okyere2015/Matlab_codes: Matlab codes for monkeypox deterministic models (v1.0.0). [Code] Zenodo. 2023. Publisher Full Text

Optimal control model of human-to-human transmission of monkeypox virus

Abstract

Keywords

Revised Amendments from Version 1

Introduction

Methods

Figure 1. The flow diagram of the extended monkeypox model.

(1)

Positivity and boundedness of solution

(2)

Theorem 1

Proof

Theorem 2

Proof

Steady states of the model

(3)

(4)

(5)

Optimal control

(6)

Analysis of the optimal control model

(7)

(8)

(9)

Theorem 3

(10)

Proof

(11)

(12)

(13)

Numerical analysis of the model

Table 1. Description of the parameters and values.

Figure 2. Dynamics of the susceptible compartment.

Figure 3. Dynamics of individuals exposed to the monkeypox.

Figure 4. Dynamics of the infected compartment.

Figure 5. Dynamics of the hospitalised compartment.

Figure 6. Dynamics of the recovered compartment.

Figure 7. Dynamics of the deceased compartment.

Control strategy I (public education)

Figure 8. Behaviour of the susceptible with optimal control.

Figure 9. Behaviour of the exposed individuals with optimal control.

Figure 10. Behaviour of the infected with optimal control.

Figure 11. Behaviour of the hospitalised individuals with optimal control.

Figure 12. Behaviour of the recovered individuals with optimal control.

Figure 13. Behaviour of the deceased with optimal control.

Control strategy II (immunization)

Figure 14. Behaviour of the susceptible with optimal control.

Figure 15. Behaviour of the exposed individuals with optimal control.

Figure 16. Behaviour of the infected with optimal control.

Figure 17. Behaviour of the hospitalised individuals with optimal control.

Figure 18. Behaviour of the recovered individuals with optimal control.

Figure 19. Behaviour of the deceased with optimal control.

Strategy III (public education vs immunization)

Figure 20. Behaviour of the susceptible with optimal control.

Figure 21. Behaviour of the exposed individuals with optimal control.

Figure 22. Behaviour of the infected individuals with optimal control.

Figure 23. Behaviour of the hospitalised individuals with optimal control.

Figure 24. Behaviour of the recovered individuals with optimal control.

Figure 25. Behaviour of the deceased compartment with optimal control.

Conclusion

Data availability

Underlying data

Extended data

Software availability

References

Comments on this article Comments (0)

Open Peer Review

Comments on this article Comments (0)

Open Peer Review

Reviewer Status

Reviewer Reports

Comments on this article

Browse by related subjects

Competing Interests Policy

Stay Updated