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Population dynamics of a mathematical model for Monkeypox

August 2023

Authors:

Timothy Ashezua

Joseph Sarwuan Tarka University

Reuben Iortyer Gweryina

University of Agriculture

Francis Shienbee Kaduna

University of Agriculture

Monkeypox, a zoonotic disease, is gradually posing a public health challenge to both developed and developing countries of the world. To this end, there is need to come up with prevention and control strategies to help curb the spread of the disease in the population. In this study, a new deterministic mathematical model for monkeypox is developed to critically look at the impact of public awareness campaign, treatment and vaccination on the transmission dynamics, prevention and control of monkeypox. Qualitative analysis of the model shows that it undergoes the phenomenon of backward bifurcation. However, this phenomenon may not occur as a results of the non-availability of monkeypox vaccine in the community. Furthermore, the model is shown to have three endemic equilibria, namely: the human-to-human, animal-to-human and animal-to-animal transmission modes, respectively, when the associated reproduction number is greater than unity. For the special case, where there is only human-to-human transmission of monkeypox (i.e. υ = ω = φ = β a = 0), the disease-free equilibrium of the model is shown to be globally asymptot-ically stable (GAS) whenever the associated reproduction number is less than unity. For the same scenario as above, the unique endemic equilibrium of the model is globally asymptotically stable whenever the reproduction number is greater than unity. Both analytical and numerical results show a relationship between the rate of vaccination of the susceptible individuals, the progression rate from the exposed stage of infection to the asymptomatic and symptomatic stages of infection, treatment rate of the symptomatic individuals, public awareness campaign and the reproduction number. Results from the sensitivity analysis of the model, using the human reproduction number number, R 0h , show that, the most sensitive parameters are the monkeypox transmission rate from human-to-human (β h), efficacy of public awareness campaign (ψ), vaccination rate for the susceptible individuals (υ), waning rate of the monkeypox vaccine (ω) and the treatment rate for the symptomatic individuals (τ). The epidemiological implication of these results is that, public awareness campaign in the population should be strengthened by the government and other critical stakeholders in order to educate the populace on the need to be vaccinated against monkeypox infection, get to a health facility immediately when blisters show up on their body and the need to develop vaccines that will not wane within a short period of time. Numerical simulations results show that, increasing the rates of vaccination of the susceptible individuals, efficacy of public awareness campaign and the treatment of symptomatic individuals could help in reducing monkey-pox burden in the population.

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International

Journal of

Mathematical

Analysis and

Modelling

(Formerly Journal of the Nigerian Society

for Mathematical Biology)

Volume 6, Issue 1 (July), 2023

ISSN (Print): 2682 - 5694

ISSN (Online): 2682 - 5708

International Journal of Mathematical Analysis and Modelling

Volume 6, Issue 1, July 2023, pages 40 - 63

Population dynamics of a mathematical model for

Monkeypox

T.T. Ashezua∗†

,R.I. Gweryina†‡

, and F.S. Kaduna†§

Abstract

Monkeypox, a zoonotic disease, is gradually posing a public health challenge to both developed

and developing countries of the world. To this end, there is need to come up with prevention

and control strategies to help curb the spread of the disease in the population. In this study, a

new deterministic mathematical model for monkeypox is developed to critically look at the impact

of public awareness campaign, treatment and vaccination on the transmission dynamics, preven-

tion and control of monkeypox. Qualitative analysis of the model shows that it undergoes the

phenomenon of backward bifurcation. However, this phenomenon may not occur as a results of

the non-availability of monkeypox vaccine in the community. Furthermore, the model is shown

to have three endemic equilibria, namely: the human-to-human, animal-to-human and animal-to-

animal transmission modes, respectively, when the associated reproduction number is greater than

unity. For the special case, where there is only human-to-human transmission of monkeypox (i.e.

υ=ω=φ=βa= 0), the disease-free equilibrium of the model is shown to be globally asymptot-

ically stable (GAS) whenever the associated reproduction number is less than unity. For the same

scenario as above, the unique endemic equilibrium of the model is globally asymptotically stable

whenever the reproduction number is greater than unity. Both analytical and numerical results

show a relationship between the rate of vaccination of the susceptible individuals, the progression

rate from the exposed stage of infection to the asymptomatic and symptomatic stages of infection,

treatment rate of the symptomatic individuals, public awareness campaign and the reproduction

number. Results from the sensitivity analysis of the model, using the human reproduction num-

ber number, R0h, show that, the most sensitive parameters are the monkeypox transmission rate

from human-to-human (βh), eﬃcacy of public awareness campaign (ψ), vaccination rate for the

susceptible individuals (υ), waning rate of the monkeypox vaccine (ω)and the treatment rate for

the symptomatic individuals (τ). The epidemiological implication of these results is that, public

awareness campaign in the population should be strengthened by the government and other criti-

cal stakeholders in order to educate the populace on the need to be vaccinated against monkeypox

infection, get to a health facility immediately when blisters show up on their body and the need

to develop vaccines that will not wane within a short period of time. Numerical simulations re-

sults show that, increasing the rates of vaccination of the susceptible individuals, eﬃcacy of public

awareness campaign and the treatment of symptomatic individuals could help in reducing monkey-

pox burden in the population.

Keywords:Monkeypox; reproduction number; public awareness campaign; vaccination; bifurcation

analysis

1 Introduction

Monkeypox, a zoonotic disease caused by a virus in the orthopoxvirus group which includes smallpox

[1, 28], is gradually posing a public health challenge to both developed and developing countries of the

∗Corresponding Author. E-mail: timothy.ashezua@uam.edu.ng

†Department of Mathematics, Joseph Sarwuan Tarkaa University, Makurdi, Nigeria

‡E-mail: gweryina.reuben@uam.edu.ng

§E-mail: kadunafrancis@gmail.com

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T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

world. Globally, the disease accounts for over 46,724 conﬁrmed cases and 13 deaths recorded with

the highest number of conﬁrmed cases recorded in the United States of America (16,925), followed by

Spain (6,284) and then Brazil (3,984)] as at 25th August, 2022, according to [6]. In Africa, monkeypox

was ﬁrst identiﬁed in humans in 1970 in the Democratic Republic of the Congo in a 9 months old boy in

a region where smallpox had just been eradicated in 1968. As at 8th August, 2022, about 59 conﬁrmed

cases and 72 deaths have been recorded in WHO African regions including Cameroon, Central African

Republic, Republic of Congo, Democratic Republic of the Congo, Liberia, Nigeria, Sierra Leone and

Ghana [29]. In Nigeria, 157 cases have been conﬁrmed and 4 deaths recorded as at 8th August, 2022

[25] . Monkeypox virus (MPV) is transmitted to humans through close contact with an infected animal

or with materials contaminated with the virus [7]. Human-to-human transmission of the disease spreads

through close contact with body ﬂuids, lesions, respiratory droplets and contaminated materials such

as beddings [28]. MPV infection passes through a number of stages. When an individual gets exposed

and infected, it can take 5 to 21 days for the ﬁrst symptoms to appear. The early symptoms of the

disease include: fever, headache, muscle aches, back ache, fatigue, chills and swollen lymph nodes as

reported in [16]]. After infection, individuals can test positive (Asymptomatic) without symptoms and

later develop symptoms (Symptomatic). Thereafter, individuals who have shown blisters are isolated

for treatment [1]. These blisters, which usually appears after 1 to 3 days of the early infection, aﬀects

the face, palms of the hand, soles of the feet, mouth and genitalia [7, 16, 28]. Persons who have tested

positive without symptoms can recover naturally due to strong immunity and for individuals with

compromised immune system can progress to the symptomatic stage of infection [1]. Upon completion

of treatment at the isolation centre, individuals can recover with permanent immunity and can equally

die due to the disease [3]. Monkeypox infection can be controlled and prevented by using the following

vaccines to curtail the spread of the disease: vaccinia vaccine (smallpox vaccine) which is 85% eﬀective

in preventing an individual from contracting the disease, vaccinia immune globulin (VIG) and antiviral

medications (in animals)[16, 29]. At the moment, since there are no known vaccines designed to prevent

susceptible individuals from contracting monkeypox infection, antiviral vaccines and drugs developed

to protect against smallpox are mostly deployed to prevent and treat monkeypox virus infections [8].

Mathematical models of infectious diseases have over the years provided useful insights into the

transmission dynamics, prevention and control of infectious diseases, see for example,[12, 13, 17].

However, for monkeypox, few mathematical models have been developed and used to understand the

transmission dynamics, control and prevention of the disease, see for example,[3, 10, 20, 21, 23, 26].

[3], formulated and analyzed a mathematical model on the transmission dynamics of pox-like infec-

tions using the basic S-I-R model for both the human and animal population. They assumed that both

the human and animal population recover from monkeypox infection after treatment with permanent

immunity. Another mathematical model that incorporates treatment and vaccination strategies to cur-

tail the spread of the disease was developed and analyzed by [26]. [10] formulated and analyzed a

mathematical model for monkeypox virus transmission dynamics that includes imperfect vaccine com-

partment for the human sub-population. Another study by [23] developed a mathematical model of

monkeypox virus transmission dynamics incorporating the quarantine class and public enlightenment

campaign parameters into the human population as a means of curtailing the spread of the disease.

Peter et al. [20], designed and analyzed a mathematical model of monkeypox transmission dynamics

using the susceptible-exposed-infected-isolated-recovered model for the human sub-population while the

susceptible-exposed-infected was used for the rodent sub-population. Recently, [21] ﬁtted a model using

the non-linear least square method on cumulative reported cases of monkeypox virus from January to

December, 2019. Further, the fractional order mathematical model of monkeypox transmission dy-

namics was formulated and analyzed with the assumption that the recovered humans confer immunity

after treatment. This study presents a new deterministic mathematical model to study the transmission

dynamics of monkeypox in a population with public awareness campaign, vaccination and treatment

as control strategies. The model is based on the following assumptions:

i. The exposed individuals are broken down into the asymptomatic and symptomatic individuals.

This is in line with the information obtained from [6] and [1].

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ii. Asymptomatic individuals can recover naturally and equally develop symptoms to progress to

the symptomatic class [1].

iii. Individuals recover after treatment with permanent immunity [3, 21].

In all the aforementioned mathematical models of monkeypox, non of them considered the assumptions

(i) and (ii) as biologically supported in [1, 9]. In the present study, the exposed class have been split

into the asymptomatic and symptomatic classes. Also, public awareness campaign, vaccination and

treatment are included as intervention strategies in our model to curtail the spread of the disease in the

population. The rest of the paper is organized as follows: The new monkeypox model is formulated

in Section 2. The mathematical analysis is carried out in Section 3. Sensitivity analysis is conducted in

Sections 4. Results of the numerical simulations are presented in Section 5. The paper is concluded in

Section 6.

2 Model Formulation

The total population at time t, denoted by NH(t), is divided into the human (NH(t)) and animal

(NA(t)) populations. The total human population is further sub-divided into the seven mutually-

exclusive compartments of the susceptible (SH(t)), vaccinated (VH(t)), exposed (EH(t)), asymp-

tomatic (AH(t)), symptomatic (IH(t)), isolated (TH(t)) and treated individuals who recover from

monkeypox infection (RH(t)). Similarly, the total animal population is sub-divided into the suscepti-

ble (SA(t)), exposed (EA(t)), infected (IA(t)) and recovered (RA(t)) animal sub-population. Thus,

N(t) = NH(t) + NA(t),

NH(t) = SH(t) + VH(t) + EH(t) + AH(t) + IH(t) + TH(t) + RH(t),

NA(t) = SA(t) + EA(t) + IA(t) + RA(t)

(1)

The susceptible population (for both human and animals) are increased by the recruitment of new

individuals (assumed susceptible) into the population at a rate Λh(Λa)for human (animal). Susceptible

individuals acquire monkeypox infection, following eﬀective contact with infected animals and humans

(i.e. those in the AH,IHand IAclasses, at a rate λH, given by

λH= (1 −εψ)βh(IH+ηAH)

+βaIA

NA,(2)

Similarly, the susceptible animals acquire the infection following eﬀective contact with an infected

animal (i.e., those in the IAclass) at a rate

λA=βrIA

,(3)

where the parameters εand ψ(0<ε<1and 0< ψ < 1) represent public awareness campaign and ef-

ﬁcacy to public awareness campaign, respectively. The term εψ represent the eﬀectiveness of the public

awareness campaign. The terms βh,βaand βrare the probabilities of transmitting monkeypox virus

from human-to-human, animal-to-animal and animal-to-animal, respectively. Furthermore, ηis the

modiﬁcation parameter associated with the reduced infectiousness of individuals in the asymptomatic

class (AH). Individuals in EHclass progresses to the infected classes (AH)and (IH)at rate θwhile ρ

is the proportion of individuals that progressed to class (AH). Similarly, the animals in EAprogresses

to the infected class IAat a rate ϕ. Individuals in the asymptomatic class either recover naturally and

move to the recovered class RHat a rate σor progress to the the symptomatic class at a rate φ. Also,

individuals in IHcompartment who show symptoms of monkeypox virus are moved to the isolation

centre (TH)for treatment at a rate τ. The parameter γhaccounts for the recovery rate of individuals in

THafter treatment, while γais the recovery rate of the animals in IAclass. Furthermore, natural death

rate µh(µa)occurs in all the epidemiological classes of human (animal) population while individuals in

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T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

IHand THclasses suﬀer an additional monkeypox induced death at rate δh.The animal population in IA

equally die due to monkeypox infection at a rate δa. Combining all these deﬁnitions and assumptions,

it follows that the new monkeypox model is given by the system of diﬀerential equations in (4). The

ﬂow diagram of the model is shown in Figure 1, and the variables of the model are tabulated in Table

1. The variables and parameters of the model are interpreted in Table 1.

Figure 1: Flow diagram for model (4)

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Table 1: Description of parameters in the monkeypox model (4).

Variables/parameters Interpretation

SHSusceptible human population

VHVaccinated human population

EHExposed human population

AHAsymptomatic human population

IHSymptomatic human population

THIsolated human population

RHRecovered human population

SASusceptible animal population

EAExposed animal population

IAInfected animal population

RARecovered animal population

ΛhRecruitment rate into the susceptible human compartment

υEﬀective vaccination rate of the susceptible humans

µhNatural death rate for the humans

δhDeath rate due to monkeypox virus for the humans

θRate of progression from EHto AH

ρProportion of the exposed persons moving to class AH

φProgression from AHto IHdue to compromised immunity

σNatural recovery rate of the asymptomatic individuals

τTreatment rate for the symptomatic individuals

ωWaning rate of the monkeypox vaccine

ηModiﬁcation parameter associated with the reduced rate of MPV transmission

for the asymptomatic individuals

εRate of public awareness campaign

ψEﬃcacy rate of public awareness campaign

εψ Eﬀectiveness of public awareness campaign

βhProbability of transmitting monkeypox from human-to-human

βaProbability of transmitting monkeypox from animal-to-human

βrProbability of transmitting monkeypox from animal-to-animal

ΛaRecruitment rate into the susceptible animal compartment

µaNatural death rate for the animal population

ϕProgression rate from EAto IA

δaDisease induced death rate for the animal population

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dSH(t)

dt = Λh−λHSH+ωVH−(υ+µh)SH,

dVH(t)

dt =υSH−(ω+µh)VH,

dEH(t)

dt =λHSH−(θρ +θ(1 −ρ) + µh)EH,

dAH(t)

dt =θρEH−(σ+φ+µh)AH,

dIH(t)

dt =φAH+θ(1 −ρ)EH−(τ+µh+δh)IH,

dTH(t)

dt =τIH−(γh+µh+δh)TH,

dRH(t)

dt =σAH+γhTH−µhRH,

dSA(t)

dt = Λa−λASA−µaSA,

dEA(t)

dt =λASA−(ϕ+µa)EA,

dIA(t)

dt =ϕEA−(γa+µa+δa)IA,

dRA(t)

dt =γaIA−µaRA.

(4)

where λHand λAare as given in equations (2) and (3), respectively. The results below holds for the

model (4)

Theorem 2.1. All solutions of the model (4) with positive initial data remain positive for all time t > 0.

Furthermore, the model is a dynamical system on the region Γ=Γ1∪Γ2⊂R7

+×R4

+with,

Γ1={(SH, VH, EH, AH, IH, TH, RH) : SH+VH+EH+AH+IH+TH+RH=NH≤Λh

µh

Γ2={(SA, EA, IA, RA) : SA+EA+IA+RA=NA≤Λa

µa

(5)

and the region Γis attracting with respect to the model (4) with initial conditions in R11

+.Proof.

Following similar approach as in [12], it is easy to see that the equations for human (susceptible and

vaccinated individuals) and animal (susceptible animals) in model (4) leads to the following ﬁrst-order

inequality equations:

dSH

dt + (λH+υ+µh)SH>0,dVH

dt + (ω+µh)VH>0and dSA

dt + (λA+µa)SA>0.

Multiplying these inequalities by the integrating factor

αSH(t) = expR[λH(τ)+(υ+µh)]dτ , αVH(t) = expR[(ω+µh)]dτ , αSA(t) = expR[λA(τ)+µa]dτ ,(6)

and observing that

αSH(t)dSH

dt + (λH(τ)+(υ+µh))SH=dSHαSH

dt ,

αVH(t)dVH

dt + (ω+µh)VH=dVHαVH

dt ,

αSA(t)dSA

dt + (λA+µa)SA=dSAαSA

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then integrating with respect to time from 0to tgives SH(t)≥0,VH(t)≥0and SA(t)≥0at all

times, respectively. However, this direct approach does not apply to the remaining equations. In any

case, having the nonnegativity of SH,VHand SAin mind, it can be shown that the remaining eight

equations of the model (4) form a monotone system. Consequently, all its solutions corresponding

to positive initial data remain positive at all times t≥0. By adding the ﬁrst seven and the last four

equations of model (4), the following conservation law is obtained

dNH

dt = Λh−µhNH−δh(IH+TH),

dNA

dt = Λa−µaNA−δaIA.

(7)

Thus, a standard comparison theorem can be used to show that the general a priori estimates below

hold

0≤NH(t)≤NH(0)exp−µht+Λh

µh1−exp−µht,

0≤NA(t)≤NA(0)exp−µat+Λa

µa1−exp−µat.

(8)

Combining these a priori estimates and the fact that, the right hand side of model (4) is locally Lipschitz,

we conclude that there exists a unique global solution in the domain Γ( see Theorem 2.1.5 in [24])).

Thus, the model (4) is a dynamical system on Γ. On the other hand, if a solution is outside the region

Γ, that is NH(t)≥Λh

µhand NA(t)≥Λa

µa, then, it follows from the above conservation law that dNH

dt ≤0

and dNA

dt ≤0. Hence, the above general a priori estimates show that NH(t)tends to Λh

µhand NA(t)

tends to Λa

µaas t→ ∞. Thus, the region Γis attracting.

3 Mathematical Analysis

3.1 Asymptotic Stability of Disease-free Equilibrium (DFE)

The DFE of the model (4) is given by E0= (S0

H, V 0

H, E0

H, A0

H, I0

H, T 0

H, R0

H, S0

A, E0

A, I0

A, R0

A),where

H=Λhk2

k1k2−υω , V 0

H=υΛh

k1k2−υω , E0

H= 0, A0

H= 0, I0

H= 0, T 0

H= 0, R0

H= 0,

A=Λa

µa

, E0

H= 0, I0

A= 0, R0

A= 0,

with k1= (υ+µh)and k2= (ω+µh).

The local stability of E0will be determined using the next generation operator method on model (4)

[27]. Using the notation in [27], it follows that matrices Fand V, for the new infection terms and the

remaining transition terms, respectively, are given by







0y1y20y3

0 0 0 0 0

0 0 0 0 βr

0 0 0 0 0





and V=







k30 0 0 0

−θρ k40 0 0

−θ(1 −ρ)−φ k50 0

0 0 0 k70

0 0 0 −ϕ k8





,

where

y1=(1 −εψ)βhηk2

υ+k2

, y2=(1 −εψ)βhk2

υ+k2

, y3=(1 −εψ)βaµaΛhk2

Λa(k1k2−υω),

k3= (θρ +θ(1 −ρ) + µh), k4= (σ+φ+µh), k5= (τ+µh+δh), k7= (ϕ+µa),

k8= (γa+µa+δa).

(9)

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It follows that the reproduction numbers of the model (4) are

R0={R0h, R0a}.(10)

with R0hand R0abeing the monkeypox induced reproduction numbers for the humans and animals,

respectively, which are given by

R0h=(1 −εψ)βhk2ηk5θρ +φθρ +k4θ(1 −ρ)

k3k4k5(υ+k2)(11)

and

R0a=βrϕ

k7k8

.(12)

The results below follows from Theorem 2 in [27].

Lemma 3.1. The DFE (E0) of the model (4) is locally asymptotically stable whenever R0h<1and R0a<1,

and unstable if otherwise.

The threshold quantity R0measures the average number of new monkeypox infections generated

by an index case in a completely susceptible population [27]. In particular, R0h(R0a)represents the

average number of new monkeypox infections in the human (animal) population generated by a single

infected human (animal) introduced into a completely susceptible human (animal) population. The epi-

demiological implication of Lemma 3.1 is that when R0is less than unity, monkeypox can be eradicated

from the population if the initial sizes of the subpopulation of the model are in the basin of attraction

of the DFE (E0). Hence, a small inﬂux of monkeypox infected humans(animals) into the community

will not generate large monkeypox outbreaks, and the disease will die out with time. To ensure that

monkeypox eradication is independent of the initial sizes of the sub-population, it is necessary to show

that the DFE is globally-asymptotically stable (GAS) if b

R0<1. This is shown for the case where

υ=ω=φ=βa= 0.

3.2 Analysis of the (human) reproduction number, R0h

In this subsection, the threshold parameter, R0his used to determine the impact of vaccination rate

(υ)on the susceptible individuals, the progression rate (θ)from the exposed to the asymptomatic and

symptomatic stages of infection and the treatment rate (τ)for the symptomatic individuals on the

control of monkeypox infection in the population. It is obvious from (10) that

lim

υ→∞ R0h= 0,

lim

τ→∞ R0h=(1 −εψ)βh(ω+µh)θηρ

(υ+ω+µh)(σ+φ+µh)(θ+µh)>0

lim

θ→∞ R0h=(1 −εψ)βh(ω+µh)ηρ(τ+µh+δh)+(σ+φ+µh)−ρ(σ+µh)

(υ+ω+µh)(σ+φ+µh)(τ+µh+δh)>0

(13)

Therefore, a monkeypox control strategy that results in high vaccination rate (υ→ ∞), the progression

rate (θ→ ∞) from the exposed stage of infection to the asymptomatic and symptomatic stages of

infection and the situation where by individuals in the symptomatic stage of infection are promptly

moved to the isolation centre for treatment (τ→ ∞) are high, can lead to the eﬀective control of

monkeypox if the results in the respective right hand side of are less than unity. From (13), it is

observed that, a near total elimination of monkeypox is achievable. In this case, the strategy is to

focus on vaccination programme for individuals in the susceptible class with a high vaccination rate

(υ→ ∞). This will certainly yield a positive results if the susceptible individuals are vaccinated, their

chances of contracting the disease will be low hence reducing the transmission rate in the population.

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By computing the partial derivatives of R0hwith respect to the parameters υand τunder investigation

reveals further their impacts on the control of monkeypox in the community. Thus, this yields

∂R0h

∂υ =−(1 −εψ)βh(ω+µh)[θρη(τ+µh+δh) + φθρ +θ(1 −ρ)(σ+φ+µh)]

(υ+ω+µh)2(σ+φ+µh)(θρ +θ(1 −ρ) + µh)(τ+µh+δh)<0,

∂R0h

∂τ =−(1 −εψ)βhθ(ω+µh)(σ+φ+µh)−ρ(σ+µh)

(υ+ω+µh)(σ+φ+µh)(θ+µh)(τ+µh+δh)2<0,

∂R0h

∂θ =(1 −εψ)µhβh(ω+µh)ηρ(τ+µh+δh)+(σ+φ+µh)−ρ(σ+µh)

(υ+ω+µh)(σ+φ+µh)(θ+µh)2(τ+µh+δh)>0.

(14)

It is obvious, from (14) that unconditionally, the partial derivatives are less than zero. Hence, eﬀective

vaccination rate against monkeypox infection for the susceptible individuals and the rate at which

individuals in the symptomatic class are promptly moved to the isolation centre for treatment will have

a positive impact in curtailing the spread of the disease in the community, irrespective of the values of

the other parameters on the right hand side of (14). Also, exhibiting the progression of the exposed to

the infectious stages could be helpful in reducing the spreading chances of the disease.

3.3 Backward Bifurcation Analysis

3.3.1 Existence of backward bifurcation

In this subsection, we wish to determine the number of equilibrium solutions the model (4) can have.

Let E∗∗ =S∗∗

H, V ∗∗

H, E∗∗

H, A∗∗

H, I∗∗

H, T ∗∗

H, R∗∗

H, S∗∗

A, E∗∗

A, I∗∗

A, R∗∗

Arepresent any arbitrary endemic

equilibrium of model (4). Then, by setting the right-hand sides of the equations of the model (4) at

equilibrium state to zero gives:

S∗∗

H=Λhk2

[k2(k1+λ∗∗

H)−υω], V ∗∗

H=υΛh

[k2(k1+λ∗∗

H)−υω], E∗∗

H=λ∗∗

HΛhk2

k3[k2(k1+λ∗∗

H)−υω],

A∗∗

H=λ∗∗

HθρΛhk2

k3k4[k2(k1+λ∗∗

H)−υω], I∗∗

H=λ∗∗

Hk2(φθρ +θ(1 −ρ)k4)

k3k4k5[k2(k1+λ∗∗

H)−υω],

T∗∗

H=λ∗∗

Hτk2Λh(φθρ +θ(1 −ρ)k4)

k3k4k5k6[k2(k1+λ∗∗

H)−υω], R∗∗

H=λ∗∗

HΛhk2[σk5k6θρ +τ γh(φθρ +θ(1 −ρ)k4)]

µhk3k4k5k6[k2(k1+λ∗∗

H)−υω],

S∗∗

A=Λa

λ∗∗

A+µa

, E∗∗

A=λ∗∗

AΛa

k7(λ∗∗

A+µa), I∗∗

A=λ∗∗

AϕΛa

k7k8(λ∗∗

A+µa), R∗∗

A=λ∗∗

AγaϕΛa

µak7k8(λ∗∗

A+µa)

(15)

where (as before),

k1= (υ+µh), k2= (ω+µa), k3= [θρ +θ(1 −ρ) + µa], k4= (σ+θ(1 −ρ) + µh),

k5= (τ+µh+δh), k6= (γh+µh+δh), k7= (ϕ+µa), k8= (γa+µa+δa).(16)

and λ∗∗

H,λ∗∗

AH and λ∗∗

Aare deﬁned by the following equations

λ∗∗

h= (1 −εψ)βh(I∗∗

H+ηA∗∗

N∗∗

,(17)

λ∗∗

A=βrI∗∗

N∗∗

,(18)

λ∗∗

AH = (1 −εψ)βaI∗∗

N∗∗

,(19)

to be the associated forces of infection for human-to-human, animal-to-animal and animal-to-human,

respectively. We shall consider the cases below for discussion.

Case I: (Human-to-human transmission)

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This is the scenario where the transmission of monkeypox is from human-to-human. In this case,

βa= 0 in equation (2) gives equation (17). Substituting the ﬁrst seven expressions in (4) into (17) and

simplifying gives

1 + λ∗∗

HM2

(υ+k2)=(1 −εψ)βhk2φθρ +θ(1 −ρ)k4+ηk5θρ

k3k4k5(υ+k2)(20)

from (20), we obtain

λ∗∗

H=1

(R0h−1) >1(21)

where M2=k2

+θρk2

k3k4

+(φθρ +θ(1 −ρ)k4)k2

k3k4k5

+τk2(φθρ +θ(1 −ρ)k4)

k3k4k5k6

+σk5k6θρ +τ γh(φθρ +θ(1 −ρ)k4)

µhk3k4k5k6

and M3=M2

(υ+k2).

Thus, a unique endemic equilibrium exists for human-to-human transmission whenever λ∗∗

H>0for

R0h>1.

Case II: (Animal-to-animal transmission)

This is the situation where the transmission of monkeypox is from animal to animal. The force of

infection at steady state for animal to animal transmission is given by (18). Substituting the last four

expressions in (4) into (18) gives

1 + M1λ∗∗

A=λrθ3

k7k8

(22)

which results in

λ∗∗

A=1

(R0a−1) >1(23)

where M1=1

+θ3

k7k8

+θ3γa

µak7k8

Hence, a unique endemic equilibrium exists for animal to animal transmission when R0a>1.

Case III: (Animal-to-human transmission)

This is the case where the transmission of monkeypox infection is from animal to human. Here, we

set βh= 0 in (2), to get equation (19). From (18), we obtain

λ∗∗

βr

=I∗∗

N∗∗

(24)

Substituting (24) into (19), gives

λ∗∗

AH =(1 −εψ)βa

βr

λ∗∗

A.(25)

Using (23) in (25), yields

λ∗∗

AH =M4(R0a−1) >1,(26)

where M4=(1 −εψ)βa

βrM1

Here too, a unique endemic equilibrium exists for animal-to-human transmission when R0a>1.

3.4 Backward bifurcation analysis

The phenomenon of backward bifurcation, which is characterized by the existence of a stable DFE

and a stable endemic equilibrium when the associated reproduction number of the model is less than

unity, can be traced as done over the years in numerous disease transmission models, (see, for example,

[4, 14]).

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Suppose E∗∗ = (S∗∗

H, V ∗∗

H, E∗∗

H, A∗∗

H, I∗∗

H, T ∗∗

H, R∗∗

H, S∗∗

A, E∗∗

A, I∗∗

A, R∗∗

A)represents any arbitrary en-

demic equilibrium of model (4), that is an equilibrium where at least one of the infected classes is

non-zero. We will investigate the existence of backward bifurcation using the centre manifold theory

[7, 8]. To apply this theory, it is convenient to deﬁne the following change of variables SH=x1,

VH=x2,EH=x3,AH=x4,IH=x5,TH=x6,RH=x7,SA=x8,EA=x9,IA=x10 and

RA=x11, so that NH=x1+x2+x3+x4+x5+x6+x7and NA=x8+x9+x10 +x11 . Further,

by adopting the vector notation X= (x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)T, the model (4) can

be written in the form dX

dt =F= (f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11)T, as follows:

dx1

dt =f1= Λh−λHx1+ωx2−k1x1,

dx2

dt =f2=υx1−k2x2,

dx3

dt =f3=λHx1−k3x3,

dx4

dt =f4=θρx3−k4x4,

dx5

dt =f5=φx4+θ(1 −ρ)x3−k5x5,

dx6

dt =f6=τx5−k6x6,

dx7

dt =f7=σx4+γhx6−µhx7,

dx8

dt =f8= Λa−λAx8−µax8,

dx9

dt =f9=λAx8−k7x9,

dx10

dt =f10 =ϕx9−k8x10,

dx11

dt =f11 =γax10 −µax11.

(27)

where

k1= (υ+µh), k2= (ω+µh), k3= [θρ +θ(1 −ρ) + µh], k4= (σ+φ+µh),

k5= (τ+µh+δh), k6= (γh+µh+δh), k7= (ϕ+µa), k8= (γa+µa+δa).(28)

and, the forces of infection are given by

λH= (1 −εψ)[βh(x5+ηx4)

+βax10

](29)

and

λA=βrx10

.(30)

Consider the case when βh=β∗

hand βr=β∗

rare chosen as the bifurcation parameters. Solving for

βh=β∗

hand βr=β∗

rfrom R0= 1, respectively yields

βh=β∗

h=k3k4k5(υ+k2)

(1 −εψ)k2k4θ(1 −ρ) + θρ(φ+ηk5)(31)

and

βr=β∗

r=k7k8

ϕ.(32)

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The Jacobian matrix of the transformed system (28), evaluated at the DFE (E0)with βh=β∗

hand

βr=β∗

r, is given by

J(E0) = A6×6B6×5

C5×6D5×5,

where







−k1ω0−q1−q20

υ−k20 0 0 0

0 0 −k3q1q20

0 0 θρ −k40 0

0 0 θ2φ−k50

0 0 0 0 τ−k6







, B =







000−q30

0 0 0 0 0

000 q30

0 0 0 0 0













000σ0γh

0 0 0 0 0 0





and D=







−µh0 0 0 0

0−µa0−βr0

0 0 −k7βr0

0 0 ϕ−k80

0 0 0 γa−µa





,

with q1=(1 −εψ)ηβ∗

hk2

(υ+k2),q2=(1 −εψ)β∗

hk2

(υ+k2),q3=(1 −εψ)Λhk2β∗

aµa

Λa(k1k2−υω).

Matrix J(E0)has a right eigenvector (associated with the zero eigenvalue) given by

w= [w1, w2, w3, w4, w5, w6, w7, w8, w9, w10, w11]T, where

w1=−(1 −εψ)β∗

hk2[k4θ(1 −ρ) + θρ(φ+ηk5)]

k4k5(υ+k2)(k1k2−υω),

w2=−(1 −εψ)υβ∗

hk2[k4θ(1 −ρ) + θρ(φ+ηk5)]

k4k5(υ+k2)(k1k2−υω), w3=w3>0, w4=θρ

w3>0,

w5=(φθρ +k4θ(1 −ρ))

k4k5

w3>0, w6=τ(φθρ +k4θ(1 −ρ))

k4k5k6

w3>0,

w7=[σk5k6θρ +τ γh(φθρ +k4θ(1 −ρ))]

µhk4k5k6

w3>0, w8=w9=w10 =w11 = 0.

(33)

Furthermore, J(E0)has a left eigenvector (associated with the zero eigenvalue), given by

v= (v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11), satisfying v.w = 1, with

v1=v2= 0, v3=v3>0, v4=β∗

hk2(1 −εψ)(φ+ηk5)

k4k5(υ+k2)>0

v5=β∗

hk2(1 −εψ)

k5(υ+k2)>0, v6=v7=v8=v9=v10 =v11.

(34)

It follows from Theorem 4.1 in [8] that if we compute the associated non-zero partial derivatives of F

at the DFE (E0)and simplifying, that

a=P11

k,i,j=1 vkwiwj

∂2fk

∂xi∂xj

(E0) = 2v3β∗

h(1 −εψ)(k1k2−υω)

Λ(υ+k2)2[A−B].Thus, the bifurcation coef-

ﬁcient, a, is greater than zero whenever A > B, where

A=ηw2

Λh

−k2w2(w5+ηw4)>0, since w2<0

and

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B=w5(w4k2−υw1+k2w3+k2w6+k2w7+w4k2η+k2w5) + w4η(−υw1+w3k2

+w6k2+w7k2)>0, since w1<0.

It can therefore, be shown that

b=P11

k,i=1 vkwi

∂2fk

∂xi∂β∗

(E0) = (1 −εψ)v3k2

(υ+k2)(w5+w4η)>0.

Hence, it follows, from Theorem 4.1 in [15], that the model (4), exhibits a backward bifurcation at

R0<1whenever A > B.

3.5 Existence of Unique Endemic Equilibrium: Special Case

Theorem 3.2. The model (4) has a unique endemic (positive) equilibrium when υ=ω=φ=βa= 0

whenever b

R0>1(i.e. when there is only human-to-human transmission of monkeypox).

Proof: To establish the existence of endemic equilibria of model (4), for the special case υ=ω=

φ=βa= 0 , let E∗

1=S∗∗

H, E∗∗

H, A∗∗

H, I∗∗

H, T ∗∗

H, R∗∗

Hrepresent any arbitrary endemic equilibrium

of model (4). The equations in (4), with υ=ω=φ=βa= 0, are solved in terms of the force of

infection at equilibrium state to get

S∗∗

H=Λh

(µh+λ∗∗

H), E∗∗

H=λ∗∗

HΛh

k3(µh+λ∗∗

H),

A∗∗

H=λ∗∗

HθρΛh

k3b

k4(µh+λ∗∗

H), I∗∗

H=λ∗∗

HΛhθ(1 −ρ)

k3k5(µh+λ∗∗

H),

T∗∗

H=λ∗∗

HΛhτθ(1 −ρ)

k3k5k6(µh+λ∗∗

H), R∗∗

H=λ∗∗

HΛh[σk5k6θρ +τ γhb

k4θ(1 −ρ)

µhk3b

k4k5k6(µh+λ∗∗

H),

(35)

where

k3= (θρ +θ(1 −ρ) + µh),b

k4= (σ+µh), k5= (τ+µh+δh),

k6= (γh+µh+δh).(36)

and λ∗∗

h(for the human-to-human)is the same as given by equation (17): Substituting (35) and (36)

into (17) gives

(1 + λ∗∗

hM1) = (1 −εψ)βhb

k4θ(1 −ρ) + ηθρk5

k3b

k4k5

which yields

λ∗∗

h=1

R0−1) >0

where

M1=1

+θρ

k3b

+τθ(1 −ρ)

k3k4k5

+(σk5k6θρ +γhτb

k4θ(1 −ρ))

µhk3b

k4k5k6

Hence, a unique endemic equilibrium (for the case υ=ω=φ=βa= 0) exists when c

R0>1.

Let,

Γ3= (SH, EH, AH, IH, TH, RH)∈Γ : EH=AH=IH=TH=RH= 0 be the stable manifold of

the DFE (E0). We claim the following

Theorem 3.3. The DFE of model (4), with υ=ω=φ=βa= 0 is globally asymptotically stable in Γ

whenever b

R0≤1.

Proof:

V1=ηk5θρ +θ(1 −ρ)b

k4EH+k3k5ηAH+k3b

k4IH(37)

The Lyapunov derivative of (37) (where a dot denotes diﬀerentiation with respect to t) gives

V1=ηk5θρ +θ(1 −ρ)b

k4˙

EH+k3k5η˙

AH+k3b

k4˙

IH(38)

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Substituting the respective right hand side of model (4) into (38) gives

V1=ηk5θρ +θ(1 −ρ)b

k4λHSH−k3EH+k3k5ηθρEH−b

k4AH

+k3b

k4θ(1 −ρ)EH−k5IH

=(1 −εψ)βh(IH+ηAH)SH

NHhηk5θρ +θ(1 −ρ)b

k4i−k3b

k4k5ηAH−k3b

k4k5IH

=(1 −εψ)βh(IH+ηAH)SH

NHhηk5θρ +θ(1 −ρ)b

k4i−k3b

k4k5(IH+ηAH)

=k3b

k4k5(IH+ηAH)h(1 −εψ)βhSH

[ηk5θρ +θ(1 −ρ)b

k4]

k3b

k4k5

−1i

(39)

It follows from (39), noting that SH(t)≤NH(t)and NH(t)≤Λh

µh

in Γfor all t > 0, that

V1≤k3b

k4k5(IH+ηAH)c

R0−1(40)

Hence, ˙

V1≤0if c

R0≤1with ˙

V1= 0 if and only if IH=AH= 0. Therefore, ˙

V1is a Lya-

punov function in Γ, and it follows from LaSalle’s invariance principle [18] that every solution to

the equation in (4) (with υ=ω=φ=βa= 0) with the initial conditions in Γconverges to E0

as t→ ∞, that is (EH(t), AH(t), IH(t), TH(t), RH(t)) →(0,0,0,0,0) as t→ ∞. Substituting

EH=AH=IH=TH=RH= 0 into the ﬁrst equation in (4) gives SH→Λh

µh

as t→ ∞. Thus,

(SH, EH, AH, IH, TH, RH)→(Λh

µh

,0,0,0,0,0) as t→ ∞ for c

R0≤1. So the DFE, E0, is globally

asymptotically stable in Γif c

R0≤1for the case υ=ω=φ=βa= 0. The above proof is numerically

illustrated in Figure 2. The epidemiological implication of Theorem 3.3 is that, when there is no vacci-

nation and progression from asymptomatic to symptomatic stages of infection (i.e. if only treatment is

the control strategy), monkeypox could still be eﬀectively controlled in the community if the threshold

value b

R0<1.

Figure 2: Simulation of model (4), showing the total number of infected population as a function of

time, using diﬀerent initial conditions. Parameter values used are as in Table 2 (so that b

R0= 0.000041)

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Theorem 3.4. The unique endemic equilibrium of model (4), with υ=ω=φ=βa= 0 is globally

asymptotically stable in Γ\Γ3whenever b

R0>1.

Proof: We consider model (4) where υ=ω=φ=βa= 0 and b

R0>1, then the associated

unique endemic equilibrium of the model exists. The following non-linear Lyapunov function of the

Goh-Volterra type is considered.

V2=SH−S∗∗

H−S∗∗

HlnSH

S∗∗

H+EH−E∗∗

H−E∗∗

HlnEH

E∗∗

H

+ηc

βhS∗∗

k4AH−A∗∗

H−A∗∗

HlnAH

A∗∗

H+c

βhS∗∗

k5IH−I∗∗

H−I∗∗

HlnIH

I∗∗

H (41)

Taking the time derivative of (40) yields

V2=˙

SH−S∗∗

SH+˙

EH−E∗∗

EH+ηc

βhS∗∗

k4˙

AH−A∗∗

AH+c

βhS∗∗

k5˙

IH−I∗∗

IH(42)

Substituting the respective expressions on the right-hand sides of equations (4) into (41) gives

V2= Λh−c

βhSH(IH+ηAH)−µhSH−S∗∗

SHhΛh−c

βhSH(IH+ηAH)−µhSHi

βhSH(IH+ηAH)−k3EH−E∗∗

EHhc

βhSH(IH+ηAH)−k3EHi

+ηc

βhS∗∗

k4hθρEH−b

k4AH−A∗∗

(θρEH−b

k4AH)i

βhS∗∗

k5hθ(1 −ρ)−k5IH−I∗∗

(θ(1 −ρ)−k5IH)i

(43)

Equation (42) reduces to

V2= Λh−c

βhSH(IH+ηAH)−µhSH−S∗∗

Λh+c

βhS∗∗

H(IH+ηAH) + µhS∗∗

βhSH(IH+ηAH)−k3EH−E∗∗

EHc

βhSH(IH+ηAH) + k3E∗∗

H+ηc

βhS∗∗

HθρEH

−c

βhS∗∗

HηAH−A∗∗

AHc

βhS∗∗

ηθρEH+c

βhS∗∗

HηA∗∗

H+c

βhS∗∗

Hθ(1 −ρ)EH

−c

βhS∗∗

HIH−c

βhS∗∗

I∗∗

θ(1 −ρ)EH+c

βhS∗∗

HI∗∗

(44)

Λh=c

βhS∗∗

H(I∗∗

H+ηA∗∗

H) + µhS∗∗

k3E∗∗

H=c

βhS∗∗

H(I∗∗

H+ηA∗∗

θρE∗∗

H=b

k4AH

θ(1 −ρ)E∗∗

H=k5I∗∗

(45)

Substituting (44) into (43) and simplifying yields

V2=µhS∗∗

H2−SH

S∗∗

−S∗∗

SH+ 3c

βhS∗∗

HηA∗∗

H−c

βh

S∗∗2

ηA∗∗

H−E∗∗

EHc

βhSHηAH

−A∗∗2

E∗∗

ηc

βhS∗∗

H+ 3c

βhS∗∗

HI∗∗

H−c

βh

S∗∗2

I∗∗

H−E∗∗

EHc

βhSHIH−c

βhS∗∗

I∗∗2

E∗∗

(46)

(Formerly Journal of the Nigerian Society for Mathematical Biology)

T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

Equation (45) is further simpliﬁed to

V2=µhS∗∗

H2−SH

S∗∗

−S∗∗

SH+ηc

βhS∗∗

HA∗∗

H3−S∗∗

−E∗∗

HSHAH

EHS∗∗

HA∗∗

−A∗∗

HEH

AHE∗∗

H

βhS∗∗

HI∗∗

H3−S∗∗

−E∗∗

HSHIH

EHS∗∗

HI∗∗

−I∗∗

HEH

IHE∗∗

H(47)

Finally, since the arithmetic mean exceeds the geometric mean, the following inequalities hold

2−SH

S∗∗

−S∗∗

≤0,3−S∗∗

−E∗∗

HSHAH

EHS∗∗

HA∗∗

−A∗∗

HEH

AHE∗∗

≤0,

3−S∗∗

−E∗∗

HSHIH

EHS∗∗

HI∗∗

−I∗∗

HEH

IHE∗∗

≤0.

Thus, ˙

V2≤0for c

R0>1. Since the relevant variables in the equations for THand RHare at the

endemic equilibrium state, it follows that these can be substituted into the equations for THand RH

giving TH(t), RH(t)→T∗∗

H, R∗∗

Has t→ ∞. Hence, V2is a Lyapunov function in Γ\Γ3.

4 Sensitivity Analysis

In order to determine the impact of the model parameters on the transmission dynamics, prevention

and control of monkeypox, it is important to conduct sensitivity analysis using the normalized forward

sensitivity index [12, 13]. Following [13], the relative importance of the input parameters associated

with the output variable Roh is evaluated using the formula:

χRoh

p=∂Roh

∂p ×p

Roh

,(48)

where pdenotes model parameters contained in Roh. Using the parameters values in Table 2 on equa-

tions (48), we obtained the sensitivity indices presented in Table 3. Results in Table 3 shows that, the

most sensitive parameters are the monkeypox transmission rate from human-to-human (βh), eﬃcacy

of public awareness campaign (ψ), vaccination rate for the susceptible individuals (υ), waning rate of

the monkeypox vaccine (ω)and the treatment rate for the symptomatic individuals (τ). The epidemi-

ological implication of these results as in Table 3 is that, public awareness campaign in the population

should be strengthened by the government and other critical stakeholders in order to educate the pop-

ulace on the need to be vaccinated against monkeypox infection, get to a health facility immediately

when blisters show up on their body and the need to develop vaccines that will not wane within a short

period of time.

5 Numerical Simulations

In this section, numerical simulations are carried out on model (4) using the parameter values in Ta-

ble 2, and initial data relevant to monkeypox transmission dynamics in Nigeria. Particularly, for the

human population, N= 217,485,651 was found in [22] while VH(0) = 5,000 was gotten from

the works of [26]. Further, we obtained EH(0) = 413,AH(0) = 256 and IH(0) = 157 from

[15]. SH(0) = 217,479,522 was estimated based on the above data information. TH(0) = 153

and RH(0) = 150 where assumed since no information was found regarding their initial conditions.

For the animal population, information regarding their initial conditions was found in the works of

[26]. Hence, we set, SA(0) = 250,EA(0) = 125,IA(0) = 75 and RA(0) = 50. The model (4) is

solved numerically using MATLAB ODE45 solver. In Figure 3, an increase in the vaccination rate

of the susceptible individuals experienced an increase in the population of the vaccinated individuals

(i.e. individuals vaccinated against monkeypox infection). This increase in the vaccination rate of the

susceptibles led to the reduction in the population of the susceptibles, asymptomatic and symptomatic

individuals. In Figure 4,we observed that, increasing the eﬃcacy rate of public awareness campaign(i.e.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

International Journal of Mathematical Analysis and Modelling

Volume 6, Issue 1, July 2023, pages 40 - 63

Table 2: The parameters values of model (4)

Parameter Nominal value Reference

Λh0.029 year−1[2]

υ0.85 year−1[28]

ω0.60 year−1Assumed

µh0.02 year−1[2]

δh0.1year−1[11]

θ0.20 year−1[20]

ρ0.6year−1Assumed

φ0.3year−1Assumed

σ0.4year−1Assumed

τ0.6year−1Assumed

η0.75 year−1Assumed

ε0.80 year−1Assumed

ψ0.70 year−1Assumed

βh0.000063 year−1[3]

(βa, βr) (0.000252,0.0027) [3]

Λa2year−1[3]

µa1.5year−1[3]

ϕ0.3year−1[26]

γa0.6year−1[3]

δa0.4year−1[3]

Table 3: Sensitivity indices of some the parameters values

Parameter Sensitivity indices

βh+1.000000

ε−1.272727

ψ−1.272727

σ−0.353535

τ−0.492424

υ−0.578231

θ+0.090909

ρ+0.090909

µh−0.106347

ω+0.559579

η+0.409091

φ−0.037879

δh−0.082071

(Formerly Journal of the Nigerian Society for Mathematical Biology)

T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

sensitizing the public on the need to be vaccinated against monkeypox infection and other safety mea-

sures) led to an increase in the number of vaccinated individuals while the population of the susceptibles,

asymptomatic and symptomatic reduced greatly. Figure 5 demonstrates the eﬀect of the progression

rate from the exposed population to the asymptomatic and symptomatic population on the number

of infected population on some stages of monkeypox infection. Here, an increase in the progression

rate (θ)led to the reduction in the population of the exposed individuals while a population increase of

the asymptomatic, symptomatic and isolated individuals are recorded. From Figure 6, increasing pub-

lic awareness campaign (ε)and its eﬃcacy (ψ)experienced a reduction in the eﬀective reproduction

number (human) below unity.

We demonstrate numerically in Figures 7, 8 and 9 the relationship between the eﬀective repro-

duction number for the human population and the progression rate (θ), the eﬀective vaccination rate

(υ)and the rate at which the symptomatic individuals are moved to the isolation centre for treatment

(τ). In Figure 7, we observed that, the eﬀective reproduction number increases with an increase in the

progression rate from the exposed stage of infection to the asymptomatic and symptomatic stages of in-

fection, respectively. This implies that if concerted eﬀorts are not put in place to check the progression

rate, it will trigger an increase in the number of monkeypox cases in the population. In Figures 8 and

9, increasing the vaccination rate for the susceptibles and the rate at which symptomatic individuals are

promptly moved to the isolation centre for treatment, respectively, decreases the eﬀective reproduction

number. If the results in Figures 8 and 9 are put to practice, it will help in controlling the spread of

monkeypox infection in the population.

Figure 3: Simulation of model (4) showing the population of susceptible, vaccinated, asymptomatic

and symptomatic individuals. Here, the eﬀective vaccination rate, υis varied from 0.25 to 0.85.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

International Journal of Mathematical Analysis and Modelling

Volume 6, Issue 1, July 2023, pages 40 - 63

Figure 4: Simulation of model (4) showing the population of susceptible, vaccinated, asymptomatic

and symptomatic individuals. Here, the eﬃcacy rate of public awareness campaign, ψis varied from

0.20 to 0.80.

Figure 5: Simulation of model (4) showing the population of exposed, asymptomatic, symptomatic and

the isolated individuals. Here, the progression rate, θis varied from 0.20 to 0.80.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

Figure 6: Contour plot of the human reproduction number, R0has a function of public awareness

campaign, εand eﬃcacy rate of public awareness campaign, ψ.

Figure 7: Plot of the human reproduction number, R0has a function of progression rate, θParameter

values used are as in Table 2.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

International Journal of Mathematical Analysis and Modelling

Volume 6, Issue 1, July 2023, pages 40 - 63

Figure 8: Plot of the human reproduction number, R0has a function of eﬀective vaccination rate, υ

Parameter values used are as in Table 2.

Figure 9: Plot of the human reproduction number, R0has a function of treatment rate of the symp-

tomatic individuals, τParameter values used are as in Table 2.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

T.T. Ashezua, R.I. Gweryina, and F.S. Kaduna

6 Conclusion

This study presents a new deterministic mathematical model for gaining insights into the transmission

dynamics of monkeypox infection in a population with treatment, vaccination and public awareness

campaign as control strategies. The disease-free equilibrium (DFE) of model (4) is locally asymptotically

stable whenever the associated reproduction number is less than unity. Rigorous qualitative analysis

of model (4) show that it undergoes the phenomenon of backward bifurcation, where a stable DFE

coexits with a stable endemic equilibrium. In a backward bifurcation situation, the eﬀective monkeypox

control is dependent on the initial sizes of the sub-population of the model. The implication is that,

the presence of backward bifurcation in the transmission dynamics of monkeypox makes its control

challenging. In the absence of this phenomenon (that is for the special case, υ=ω=φ=βa= 0 where

there is only human-to-human transmission of the disease), the disease-free equilibrium of the model

(4) is shown to be globally-asymptotically stable (GAS) when the associated reproduction number,

R0≤1. Using a non-linear Lyapunov function for the same special case as above, we showed that the

unique endemic equilibrium point of the model is globally asymptotically stable when the associated

reproduction number, c

R0>1. Results from the sensitivity analysis of the model, using the human

reproduction number number, R0h, show that, the most sensitive parameters are the monkeypox

transmission rate from human-to-human (βh), eﬃcacy of public awareness campaign (ψ), vaccination

rate for the susceptible individuals (υ), waning rate of the monkeypox vaccine (ω)and the treatment

rate for the symptomatic individuals (τ). The epidemiological implication of these results is that,

public awareness campaign in the population should be strenghtened by the government and other

critical stakeholders in order to educate the populace on the need to be vaccinated against monkeypox

infection, get to a health facility immediately when blisters show up on their body and the need to

develop vaccines that will not wane within a short period of time

Numerical simulations results reveal that, increasing the rate of vaccination of the susceptible indi-

viduals, the eﬃcacy rate of the public awareness campaign and the movement rate of the symptomatic

individuals to the isolation centre for treatment, respectively, could help in the reduction of monkeypox

cases in the population. So far, few mathematical models have been formulated to study the transmission

dynamics, prevention and control of monkeypox. In view of this, we suggest the following extensions

of our model for further study:

i. Carry out optimal control and cost-eﬀectiveness analysis of the control strategies.

ii. Further theoretical results, such as the global asymptotic stability of the disease-free and unique

endemic equilibrium, which exists for R0a>1(for the special case of animal-to-human trans-

mission), can be explored.

Acknowledgements

The authors sincerely appreciates the Department of Mathematics, Joseph Sarwuan Tarka University,

Makurdi, Nigeria for providing a very conducive research environment.

Funding

The authors did not enjoy funding from any agency.

Competing interests

The authors declare that they have no ﬁnancial or competing interests.

(Formerly Journal of the Nigerian Society for Mathematical Biology)

International Journal of Mathematical Analysis and Modelling

Volume 6, Issue 1, July 2023, pages 40 - 63

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S_H

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E_H

(exposed),

I_H

(infected),

C_H

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V_H

(vaccinated), and

R_H

(recovered). For rodents, the compartments are

S_M

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E_M

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I_M

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\sigma = 0.2, 0.4, 0.6, 0.8, 1

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\phi_H

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\rho

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\phi_M

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\mathbb {R}_{0}

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and R0h<1

\mathbb {R}_{0h} <1

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and R0h>1

\mathbb {R}_{0h} >1

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