Wiley
Journal of Applied Mathematics
Volume 2024, Article ID 5533885, 44 pages
https://doi.org/10.1155/2024/5533885
Research Article
Mathematical Analysis of Malaria Epidemic: Asymptotic
Stability With Cost-Effectiveness Study
Sacrifice Nana-Kyere , Baba Seidu , and Kwara Nantomah
Department of Mathematics, C. K. Tedam University of Technology and Applied Sciences, Navrongo, Ghana
Correspondence should be addressed to Sacrifice Nana-Kyere; snanakyere.stu@cktutas.edu.gh; nana.sacrifice@vvu.edu.gh
Received 27 January 2024; Revised 26 March 2024; Accepted 21 May 2024
Academic Editor: Waleed Adel
Copyright © 2024 Sacrifice Nana-Kyere et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Malaria is an old, curable vector-borne disease that is devastating in the tropics and subtropical regions of the world. The disease
has unmatched complications in the human host, especially in children. Mathematical models of infectious diseases have been
the steering wheel, driving scientists towards elucidation of the dynamic behaviour of epidemics and providing tailored
strategic management of diseases. With the ongoing vaccination programs for vector-borne diseases, the research proposes a
nonlinear differential equation model for the malaria disease that provides public health with a shift from the classical
understanding of nonpharmaceutical preventive malaria control to pharmaceutical measures of vaccines. The asymptotic
dynamic behaviour of the model is studied at the model’s equilibria. The bifurcation type invoked at the disease-free state is
analysed, and the result revealed that the convention that R0 < 1 is the condition for eradicating the disease is not always
sufficient when the system undergoes backward bifurcation. Furthermore, sensitivity analysis was investigated to quantify the
amount of influence each parameter has on R0 . With the Latin hypercube sampling and partial rank correlation coefficient
method, the uncertainty in R0 is computed with a 95% confidence interval, with the mean, and 5th and 95th percentiles,
respectively, simulated as 0.143788, 0.01545, and 0.41491. An intervention model was derived from the nonintervention
model to experiment with and evaluate the respective effects of the various pairings of interventions on the dynamics of the
disease. Lastly, an in-depth cost analysis was studied to identify the most cost-effective intervention regarding rewarding the
desired outcome. From the analysis, we recommend that besides the nonpharmaceutical measure of bed nets and insecticide
spray, public health should target the pharmaceutical intervention of vaccine as it can close the gap in malaria prevention.
Keywords: ACER; ICER; LHS-PRCC; optimal control; vector-borne diseases
1. Introduction
Malaria is one of the most curable vector-borne diseases in
the world, predominating in tropical and subtropical
regions where the weather and change in climate favour
the disease transmission [1]. Even though many eradication
programs have been carried out, the disease remains
endemic in these climate-favoured regions. The disease,
over the years, has been a sting on the human race, causing
both economic and social burdens through illness and
death, and is recognised as one of the longest-known
vector-borne health concerned diseases. In the 2022 report
of the World Health Organization [2], an estimated 249
million and 608,000 malaria cases and deaths were reported.
The estimates mean that in 2022, a hike of 5 million in
additional cases was reported. In the WHO African region,
233 million cases of global Malaria cases were reported. The
report constitutes 94% of malaria cases. The burden of the
disease in Africa is alarming that billions of dollars are spent
yearly by governments for prevention and treatment
2
services [3]. The disease is noted for its sensitivity to changing climatic conditions, with hot temperatures favouring the
growth of the vector and multiplying transmissibility [4].
Recently, death related to malaria incidence has lowered significantly. However, cases of malaria incidence are on a
surge in some metropolitans and local communities. The
current epidemic surge in both metropolitans and local
communities substantially reflects the changing dynamics
of malaria transmission, with communities identified as free
malaria regions witnessing higher cases and places marked
as endemic areas having fewer cases. Areas of disease dominance include Africa, Asia, the Americas, and the Caribbean, with the disease endemic in regions with escalating
sanitation problems. After a hurdle of many struggles, a significant landmark has been reached in vaccine development.
In October 2021, the World Health Organization regulatory
body endorsed the first-ever malaria vaccine, a breakthrough for vector-borne diseases [5]. Expectedly, the vaccine roll-out with other known malaria control measures
such as sleeping under an insecticide-treated net and indoor
and outdoor residual spray has pushed down the number of
infected and hospitalized [6].
Mathematical modelling has been the framework for
providing dynamic insight into disease transmission and
phenomenon to stakeholders and intervention strategies
needed to put in place for the optimum solution ([7–13]).
The study of Tumwiine, Mugisha, and Luboobi [14] proposed a nonlinear model for malaria disease that captured
human recruitment via repeated immigration. The study
determined the model’s equilibria and carried out a global
asymptotic stability analysis of the model’s equilibria. Sabbar, Yavuz, and Özköse [15] considered a general epidemic
model that analysed the effect of stochasticity on the
dynamic behaviour of extinction disease. Penny et al. [16]
examined the cost-effective analysis of the malaria vaccine
and its impact on the transmission of the disease by considering four models of malaria. Studies of Romero-Leiton and
Ibargüen-Mondragón [17] considered a mathematical
model of malaria disease that is segregated into eight compartments that determined to elucidate the dynamic behaviour of the epidemic in Tumaco, Colombia. The study
examined the local and global dynamic stability behaviour
at the model’s steady states. Further, the condition that leads
to backward or forward bifurcation is determined. The
model is then modified to optimal control by capturing
time-dependent controls. Lastly, economic cost analysis is
studied of the proposed strategies to find the optimum strategy. Research conducted by B. Traoré, Sangaré, and S.
Traoré [18] proposed a nonlinear malaria model that factored the different developmental stages of the vector into
consideration. The study, in addition, examined the impact
of change in climate on the metamorphic life cycle of the
vector. The global qualitative analysis of the model was studied to explore the dynamic behaviour of the epidemic’s
steady states. The analysis showed that the global behaviour
is dependent on the humans’ R0 and the vector’s effective
reproduction number, Rv . In [19], the authors constructed
a malaria model that explored the dynamic behaviour at
the endemic state. The R0 was derived analytically, and
Journal of Applied Mathematics
the existence of the malaria-free point when R0 is less than
one was proved. A perturbation analysis of the model was
explored to approximate the endemic point. Stochastic perturbation is then applied to the model, resulting in a stochastic system. A numerical solution is provided for the models.
The analyses showed that due to the stochastic variability,
the infected people in the population are not constant but
undergo fluctuations. In the study of Oke et al. [20], a deterministic malaria model was proposed that used differential
equation tools to analyse the stability at the equilibria. The
model determined controls for the dynamical system such
that the objective function is optimized over a time frame.
The impact of the identified controls on the infected individuals was investigated. The investigation results indicated that
the pairing of the controls—treated bednet, medication, and
insecticide spraying—substantially impacts the infectives by
significantly minimizing it. In [21], the authors modelled a
transmission model of malaria disease that primarily
assessed the linearity of the R0 ’s parameters and their
impact on the R0 . Further, measures that target the biting
rate of the vector were identified. The model result proved
that interventions that targeted the biting rate of the vector
successfully controlled the epidemic. In the work of Huo
and Qiu [22], a transmission model of malaria disease which
considered the returning of infected humans not only to the
susceptible compartment but to the infective as well is proposed. The model’s stability analysis at the steady states
was carried out, which showed the persistence of the infection when R0 > 1 and the global asymptotic dynamic behaviour of the disease when R0 ≤ 1. Orwa, Mbogo, and Luboobi
[23] considered a nonlinear compartmental malaria model
that captured the liver stage to the blood stage of the parasite
development and examined the impact of immune response
and treatment on the disease transmission. The qualitative
model analysis was studied, which revealed the asymptotic
stability of the disease-free state locally and globally. In
Ducrot et al.’s study [24], a malaria model that captured
the two natures of the human host—semi-immune and nonimmune—was proposed. The condition that leads to backward or forward bifurcation is examined. Finally, measures
to curb the disease were explored for the associated subgroup of semi-immune, nonimmune, and vector. From the
studies of Yang [25], a compartmental model of malaria
transmission is constructed that examines the impact of
global warming on the dynamics of the disease. The model
carried out an explicit derivation of the R0 . The analytic
method of sensitivity analysis was applied to the model to
assess the different scenarios when the model’s parameters
are varied. The article by Nana-Kyere et al. [26] proposed
an optimal intervention model with time-dependent controls that were qualitatively and numerically solved to find
the optimum eradication measure for the disease. Arquam,
Singh, and Cherifi [27] proposed and examined a datainspired SIR vector-borne model that assessed the temperature variations’ effect on vector-borne diseases. The study of
Iggidr, Sallet, and Souza [28] examined the dynamics of
multigroup vector-borne diseases, an extension of the Bailey–Dietz model. The research of Kuniyoshi and Santos
[29] employed the variant SIR model type to analyse the
Journal of Applied Mathematics
effect of insecticide resistance on vector-borne diseases. The
authors in [30] considered a SIR model for a vectortransmitted disease incorporating an age-structured vaccinated population to examine the impact of vaccines on the
disease dynamics. They applied the results from the analysis
to study dengue fever disease. Using the classical SIR model,
the authors in [31] assessed the impact of mathematical
models on intervention provision directed at managing
vector-borne diseases. In the study of [32], a compartmentalized SEIR-SEI vector-borne model of host–vector is utilised to predict dengue outbreak. The model also estimated
the epidemic’s important parameters by fitting it to actual
dengue fever data. Smith et al. [33] developed a mathematical model for the malaria epidemic that assessed vaccines’
impact on the disease dynamics by fitting the model to field
malariology datasets from several portions of Africa. In a
related study of [34], a climate model that assessed the rainfall and temperature variability influence on the vector–host
malaria model is considered. The model was fitted to malaria
transmission data of Limpopo Province, South Africa, to
determine all the malaria spikes in the province. The study
of White et al. [35] employed the clinical trial data to estimate the efficacy of the RTS malaria vaccine profile using
mathematical models. The study further illustrated how
clinical trial estimated parameters can be utilised to forecast
vaccination campaign impacts on the disease. Slater, Okell,
and Ghani [36] explored the effects of integrating two factors: pharmacokinetic and pharmacodynamic in drug design
on the malaria transmission dynamics in humans using
mathematical models. The study demonstrated that the
two therapies are the key to eliminating the disease. In Olaniyi et al.’s study [37], a malaria model that accounted for
the different transmission trends in a social hierarchystructured population was proposed. A new optimal control
model was developed to examine the proposed strategies’
efficiency. The cost-effectiveness analysis was considered to
determine the most cost-effective strategy of the considered
strategies. In [38], a mathematical model describing vector–
host transmissibility dynamics is considered that focused primarily on the maturity delay of the vector. In [39], a malaria
transmissibility model was proposed that examined the effect
of the vaccine on the model’s dynamics. The analyses showed
that a vaccine would minimize secondary infections when it
minimizes the infectious period of the disease. Using the
curve fitting technique, Nana-Kyere, Seidu, and Nantomah
[40] calibrated a mathematical model in the context of Ghana
that investigated the substantial intervention program for
containing the disease in Ghana.
With the ongoing vaccination program for vector-borne
diseases, the research primarily focuses on deriving a mathematical model for the malaria epidemic that would provide
public health with a shift from the classical understanding of
nonpharmaceutical preventive malaria control to the pharmaceutical measure of vaccines by quantifying the potency
of vaccines in eradicating the disease. The research would
provide the public health decision body, a valuable insight
into the significant contribution of vaccines, by reducing
the overall disease burden and incidence, hospitalization,
and death resulting from malaria disease.
3
The design of the research is as follows: The derivation
of the Malaria model and demonstration of the model’s
positivity and boundedness are presented in Section 2.
Section 3 is devoted to the computation of R0 , assessment
of the local stability of model system (1) at the model’s
equilibria, and exploring the global stability of model system (1) at model’s equilibria, bifurcation study, global sensitivity analysis, and vaccine efficacy assessment. Section 4
redesigns the model system (1) to the intervention model
to explore interventions that would help mitigate the disease. The section furthermore provided numerical simulations for the models to complement the analytic analyses.
In Section 5, cost-effective analysis is studied to let stakeholders know which interventions to prioritize in their
attempt to curb the menace. Section 6 provides a thorough
discussion and conclusion based on the outcome of the
investigation.
2. Derivation of the Malaria Model
The present section formulates a nonlinear differential
equation model for malaria disease that segregates the population into susceptible humans, SH ; exposed humans, EH ;
vaccinated humans, V H ; asymptomatic humans, AH ;
mildly infected humans, I H1 ; severely infected humans,
I H2 ; and recovered humans, RH . The model assumes that
the mildly infected are individuals who are showing symptoms of the disease but are not in the acute stage, while the
severely infected individuals are those in the acute stage of
the disease. Hence, the human population N H is denoted
as follows: N H = SH + V H + EH + AH + I H1 + I H2 + RH . The
vector has a population given by N V = SV + EV + I V , with
SV , denoting susceptible mosquitoes, EV , the exposed mosquitoes, and I V , the infected mosquitoes. The model
assumes that recruitments of susceptible humans and susceptible mosquitoes into the human and vector populations are, respectively, given by Λa and Λb . β1 and β2
represent the pathways of transmission from mosquitoes
to humans and humans to mosquitoes. α denotes the biting rate of mosquitoes. The transition from the exposed
to the asymptomatic human is given by the rate ψa . ϕa also
denotes the transition from asymptomatic to mildly
infected humans. σa is the rate at which the mildly infected
humans progress to severely infected human status. γa represents the recovery rate of the severely infected humans.
Further, σ1 represents the death of severely infected
humans owing to the disease. δ1 , δ2 are the recovery rates
of the asymptomatic and mildly infected humans. μδ is
the natural death rate of humans. qm is the rate at which
exposed mosquitoes progress to the infected compartment.
In addition, μd is the mosquitoes’ natural death rate. ϕ1 is
the vaccination rate of susceptible humans who are vaccinated, and ϕ2 is the rate at which the vaccinated humans
enter the recovery compartment. Finally, d1 , d 2 , d3 , d 4 , η1 ,
and η2 are modification parameters. The following assumptions are presented in the schematic diagram of Figure 1.
The description of the parameter values is further given
in Table 1.
4
Journal of Applied Mathematics
��
��
��
�1
�b
VH
��
�1
E�
�d
SV
d1
��
��
V�
�2
�1
A�
HV
d3
d4
�a
�2
d2
��
EV
�d
�2
I�1
��
qm
��
I�2
�1
��
IV
�d
��
R�
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Figure 1: Schematic of the malaria model.
λVH is the per-capita contact transmission rate from
mosquitoes to humans given by
d
η E + η2 I V
SH
S = Λa − ϕ1 SH − μδ SH − αβ1 1 V
NH
dt H
d
V = ϕ1 SH − ϕ2 V H − μδ V H
dt H
d
η E + η2 I V
SH − ψa + μδ EH
E = αβ1 1 V
NH
dt H
λVH = αβ1
η1 EV + η2 I V
NH
λHV , the per-capita infection rate from humans to
mosquitoes given by
d
A = ψa EH − ϕa AH − δ1 AH − μδ AH
dt H
d
I = ϕa AH − δ2 I H1 − σa I H1 − μδ I H1
dt H1
d
I = σa I H1 − γa I H2 − σ1 I H2 − μδ I H2
dt H2
d
R = γa I H2 + δ1 AH + δ2 I H1 + ϕ2 V H − μδ RH
dt H
d
d E + d 2 AH + d3 I H1 + d4 I H2
S = Λb − μd SV − αβ2 1 H
SV
NH
dt V
λHV = αβ2
d1 EH + d 2 AH + d 3 I H1 + d 4 I H2
NH
2.1. Model Analysis: Positivity and Boundedness
Theorem 1. Let SH , V H , EH , AH , I H1 , I H2 , RH , SV , EV , I V
be the set of positive solution of the state Equation (1) with
nonnegative parameters and initial condition given by SH
≥ 0, V H ≥ 0, EH ≥ 0, AH ≥ 0, I H1 ≥ 0, I H2 ≥ 0, RH ≥ 0, SV ≥ 0,
EV ≥ 0, I V ≥ 0 .
d
d E + d2 AH + d 3 I H1 + d 4 I H2
SV − qm EV − μd EV
E = αβ2 1 H
NH
dt V
d
I = qm EV − μd I V ,
dt V
1
Proof 1. To prove the positivity of model Equation (1), we
apply the method demonstrated in [41]. Thus, we let U =
Journal of Applied Mathematics
5
with ϱ1 = ϕ1 + μδ , ϱ2 = ϕ2 + μδ , ϱ3 = ψa + μδ , ϱ4 = ϕa
+ δ1 + μδ , ϱ5 = δ2 + σa + μδ , ϱ6 = γa + σ1 + μδ , ϱ7 = qm
+ μd , y1 = Ω2 + Ω3 + Ω4 + Ω5 , and
Table 1: Malaria model parameters and description.
Parameter
Description
α
Biting rate of mosquitoes.
ϕ1
Vaccination rate.
B = Λa , 0, 0, 0, 0, 0, 0, Λb , 0, 0 T
d1 , d2 , d3 , d4
Modification parameters.
η1 , η2
Modification parameters.
Λa
Recruitment rate of humans.
β1
σa
Transmission probability from mosquitoes to
humans.
Transmission probability from humans to
mosquitoes.
Transition rate from the exposed to the
asymptomatic compartment.
Transition rate from the asymptomatic to the mildly
infected compartment.
Transition rate from the mildly infected to the
severely infected compartment.
γa
Recovery rate of the severely infected humans.
σ1
Disease-induced death rate.
δ1 , δ2
Recovery rate of the asymptomatic and mildly
infected humans.
μδ
Death rate of humans.
β2
ψa
ϕa
Now, from the vaccinated equation of model Equation
(1), solving with the integrating factor method gives
t
V H t = e− ϕ2 +μδ t V H 0 + ϕ1 SH t e− ϕ2 +μδ t dξ
0
The asymptomatic equation of model Equation (1) also
gives
t
AH t = e− ϕa +δ1 +μδ t AH 0 + ψa EH t e− ϕa +δ1 +μδ t dξ
0
Further, solving the mildly infected equation of model
Equation (1) with the integrating factor method gives
Λb
Recruitment rate of mosquitoes.
qm
ϕ2
Transition rate from the exposed mosquitoes to the
infected compartment.
Rate at which the vaccinated humans enter the
recovery compartment.
μd
Death rate of mosquitoes.
t
I H1 t = e− δ2 +σa +μδ t I H1 0 + ϕa AH t e− δ2 +σa +μδ t dξ
0
Then, from the severely infected equation of model
Equation (1), we get
t
SH , V H ,EH , AH , I H1 , I H2 , RH , SV , EV , I V T , Ω0 = α β1 η1 /N H
EV , Ω1 = α β1 η2 /N H I V , Ω2 = α β2 d 1 /N H EH , Ω3 = α β2 d 2 /
N H AH , Ω4 = α β2 d3 /N H I H1 , and Ω5 = α β2 d4 /N H I H2 .
Then, representing malaria model Equation (1) as a nonlinear differential equation, we get
dU
= Q1 U + B
dt
2
where
Q1 =
−ϱ1 − Ω0 + Ω1
0
0
0
0
0
0
0
0
0
ϕ1
−ϱ2
0
0
0
0
0
0
0
0
Ω0 + Ω1
0
−ϱ3
0
0
0
0
0
0
0
0
ψa
0
−ϱ4
0
0
0
0
0
0
0
0
0
ϕa
−ϱ5
0
0
0
0
0
0
0
0
0
σa
−ϱ6
0
0
0
0
0
ϕ2
0
δ1
δ2
γa
−μδ
0
0
0
0
0
0
0
0
0
0
−y1 − μd
0
0
0
0
0
0
0
0
0
y1
−ϱ7
0
0
0
0
0
0
0
0
0
qm
−μd
I H2 t = e− γa +σ1 +μδ t I H2 0 + σa I H1 t e− γa +σ1 +μδ t dξ
0
It follows that d/dt V H ≥ 0, at t = 0, for V H 0 = 0, d/
dt AH ≥ 0, at t = 0, for AH 0 = 0, d/dt I H1 ≥ 0, at t = 0,
for I H1 0 = 0, and d/dt I H2 ≥ 0, at t = 0, for I H2 0 = 0.
Hence, from comparative analysis, the same can be concluded for the remaining state variables, SH t , I H2 t , SV t ,
EV t , I V t , which guarantee the positivity of the state variables throughout the study. Hence, based on this assertion,
we argue that the off-diagonal entries of matrix Q1 are nonnegative and that of B ≥ 0. Hence, the malaria equation (1)
is positively invariant in R10
+ .
Theorem 2. The malaria model (1) has a nonnegative solution and bounded positively within the invariant region,
ϖ ∈ R10 ;
ϖ=
≤
SH , V H , EH , AH , I H1 , I H2 , RH , SV , EV , I V ∈ R10
+ , NH t
Λa
Λ
,N t ≤ b
μδ V
μd
6
Journal of Applied Mathematics
Proof 2. Let N H t = SH + V H + EH + AH + I H1 + I H2 + RH
and N V t = SV + EV + I V . Then, we can deduce the following:
d
N t = Λa − μ δ N H ,
dt H
d
N t = Λb − μd N V
dt V
3. Malaria-Free State and Computation of the
Basic Reproduction Number R0
The R0 rate, one of the look-out threshold quantities, measures on average the infected individuals; an infected person
has been infected throughout infectivity. The R0 rate above
one insinuates that the number of cases is rising, and the
below one rate means the epidemic is waning. As owned
by Seidu, Makinde, and Asamoah [42], the subsection uses
the recently developed novel algebraic technique to derive
the R0 . Thus, considering the Malaria-free state given by
D0f = SH 0 , V H 0 , EH 0 , AH 0 , I H10 , I H20 , RH 0 , SV 0 , EV 0 , I V 0
R0 =
J D0f =
5
Hence, it follows that dN H t /dt ≤ 0 and dN V t /d
t ≤ 0 if N H 0 ≥ Λa /μδ and N V 0 ≥ Λb /μd . Hence,
the region given by ϖ is positive invariant. Further, N H
0 > Λa /μδ and N V 0 > Λb /μd ; then, the solution
approaches ϖ in the final time, or N H t and N V t
approach Λa /μδ and Λb /μd asymptotically. Hence, the
region given by ϖ attracts all the solutions in R10
+ .
6
Λa
Λa ϕ1
,
, 0, 0, 0, 0,
ϕ1 + μδ
ϕ1 + μδ ϕ2 + μδ
Λa ϕ1 ϕ2
Λ
, b , 0, 0
μδ ϕ1 + μδ ϕ2 + μδ μd
The Jacobian matrix of the infected subsystem evaluated
at the malaria-free fixed point is obtained as
4
In addition, the vector-related equation of (3) becomes
Λ
N V t = b 1 − e−μd t + N V 0 e−μd t
μd
D0f =
3
From (3), solving the equation related to humans gives
Λ
N H t = a 1 − e−μδ t + N H 0 e−μδ t
μδ
where
−ϱ3
0
0
0
αβ1 η1 SH 0
N H0
αβ1 η2 SH 0
N H0
ψa
−ϱ4
0
0
0
0
0
ϕa
−ϱ5
0
0
0
0
0
σa
−ϱ6
0
0
αβ2 d1 SV 0
N H0
αβ2 d 2 SV0
N H0
αβ2 d3 SV 0
N H0
αβ2 d 4 SV 0
N H0
−ϱ7
0
0
0
0
0
qm
−μd
Following this method, the determinant of J D0f can be
rewritten as Det J D0f = B − D, where B and D are the
respective parts of the determinant containing infectivity/
transmission terms and transition terms obtained as follows.
B=−
α2 β1 β2 SV 0 SH 0
N H0 2
η1 μd + η2 qm ϱ6 ϱ5 ϱ4 d1 + ϱ6 ϱ5 ψa d2
+ ϕa ϱ6 ψa d 3 + ϕa σa d 4 ψa
D = μd ϱ6 ϱ3 ϱ4 ϱ7 ϱ5
Using Remark 3.1 of [42], the basic reproduction number of the model is obtained as
R0 =
β1 β2 SV 0 SH 0 α2 η1 μd + η2 qm ϱ6 ϱ5 ϱ4 d1 + ϱ6 ϱ5 ψa d 2 + ϕa ϱ6 ψa d3 + ϕa σa d4 ψa
μd ϱ6 ϱ3 ϱ4 ϱ7 ϱ5 N H 0 2
7
Upon substituting the point D0f into Equation (7), the
R0 becomes
β1 β2 α2 η1 μd + η2 qm ϱ6 ϱ5 ϱ4 d 1 + ϱ6 ϱ5 ψa d 2 + ϕa ϱ6 ψa d3 + ϕa σa d 4 ψa Λb μ2δ
ϱ1 ϱ3 ϱ4 ϱ5 ϱ6 ϱ7 Λa μ2d
3.1. Local Stability Analysis of Malaria-Free State
Theorem 3. The Malaria-free point D0f of model (1) is LAS
when R0 < 1 and unstable when R0 > 1.
8
Proof 3. The Malaria model (1) is linearized at D0f to give the
following results.
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7
0
0
0
0
0
0
0
−αβ1 η1
ϕ1
−ϱ2
0
0
0
0
0
0
0
0
0
−ϱ3
0
0
0
0
0
αβ1 η1
0
0
ψa
−ϱ4
0
0
0
0
0
0
0
0
0
ϕa
−ϱ5
0
0
0
0
0
0
0
0
0
σa
−ϱ6
0
0
0
0
0
ϕ2
0
δ1
δ2
γa
−μδ
0
0
0
0
0
−b1 d1
−b2
−b3
−b4
0
−μd
0
0
0
0
b1 d 1
b2
b3
b4
0
0
−ϱ7
0
0
0
0
0
0
0
0
0
qm
−μd
From (9), the first four eigenvalues are derived as Φ1 =
−μd , Φ2 = −μd , Φ3 = −ϱ1 , and Φ4 = −ϱ2 . The eigenvalues that
remain are derived by the submatrix SM as follows:
SM =
SH 0
N H0
−ϱ1
SH 0
N H0
SH 0
N H0
−ϱ3
0
0
0
αβ1 η1
ψa
−ϱ4
0
0
0
0
0
ϕa
−ϱ5
0
0
0
0
0
σa
−ϱ6
0
0
b1 d1
b2
b3
b4
−ϱ7
0
0
0
0
0
qm
−μd
αβ1 η2
10
Following the Routh–Hurwitz criterion [43, 44], submatrix SM will have negative real eigenvalues if Tr SM < 0 and
−αβ1 η2
SH 0
N H0
0
SH 0
N H0
αβ1 η2
SH 0
N H0
Det > 0. Hence, from submatrix SM , it can be deduced that
the trace Tr SM = − ϱ3 + ϱ4 + ϱ5 + ϱ6 + ϱ7 < 0 and Det =
ϱ23 ϱ24 ϱ25 ϱ26 ϱ27 μd 1 − R0 .
It follows from the above derivations that when R0 < 1,
then all eigenvalues of submatrix SM would have a negative
real part. Hence, the malaria model (1) would be locally
asymptotically stable at D0f , but otherwise unstable.
3.2. Local Stability Analysis of Malaria-Present Equilibrium.
The malaria-present equilibrium is a point with at least
one nonzero infected state. The subsection determines
D∗f = S∗H , V ∗H , E∗H , A∗H , I ∗H1 , I ∗H2 , R∗H , S∗V , E∗V , I ∗V , representing the malaria-present steady states of model (1). Thus,
the equations of the malaria model (1) are expressed in
terms of the force of infection of the Malaria-present
states as
Λa
,
ϱ1 + μδ + λ∗VH
A∗H =
λ∗VH Λa ψa
,
ϱ1 + μδ + λ∗VH ϱ3 ϱ4
S∗V =
Λb
,
μd + λ∗HV
R∗H =
λ∗VH Λa ψa ϕa σa γa ϱ2 + λ∗VH Λa ψa δ1 ϱ2 ϱ5 ϱ6 + λ∗VH Λa ψa ϕa δ2 ϱ2 ϱ6 + Λa ϕ1 ϕ2 ϱ3 ϱ4 ϱ5 ϱ6
μδ ϱ1 + μδ + λ∗VH ϱ2 ϱ3 ϱ4 ϱ5 ϱ6
I ∗H1 =
Λa ϕ1
,
ϱ1 + μδ + λ∗VH ϱ2
λ∗VH Λa
ϱ1 + μδ + λ∗VH ϱ3
S∗H =
V ∗H =
λ∗VH Λa ψa ϕa
,
ϱ1 + μδ + λ∗VH ϱ3 ϱ4 ϱ5
E∗V =
9
λ∗HV Λb
,
ϱ7 μd + λ∗HV
E∗H =
I ∗H2 =
λ∗VH Λa ψa ϕa σa
ϱ1 + μδ + λ∗VH ϱ3 ϱ4 ϱ5 ϱ6
I ∗V =
λ∗HV Λb qm
ϱ7 μd + λ∗HV μd
8
Journal of Applied Mathematics
The subpopulations S∗H , V ∗H , E∗H , A∗H , I ∗H1 , I ∗H2 , R∗H are
summed up to obtain
N ∗H =
Λa μδ ϱ2 ϱ3 ϱ4 ϱ5 ϱ6 + Λa ϕ1 μδ ϱ3 ϱ4 ϱ5 ϱ6 + λ∗HV Λa μδ ϱ2 ϱ4 ϱ5 ϱ6 + λ∗HV Λa ψa μδ ϱ2 ϱ5 ϱ6
μδ ϱ1 + μδ + λ∗HV ϱ2 ϱ3 ϱ4 ϱ5 ϱ6
∗
λ Λ ψ ϕ μ ϱ ϱ + λ∗HV Λa ψa ϕa σa μδ ϱ2 + λ∗HV Λa ψa δ1 ϱ2 ϱ5 ϱ6 + λ∗HV Λa ψa ϕa σa γa ϱ2
+ HV a a a δ 2 6
μδ ϱ1 + μδ + λ∗HV ϱ2 ϱ3 ϱ4 ϱ5 ϱ6
∗
λ Λ ψ ϕ δ ϱ ϱ +Λ ϕ ϕ ϱ ϱ ϱ ϱ
+ HV a a a 2 2 6∗ a 1 2 3 4 5 6
μδ ϱ1 + μδ + λHV ϱ2 ϱ3 ϱ4 ϱ5 ϱ6
Now, with the force of infections given by
11
that R0 < 1 is an insufficient condition to eliminate the disease. Following the investigation, the mitigation of the disease would be challenging unless much effort and better
intervention measures are applied. To confirm backward
bifurcation possibility in malaria model (1), we let the discriminant Θ21 − 4Θ0 Θ2 = 0 and derive the critical value of
λ∗VH = αβ1
η1 E∗V + η2 I ∗V
N ∗H
12
λ∗HV = αβ2
d 1 E∗H + d2 A∗H + d 3 I ∗H1 + d4 I ∗H2
N ∗H
13
R0 , denoted by Rc < 1. The derivation results in Rc =
The malaria model (1) satisfies the following polynomial
equation at the malaria-present point D∗f :
1 − Θ21 /4Θ0 ϱ22 ϱ23 ϱ24 ϱ25 ϱ26 ϱ27 μd , which confirms the following
theorem.
Θ0 λ∗VH 2 + Θ1 λ∗VH + Θ2 = 0
Theorem 5. The malaria model (1) exhibits backward
bifurcation when D3 of Theorem 4 holds, resulting in
Rc < R0 < 1.
14
where
Θ0 = Λa ϱ2 μδ ϱ4 ϱ5 ϱ6 + ψa μδ ϱ5 ϱ6 + ψa ϕa μδ ϱ6
+ ψa δ1 ϱ5 ϱ6 + ψa ϕa σa γa + ψa ϕa δ2 ρ6 + ψa ϕa σa μδ ϱ2
Θ1 = Λa ϱ3 ϱ4 ϱ5 ϱ6 ϱ2 μδ + ϕ1 μδ + ϕ1 ϕ2
Θ2 = ϱ22 ϱ23 ϱ24 ϱ25 ϱ26 ϱ27 μd 1 − R0
The polynomial Equation (14) is used to demonstrate
the likelihood of multiple endemic equilibria even with R0
< 1. As observed, the coefficient Θ0 always remains nonnegative for nonnegative parameter values, and Θ2 will be nonpositive when R0 > 1. Hence, the underlying theorem holds.
Theorem 4. The malaria model (1) has
D1: a unique endemic steady-state point D∗f if Θ2 < 0 ⇔
R0 > 1;
D2: a unique endemic steady-state D∗f if Θ2 < 0, and
either Θ2 = 0 or Θ21 − 4Θ0 Θ2 = 0;
D3: two endemic steady states if Θ2 > 0, Θ1 < 0 and
Θ21 − 4Θ0 Θ2 > 0;
D4: no endemic steady state otherwise.
The discourse from Theorem 4 implies the following. In
D1 , the malaria model (1) has a unique endemic state when
R0 > 1. Further, D3 indicates a backward bifurcation possibility. Thus, the coexistence of the disease-free and endemic
steady states when R0 < 1. Backward bifurcation implies
3.3. Global Stability Analysis of Malaria-Free State. Consider
the malaria model system (1). If the model system (1) is
globally asymptotically stable at the malaria-free point D0f ,
then the system will always remain stable at D0f and the disease will not persist, regardless of the amount of the perturbation. The subsection would investigate the Castillo-Chavez
approach, which has been shown in the research of [41, 45],
to study the global asymptotic stability of the malaria model
at the malaria-free equilibrium. The analysis is presented as
follows:
dχ1
= h1 χ 1 , χ 2
dt
15
dχ2
= h2 χ 1 , χ 2 , h 2 χ 1 , 0 = 0
dt
16
with χ1 representing the population of uninfected. Thus,
χ1 = SH , V H , RH , SV , and χ2 denotes the infected, with χ2
= EH , AH , I H1 , I H2 , EV , I V . The disease-free state of model
system (1) is given by D0f = χ01 , 0 .
Hence, the point χ01 , 0 is a globally asymptotically stable equilibrium for the model (1) if the following criteria
are satisfied.
Z 1 Given dχ1 /dt = h1 χ1 , 0 , χ01 is GAS.
Z 2 h χ1 , χ2 = Zχ2 − h2 χ1 , χ2 , where h2 χ1 , χ2 ≥ 0
for χ1 , χ2 ∈ ζ
Journal of Applied Mathematics
9
Theorem 6. The point D0f = χ01 , 0 is GAS whenever R0 < 1.
In addition,
d
S + V H + RH = Λa − μδ SH + V H + RH
dt H
Proof 4. The model system (1) is considered to derive h1
χ1 , χ2 and h2 χ1 , χ2 as
Λa − ϕ1 SH − μδ SH − αβ1
Solving Equation (19) gives
η1 E V + η2 I V
SH
NH
SH + V H + RH t =
ϕ1 SH − ϕ2 V H − μδ V H
h 1 χ1 , χ2 =
γa I H2 + δ1 AH + δ2 I H1 + ϕ2 V H − μδ RH
Λb − μd SV − αβ2
αβ1
σa I H1 − γa I H2 − σ1 I H2 − μδ I H2
SH t =
Λa
ϕ1 + μδ
VH t =
Λa ϕ 1
ϕ1 + μδ ϕ2 + μδ
qm EV − μd I V
SV t =
First, to prove condition Z 1 , the system dχ1 /dt = h1 χ1 ,
0 is structured as follows:
Λb
μd
which implies the convergence of χ01 . Hence, χ1 = χ01 is GAS,
which verifies condition one. What happens next is that the
assessment of condition Z 2 is done with h2 χ1 , χ2 = Zχ2
− h2 χ1 , χ2 . Thus, we have
αβ1 η1
SH 0
NH
αβ1 η2
SH 0
NH
−ϱ3
0
ψa
−ϱ4
0
0
0
0
0
ϕa
−ϱ5
0
0
0
0
0
σa
−ϱ6
0
0
SV
αβ2 d1 0
NH
SV
αβ2 d 2 0
NH
SV
αβ2 d 3 0
NH
SV
αβ2 d4 0
NH
−ϱ7
0
0
0
0
qm
−μd
17
h2 χ1 , χ2 =
0
which gives
χ01 =
21
Λa ϕ1 ϕ2
RH t =
μδ ϕ1 + μδ ϕ2 + μδ
d 1 EH + d2 AH + d 3 I H1 + d 4 I H2
SV − qm EV − μd EV
NH
d
S = Λ − ϕ1 + μδ SH
dt H
d
V = ϕ1 SH − ϕ2 + μδ V H
dt H
d
R = ϕ2 V H − μδ RH
dt H
d
S = Λ b − μ d SV
dt V
e−μδ t
Now, from (18), as t ⟶ ∞,
η1 E V + η2 I V
SH − ψa + μδ EH
NH
ϕa AH − δ2 I H1 − σa I H1 − μδ I H1
αβ2
Λa
Λa
−
− SH + V H + R H 0
μδ
μδ
20
d 1 EH + d2 AH + d 3 I H1 + d 4 I H2
SV
NH
ψa EH − ϕa AH − δ1 AH − μδ AH
h 2 χ1 , χ2 =
19
Λa
Λa ϕ1
Λa ϕ1 ϕ2
Λ
,
,
, b
ϕ1 + μδ
ϕ1 + μδ ϕ2 + μδ μδ ϕ1 + μδ ϕ2 + μδ μd
EH
h 2 χ1 , χ2
AH
0
I H1
and
0
0
0
−
I H2
0
EV
0
IV
0
22
Λa
−
SH t =
ϕ1 + μδ
Λa
− SH 0 e− ϕ1 +μδ t
ϕ1 + μδ
ϕ1 SH t
−
ϕ2 + μδ
ϕ 1 SH t
− V H 0 e− ϕ2 +μδ t
ϕ2 + μδ
VH t =
Hence, h2 χ1 , χ2 is derived as follows:
h2 χ1 , χ2
ϕV t
ϕV t
RH t = 2 H
− 2 H
− RH 0 e−μδ t
μδ
μδ
SV t =
η2 EV
S − SH
N H H0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
d E
αβ2 1 H SV 0 − SV
NH
d A
αβ2 2 H SV 0 − SV
NH
d I
αβ2 3 H1 SV 0 − SV
NH
d I
αβ2 4 H2 SV 0 − SV
NH
0
0
0
0
αβ1
=
Λb
Λb
−
− SV 0 e−μd t
μd
μd
18
η1 E V
S − SH
N H H0
αβ1
10
Journal of Applied Mathematics
From model Equation (1), it can be confirmed that the
population of the humans is bounded by SH 0 and that of
the vector by SV 0 . Hence, αβ1 η1 EV /N H SH ≤ αβ1 η1 EV /N H
SH 0 , αβ1 η2 EV /N H SH ≤ αβ1 η2 EV /N H SH 0 , αβ2 d1 EH /N H
SV ≤ αβ2 d1 EH /N H SV 0 , αβ2 d 2 AH /N H SV ≤ αβ2 d2 AH /N H
SV 0 , αβ2 d3 I H1 /N H SV ≤ αβ2 d3 I H1 /N H SV 0 and αβ2 d4 I H2 /
P f y = diag 0, EH /AH − EH AH /A2H , EH /AH −EH AH /A2H and
′
P f y P y − = diag 0, EH /EH − AH /AH , EH /EH − AH /AH .
Further,
Py A j P y −′ =
2
N H SV ≤ αβ2 d4 I H2 /N H SV 0 which implies that h2 χ1 , χ2
is positive definite. It can also be noted that h2 χ1 , χ2 is an
M-matrix with the off-diagonal entries nonnegative. Hence,
the requirement of the two conditions is met, which is proof
of the global asymptotic stability of D0f .
Proof 5. In the study of [46], we refer to Theorem 5 and
Lemma 3 and Lemma 4 to deduce the GAS of model (1) at
the malaria-present state D∗f . We let SH , EH , AH represent the subsystem of model (1) as follows:
d
η E + η2 I V
SH
S = Λa − ϕ1 SH − μδ SH − αβ1 1 V
NH
dt H
d
η E + η2 I V
SH − ψa + μδ EH
EH = αβ1 1 V
NH
dt
23
d
A = ψa EH − ϕa AH − δ1 AH − μδ AH
dt H
−ϕ1 − μδ − αβ1
αβ1
0
η1 EV + η2 I V
NH
− ψa + μδ
0
0
ψa
− ϕ a + δ1 + μ δ
Let
g1 = −ϕ1 − μδ − αβ1 η1 EV + η2 I V /N H , g2 = αβ1 η1 EV +
η2 I V /N H
2
It follows that the second additive matrix, A j , becomes
2
EH
AH
g1 − ϱ4
0
0
g2
−ϱ3 − ϱ4
ψa
2
ρ=
ρ11
ρ12
ρ21
ρ22
where
ρ11 = g1 − ϱ3 , ρ12 = 0
ρ22 =
0 , ρ21 =
ψa EH /AH
0
g1 − ϱ4 + EH /EH − AH /AH
0
g2
−ϱ3 − ϱ4 + EH /EH − AH /AH
Now, let Y = Y 1 , Y 2 , Y 3 be a vector in R3 , with its norm
• given by Y = max Y 1 , Y 2 , Y 3 . Let L A be a
Lozinki measure with respect to Y . Then, the approximating method as in [47] is applied such that L A ≤
sup z 1 , z 2 , where sup z 1 , z 2 = sup L ρ11 + ρ12 , L ρ22
+ ρ21 and ρ12 , ρ21 denote matrix norms with respect to
the vector norm L1 . L also denotes the Lozinki measure concerning the L1 . Then,
ρ12 = max 0, 0 = 0
ρ21 = max
0
Aj =
0
L ρ11 = g1 − ϱ3
Now, the Jacobian of submatrix (23) is derived as follows:
η1 E V + η2 I V
NH
0
Hence, matrix ρ = Py A j P y −′ + P f y P y −′ is derived
as follows:
3.4. Global Stability Analysis of Malaria-Present State. The
subsection is geared towards determining the global dynamic
behaviour of model (1) at the malaria-present state.
Theorem 7. With R0 > 1, the malaria-present point D∗f of
model system (1) is GAS, otherwise unstable.
g1 − ϱ3
g1 − ϱ3
0
0
ψa
g1 − ϱ4
0
0
g2
−ϱ3 − ϱ4
Now, the function P y = SH , EH , AH is derived as P
y = diag 1, EH /AH , EH /AH , and P y −′ = diag 1, AH /EH
, AH /EH . Hence, the rate of change of P f y is given by
L ρ22 = max
ψa
EH
E
, 0 = ψa H
AH
AH
g1 − ϱ4 + g2 +
−ϱ3 − ϱ4 +
E H AH
−
,
E H AH
E H AH
−
E H AH
Hence, L ρ22 = −ϱ3 − ϱ4 − EH /EH − AH /AH . It follows
from Equation (23) that
z 1 = L ρ11 + ρ12
=
EH′
− ϱ1
EH
z 2 = L ρ22 + ρ21
=
EH′
− ϱ3
EH
Journal of Applied Mathematics
11
Now,
L A ≤ sup z 1 , z 2 =
EH′
− ϱ3
EH
24
Following the above deductions, Equation (24) is put into
(25) below.
1 t
L A dt
t 0
25
1 t EH′
− ϱ3 dt
t 0 EH
26
Π = lim supsup
t⟶∞
Hence,
Π = lim supsup
t⟶∞
Π=
1
E t
ln
− ϱ3 < 0
t
E 0
27
It follows from (27) that the subsystem (23) is GAS around
the interior point of S∗H , E∗H , A∗H . Hence, for the remaining subsystem of (1) given below,
d
V = ϕ1 SH − ϕ2 V H − μδ V H
dt H
d
I = ϕa AH − δ2 I H1 − σa I H1 − μδ I H1
dt H1
d
I = σa I H1 − γa I H2 − σ1 I H2 − μδ I H2
dt H2
d
R = γa I H2 + δ1 AH + δ2 I H1 + ϕ2 V H − μδ RH
dt H
d
d E + d 2 AH + d 3 I H1 + d4 I H2
S = Λb − μd SV − αβ2 1 H
SV
NH
dt V
d
d E + d 2 AH + d 3 I H1 + d4 I H2
SV − qm EV − μd EV
E = αβ2 1 H
NH
dt V
d
I = qm EV − μd I V
dt V
28
When the subsystem (28) is solved with the initial conditions of V H 0 , I H 1 0 , I H 2 0 , RH 0 , SV 0 , EV 0 , and
I V 0 , then as t ⟶ ∞, V H ⟶ V ∗H , I H 1 ⟶ I ∗H 1 , I H 2 ⟶
I ∗H 2 , RH ⟶ R∗H , SV ⟶ S∗V , EV ⟶ E∗V , and I V ⟶ I ∗V .
Hence, the GAS of the endemic point D∗f is ascertained.
3.5. Sensitivity Analysis. The R0 , known to be one of the
essential threshold quantities, is associated with many
parameters which are susceptible to change. Owing to the
susceptibility of these parameters, sensitivity analysis
becomes inevitable in epidemiological modelling as it sheds
light on which parameters significantly affect the R0 and
needs to be worked on. The susceptibility in the model
parameters that are directly involved in the computation of
the R0 creates variability in the estimate of R0 . However,
the statistical methods—the Latin hypercube sampling
(LHS) and partial rank correlation coefficient (PRCC)—are
considered to assess the variability in the R0 (see [48, 49]).
With a MATLAB simulation, the method enables us to dive
into the variability analysis of the malaria model (1) parameter. For the R0 of Malaria model (1) given by Equation (7),
the LHS quantifies the magnitude of the influence of the
parameters, and by the PRCC, we find influential parameters
by their computed values. By considering a 1000 sample, the
LHS analyses the uncertainty of R0 as displayed by the histogram in Figure 2. From Figure 2, the uncertainty in R0 is
exhibited by the histogram with the uncertainty computed
using a 95% confidence interval as shown by the broken
lines. The distribution depicts the estimated values of R0 ,
the mean, and 5th and 95th percentiles, respectively, simulated as 0 143788, 0 01545, 0 41491. Generally, the distribution of R0 becomes widely spread with higher uncertainty.
Figure 3 is the PRCC diagram, and it displays the sensitivity
of R0 to the considered model parameters of the malaria
model (1). From the simulation of the PRCC, the most
crucial parameters are those with significantly small p value
p ≤ 0 005 and big PRCC value, as it indicates a stronger
relationship between the parameters. The PRCC-generated
findings of Figure 3 showed that the parameters μδ , α, β1 ,
η1 , η2 , Λb , μd , β2 , d 1 , d 2 , d 3 , d 4 and qm have PRCC bars in a
positive direction. These parameters exert to some extent a
certain amount of influence on the R0 . However, the
parameters with the most significant influence are the ones
with their bars exceeding the threshold marked by the red
bar. The significance of decreasing the values of the parameters with high positive bars is that it decreases the R0 ,
bringing it down to less than one. Also, the parameters Λa
, ϕ1 , ψa , ϕa , δ1 , δ2 , γa , σa have PRCC bars in a negative direction, which means when their values are increased, they will
help lower the R0 . Hence, the interventions should aim at
lowering the positive PRCC parameter values and increase
the parameters Λa , ϕ1 , ψa , ϕa , δ1 , δ2 , γa , σa .
The observation of Figures 4(a), 4(b), 4(c), 4(d), 4(e),
and 4(f) showed a positive effect of the identified parameters
on R0 . Hence, increasing awareness to prevent mosquito bites
through the use of treated bednet, spraying the environment
to reduce drastically the mosquito population with insecticide
spray, and adhering to pharmaceutical interventions such as
seeking early medical attention at the early onset of the disease
and implementation of effective vaccination programs would
reduce R0 . Furthermore, the analysis was carried further by
generating a 3D plot for some selected parameters. The generated figure of Figure 5(a) showed that increasing the value of α
from zero to one increases the R0 , but the R0 remains the
same even if μδ is increased from zero to one. From
Figure 5(b), we noticed that as β1 increased from zero to 1,
the R0 increases. But R0 remains the same as ψa increased
from zero to one. Lastly, we observed from Figure 5(c) that
as β2 increases from zero to one, it causes R0 to increase but
R0 remains the same when ψa is increased from zero to one.
3.6. Bifurcation. Over time, it has become established that
R0 < 1 guarantees the eradication of diseases. This claim,
however, has been refuted by several recent epidemiological
12
Journal of Applied Mathematics
2000
X 0.143788
Y 2000
X 0.137358
Y 1900
1800
(estimate)
R0 (mean)
1600
R0 (5th)
1400
Frequency
X 0.384917
Y 1500
X 0.0354541
Y 1500
1200
1000
800
600
400
R0 (95th)
200
0
0
0.1
0.2
0.3
0.4
0.5
R0
0.6
0.7
0.8
0.9
1
Figure 2: Uncertainty assessment of R0 displayed by the histogram with the broken line of the diagram indicating a 95% confidence
interval, an estimate line (dashed red line), and the mean (continuous line) of R0 .
PRCC for R0
1
∗
∗
∗
∗
∗
0.8
∗
∗
∗
0.6
∗
Sensitivity measure
0.4
∗
0.2
∗
∗
0
∗
∗
∗
�a
�1
–0.2
–0.4
–0.6
–0.8
∗
∗
∗
��
�1
–1
�
�
�
�1
�1
�2
�a
�2
�a
��
�1
�b
�d
�2
d1
d2
d3
d4
qm
Parameter
PRCC
Significant (p < 0.05)
Figure 3: Sensitivity measure of R0 in respect to changes in the model’s parameters using the PRCC index. Bars with “∗ ” are the parameters
that need to be targeted in control intervention programs. Thus, the parameters with positive bars needs to be minimized and those with
negative bars needs to be increased.
studies that involve backward bifurcation. The phenomena in
several disease modelling research have demonstrated the
potential for the disease to persist even in the presence of stable
disease-free equilibrium and R0 < 1. A stable disease-free and
endemic equilibrium coexisting in the system contributes to
the disease’s persistence. Backward bifurcation has a significant
negative impact on disease control since it would need a more
aggressive strategy to bring the mathematical R0 < 1 when the
illness starts to spread. The subsection explores the possibility
of backward bifurcation in model system (1), by exploring
Journal of Applied Mathematics
13
0.16
0.9
0.1
4
2
0.1
8
0.0
0.1
0.16
0.8
0.14
0.14
0.7
0.1
2
0.12
0.0
6
0.1
0.0
0.6
0.14
8
0.12
0.5
0.1
0.1
�
0.12
0.08
0.08
0.0
0.1
6
0.4
0. 0
0.1
4
0.08
0.06
0.3
0.06
0.2
0.08
0.08
0.06
0.04
0.04
0.06
0.06
0.04
0.02
0.1
0.02
0.04
0.02
0.1
0.2
0.02
0.3
0.04
0.02
0.4
0.5
0.04
0.02
0.6
0.7
0.8
0.9
�1
(a) Contour plot of R0 as a function α and β1
0.9
0.13
0.8
0.12
0.1
0.11
3
0.09
0.08
0.07
0.06
0.05
0.04
0.1
0.7
2
0.1
0.11
0.1
0.6
0.09
�2
0.5
0.08
2
0.1
1
0.1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.4
0.07
0.3
0.06
0.2
0.05
1
0.1
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.1
0.04
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
�
(b) Contour plot of R0 as a function α and β2
0.095
0.9
0.0
85
5
0.07
8
0.0
0.8
0.0
95
0.0
0.7
0.0
8
5
0.07
0.6
0.09
9
0.085
75
0.0
0.0
0.0
8
9
�1
0.5
0.0
85
0.08
0.4
0.0
8
0.0
7
0.0
0.3
0.08
5
75
0.075
0.08
0.2
0.0
75
0.08
0.1
0.07
0.0
7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1
(c) Contour plot of R0 as a function ϕ1 and δ1
Figure 4: Continued.
0.8
0.9
14
Journal of Applied Mathematics
0.22
0.9
4
6
2
0.2
0.1
8
0.1
0.1
2
0.1
0.1
0.8
0.2
0.2
0.7
0.18
0.2
0.18
0.6
0.1
0.1
0.1
4
0.16
2
0.1
0.5
6
0.18
�2
0.16
0.14
0.14
0.4
0.16
0.12
0.14
0.1
0.3
0.12
0.12
0.2
0.14
0.12
0.1
0.1
0.12
0.08
0.1
0.1
0.1
0.08
0.1
0.2
0.08
0.1
0.08
0.3
0.4
0.08
0.5
0.6
0.7
0.8
0.9
1
(d) Contour plot of R0 as a function ϕ1 and δ2
0.1
0.9
95
85
0.0
9
0.095
75
0.0
8
0.0
0.8
0.1
0.0
0.0
0.7
0.0
95
0.0
9
0.09
0.0
8
0.6
5
0.0
8
0.09
0.085
0.08
5
0.0
�
a
75
0.0
0.5
7
0.4
0.0
8
0.08
0.085
0.0
75
0.3
0.08
0.075
0.2
0.0
7
0.08
0.07
5
0.1
0.07
0.075
0.07
0.1
0.2
0.3
0.075
0.4
0.5
0.6
0.7
0.8
0.9
1
(e) Contour plot of R0 as a function ϕ1 and γa
0.14
0.9
0.1
2
3
0.1
0.0
8
0.0
0.13
9
0.8
0.1
0.1
1
0.1
0.11
0.1
3
2
0.1
7
0.0
0.6
0.12
4
0.7
9
0.0
8
0.0
0.1
�2
1
0.1
0.1
0.5
0.09
0.4
0.08
7
0.0
6
0.0
0.1
2
0.07
0.1
1
0.1
9
0.0
8
0.0
0.3
0.2
0.06
0.1
2
0.05
0.0
0.0
9
7
6
5
0.0
0.0
0.1
0.1
1
0.1
0.04
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
a
(f) Contour plot of R0 as a function ϕa and δ2
Figure 4: Contour plot of R0 as a function of ϕa , α, β1 , β2 , ϕ1 , δ1 , δ2 , ϕ1 , γa , and δ2 .
Journal of Applied Mathematics
15
0.18
0.2
0.16
0.14
0.15
R0
0.12
0.1
0.1
0.05
0.08
0.06
0
1
0.9
0.04
0.8
0.7
0.6
0.5
0.4
0.3
0.2
00.11
�
0
0.1
00.22
0.7
0.6
0.5
0.4
0.3
0.8
0.9
1
0.02
0
�
�
(a) Sensitivity plot of R0 in terms of α and μδ
0.1
0.09
0.12
0.08
0.1
0.07
R0
0.08
0.06
0.06
0.05
0.04
0.04
0.02
0.03
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
00.11
�a
0
0
00.11
0.2
0.3
0.5
0.4
0.6
0.7
0.8
0.9
1
0.02
0.01
0
�1
(b) Sensitivity plot of R0 in terms of ψa and β1
0.078
0.08
0.076
0.075
R0
0.074
0.072
0.07
0.07
0.065
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
00.22
�a
0.1
0
0
0.1
0.2
00.33
0.5
0.4
0.6
0.7
0.8
0.9
1
0.068
0.066
�2
(c) Sensitivity plot of R0 in terms of ψa and β2
Figure 5: Sensitivity plot of R0 as a function of α, β1 , β2 , ψa , and μδ .
16
Journal of Applied Mathematics
the theory of central manifold as applied in [50]. Thus, suppose
β1 = β∗1 is considered as the bifurcation parameter, then from
(7), β1 = β∗1 at R0 becomes
β∗1 =
αμδ η1 μd + η2 qm
Now, following matrix (9), the right eigenvector w =
w1 , w2 , ⋯, w10 T is given by
ϱ21 ϱ23 ϱ24 ϱ25 ϱ26 ϱ27 μ2d
b3 ψa ϱ3 ϱ6 + b1 d 1 ϱ4 ϱ5 ϱ6 + b2 ψa ϱ5 ϱ6 + b4 ϱ3 ψa σa
29
w 1 = η1
ϱ3 μd
w
ϱ1 η1 μd + η2 qm 3
w 2 = ϕ1 η1
ϱ3 μd
w
ϱ1 ϱ2 η1 μd + η2 qm 3
w4 =
ψa
w
ϱ4 3
w5 =
ϕa ψ a
w
ϱ4 ϱ5 3
w6 =
σa ϕa ψa
w
ϱ4 ϱ5 ϱ6 3
w7 =
ϕ2 ϕ1 η1 ϱ3 ϱ4 ϱ5 ϱ6 μδ μd + δ1 ψa ϱ1 ϱ2 ϱ5 ϱ6 T 1 + δ2 ϕa ψa ϱ1 ϱ2 ϱ6 T 1 + γa σa ϕa ψa ϱ1 ϱ2 T 1
w3
ϱ1 ϱ2 ϱ4 ϱ5 ϱ6 T 1
w8 =
−αβ2 Λb d1 ϱ4 ϱ5 ϱ6 + d2 ψa ϱ5 ϱ6 + d3 ϕa ψa ϱ4 + d 4 σa ϕa ψa
w3
μ2d ϱ4 ϱ5 ϱ6
w9 =
ϱ3 μd
w
αβ1 η1 μd + η2 qm 3
w3 > 0
w10 = qm
ϱ3
w
αβ1 T 1 3
with
T 1 = η1 μd + η2 qm
v6 = αβ2 Λb
d4
v
μd ϱ6 N H 9
v5 = αβ2 Λb
σa d 4 + d2 ϱ6
v9
μd ϱ6 N H
v4 = αβ2 Λb
ϕa σa d 4 + ϕa d2 ϱ6 + d 2 ϱ5 ϱ6
v9
μd ϱ4 ϱ5 ϱ6
In addition, matrix (9) has a left eigenvector v = v1 , v2 ,
⋯, v10 , which satisfies the condition v w = 1, where
v1 = v2 = v7 = v8 = 0
v9 > 0
v3 = αβ2 Λb
v10 =
ϕa ϕa σa d 4 + ψa ϕa d2 ϱ6 + ψa d2 ϱ5 ϱ6 + d1 ϱ4 ϱ5 ϱ6
v9
ϱ3 ϱ4 ϱ5 ϱ6
ϱ3 ϱ4 ϱ5 ϱ6 ϱ7 − α2 β1 β2 η1 Λb ψa ϕa σa d 4 + ψa ϕa d2 ρ6 + ψa d 2 ϱ5 ϱ6 + d 1 ϱ4 ϱ5 ϱ6
ϱ3 ϱ4 ϱ5 ϱ6
v9
Journal of Applied Mathematics
17
Following Theorem 4.1 in [50], the nonzero partial
derivatives in relation to the equation a and b given by
wi w j ∂2 f k β∗1 , 0
∂xi ∂x j
k,i, j=1
30
wi ∂2 f k β∗1 , 0
∂xi ∂β∗1
k,i=1
31
n
a = vk 〠
and
n
b = vk 〠
are derived as follows:
∂2 f3
∂2 f3
w
=
= αβ1 η1 v3 9 w1
∂x1 ∂x9 ∂x9 ∂x1
NH
∂2 f3
∂2 f3
w
=
= αβ1 η2 v3 10 w1
∂x1 ∂x10 ∂x10 ∂x1
NH
undergo forward bifurcation when a is negative and backward
bifurcation when a is positive.
3.7. Vaccine Effect Assessment. As mentioned in the proceeding objective, the impact of vaccines on the malaria disease
dynamics is assessed by conducting a comprehensive analysis
using MATLAB 18 to generate graphs for a detailed study. The
simulated plot of the graphs revealed that increasing the vaccination parameter ϕ1 reduces the number of the exposed,
asymptomatic, mildly, and severely infected humans. As noted
in Figures 6(a), 6(b), 6(c), and 6(d), when the dose of the vaccination rate ϕ1 increasingly takes the set of values of 0.059,
0.259, 0.559, 0.759, and 0.959, it decreases the exposed, asymptomatic, mildly, and severely infected human cases. The analysis indicates that vaccines hold substantial public health
implications by shifting the classical understanding of nonpharmaceutical preventive malaria control such as insecticide
net usage and spraying the environment to control the vector
to the pharmaceutical measure of vaccines.
∂2 f9
∂2 f9
w
=
= αβ2 d 1 v9 3 w8
∂x3 ∂x8 ∂x8 ∂x3
NH
4. The Intervention Model Development
∂2 f9
∂2 f9
w
=
= αβ2 d 2 v9 3 w8
∂x4 ∂x8 ∂x8 ∂x4
NH
The section determines intervention for the malaria model
(1) that would halt the transmission of the disease. The triplet interventions u1 t , u2 t , u3 t are identified such that
∂2 f9
∂2 f9
w
=
= αβ2 d 3 v9 3 w8
∂x5 ∂x8 ∂x8 ∂x5
NH
∂2 f9
∂2 f9
w
=
= αβ2 d 4 v9 3 w8
∂x6 ∂x8 ∂x8 ∂x6
NH
• u1 t is the bednet usage of the susceptible human to
derive protection from a mosquito bite.
• u2 t is the treatment of the infected humans.
• u3 t is the elimination intervention of the vector
through insecticide spray.
Hence,
w9
w
w
w1 + αβ1 η2 v3 10 w1 + αβ2 d1 v9 3 w8
NH
NH
NH
w3
w3
w
+ αβ2 d 2 v9
w8 + αβ2 d 3 v9
w8 + αβ2 d4 v9 3 w8
NH
NH
NH
a = 2 αβ1 η1 v3
Now, it follows that deriving b using Equation (31), we get
∂ f3
Λa
= v3 η1 w9
∂x9 ∂β1
ϱ1 N H
2
∂2 f3
Λa
= v3 η2 w10
∂x10 ∂β1
ϱ1 N H
Thus,
b = v3 η1 w9
b = v3
Λa
Λa
+ v3 η2 w10
ϱ1 N H
ϱ1 N H
Λa
η w + η2 w10
ϱ1 N H 1 9
From the computations, b is positive as conventionally the
case. The result of a determines the dynamics of the malaria
model (1) around D0f . From [50], the malaria model (1) will
With the intervention of u1 t , u2 t , u3 t identified
for the malaria model (1), the structured nonautonomous
malaria intervention model becomes
d
η E + η2 I V
SH
S = Λa − ϕ1 SH − μδ SH − 1 − u1 αβ1 1 V
NH
dt H
d
V = ϕ1 SH − ϕ2 V H − μδ V H
dt H
d
η E + η2 I V
E = 1 − u1 αβ1 1 V
SH − ψa + μδ EH
dt H
NH
d
A = ψa EH − ϕa AH − δ1 AH − μδ AH
dt H
d
I = ϕa AH − δ2 I H1 − σa I H1 − μδ I H1 − u2 I H1
dt H1
d
I = σa I H1 − γa I H2 − σ1 I H2 − μδ I H2 − u2 I H2
dt H2
d
R = γa I H2 + δ1 AH + δ2 I H1 + ϕ2 V H + u2 I H1 + u2 I H2 − μδ RH
dt H
d
d E + d2 AH + d 3 I H1 + d4 I H2
SV − u3 SV
S = Λb − μd SV − 1 − u1 αβ2 1 H
NH
dt V
d
d E + d2 AH + d3 I H1 + d 4 I H2
SV − qm EV − μd EV − u3 EV
E = 1 − u1 αβ2 1 H
NH
dt V
d
I = qm EV − μd I V − u3 I V
dt V
32
18
Journal of Applied Mathematics
×105
9
8
7
6
EH
5
4
3
2
1
0
0
20
40
60
Time (days)
80
= 0.059
1
= 0.759
1 = 0.259
1
= 0.959
1
100
1 = 0.559
(a) Numerical trajectory depicting vaccine impact on the exposed humans
8
×104
7
6
AH
5
4
3
2
1
0
0
20
40
60
80
Time (days)
= 0.059
1
= 0.759
1 = 0.259
1
= 0.959
1
1 = 0.559
(b) Numerical trajectory depicting vaccine impact on the asymptomatic humans
Figure 6: Continued.
100
Journal of Applied Mathematics
2.5
19
×105
2
IH1
1.5
1
0.5
0
0
40
20
60
80
100
Time (days)
1
= 0.059
1
= 0.759
1
= 0.259
1
= 0.959
1 = 0.559
(c) Numerical trajectory depicting vaccine impact on the mildly infected humans
10000
9000
8000
7000
IH2
6000
5000
4000
3000
2000
1000
0
0
20
40
60
Time (days)
1
= 0.059
1
= 0.759
1
= 0.259
1
= 0.959
1
= 0.559
80
100
(d) Numerical trajectory depicting vaccine impact on the severely infected humans
Figure 6: Numerical trajectory depicting vaccine impact on the exposed, asymptomatic, mildly, and severely infected humans.
The target of intervention model (32) is to minimize the
number of infected human compartments and eradicate the
various phases of the vector while minimizing the expense
associated with putting the identified interventions into practice. As a result, the objective functional (33) is minimized
under the constraints of (32), a nonautonomous equation.
20
Journal of Applied Mathematics
J=
and
T
H 1 EH + H 2 AH + H 3 I H1 + H 4 I H2
0
+ H 5 SV + EV + I V +
1
h u2 + h2 u22 + h3 u23
2 1 1
33
dt
Equation (33) integrand has coefficients of the state variables which serve as balancing coefficients that measure the
intervention cost on the interval 0, T . We seek an optimal
control ui , for i = 1, ⋯, 3 such that
J u∗1 , u∗2 , u∗3 = min J u1 , u2 , u3
U
34
In connection with minimizing the objective functional (33)
to the model system (32), the Pontryagin maximum principle
[51] converts the (33) and (32) into a minimization problem
and finds its solution. We let S∗H , V ∗H , E∗H , A∗H , I ∗H1 , I ∗H2 , R∗H ,
S∗V , E∗V , and I ∗V denote the state variable and u1 t , u2 t and
u3 t the related controls. The existence of an optimal control
problem is shown as follows:
System (32) is structured into the form
X ′ = G1 X + G2 u, X
G1 =
35
−ϱ1
0
0
0
0
0
0
0
0
0
ϕ1
−ϱ2
0
0
0
0
0
0
0
0
0
0
−ϱ3
0
0
0
0
0
0
0
0
0
ψa
−ϱ4
0
0
0
0
0
0
0
0
0
ϕa
−ϱ5 − u2
0
0
0
0
0
0
0
0
0
σa
−ϱ6 − u2
0
0
0
0
0
ϕ2
0
δ1
δ2 + u2
γa + u2
−μδ
0
0
0
0
0
0
0
0
0
0
−μd − u3
0
0
0
0
0
0
0
0
0
0
−ϱ7 − u3
0
0
0
0
0
0
0
0
0
qm
−μd − u3
Considering Equation (35),
becomes
G2 u, X 1 − G2 u, X 2
≤2 η1 + η2 1 − u1 αβ1 SH 1 − SH 2
+ 2 d 1 + d 2 + d 3 + d 4 1 − u1 αβ2 SV 1 − SV 2
+ 2 1 − u1 αβ1 η1 EV 1 − EV 2 + 2 1 − u1 αβ1 η1 I V 1 − I V 2
+ 2 1 − u1 αβ2 d 1 η1 EH 1 − EH 2 + 2 1 − u1 αβ2 d 2 η1 AH 1 − AH 2
+ 2 1 − u1 αβ2 d 3 η1 I H 11 − I H 21 + 2 1 − u1 αβ2 d4 η1 I H 21 − I H 22
where
36
SH
and
VH
G1 X 1 − G1 X 2 ≤ ϱ1 + ϕ1
EH
+ ϱ3 + ψ a
AH
X=
I H1
I H2
SH 1 − SH 2 + ϱ2 + ϕ2
E H 1 − E H 2 + ϱ4 + ϕ a + δ 1
+ ϱ5 + σa + δ2 + 2u2
+ ϱ6 + γa + 2u2
, G2 u, X
RH
V H1 − V H2
AH 1 − AH 2
I H 11 − I H 12
I H 21 − I H 22 + μδ RH 1 − RH 2
+ μd + u 3
S V 1 − S V 2 + ϱ 7 + qm + u 3
+ μd + u 3
IV1 − IV2
EV 1 − EV 2
SV
37
EV
IV
η1 E V + η2 I V
SH
NH
Λa − 1 − u1 αβ1
0
1 − u1 αβ1
=
Λb − 1 − u1 αβ2
1 − u1 αβ2
η1 E V + η2 I V
SH
NH
Following Equations (36) and (37), the entire system K
X = G1 X + G2 u, X is given by
≤2 η1 + η2 + ϱ1 + ϕ1 1 − u1 αβ1 SH 1 − SH 2
+ 2 d 1 + d 2 + d 3 + d4 + μd + u3 1 − u1 αβ2 SV 1 − SV 2
0
+ 2 1 − u1 ϱ7 + qm + u3 αβ1 η1 EV 1 − EV 2
0
+ 2 1 − u1 μd + u3 αβ1 η1 I V 1 − I V 2
0
+ 2 1 − u1 ϱ3 + ψa αβ2 d 1 η1 EH 1 − EH 2
d 1 EH + d 2 AH + d 3 I H1 + d 4 I H2
SV
NH
d 1 EH + d 2 AH + d 3 I H1 + d 4 I H2
SV
NH
0
+ 2 1 − u1 ϱ4 + ϕa + δ1 αβ2 d2 η1 AH 1 − AH 2
+ 2 1 − u1 ϱ6 + γa + 2u2 αβ2 d3 η1 I H 11 − I H 21
+ 2 1 − u1 ϱ5 + σa + δ2 + 2u2 αβ2 d 4 η1 I H 21 − I H 22
+ μδ RH 1 − RH 2 + ϱ2 + ϕ2 V H 1 − V H 2
Journal of Applied Mathematics
Hence,
K X 1 − K X 2 ≤ A1 SH 1 − SH 2 + A2 SV 1 − SV 2 + A3 EV 1 − EV 2
+ A4 I V 1 − I V 2 + A5 EH 1 − EH 2 + A6 AH 1 − AH2
+ A7 I H 11 − I H 21 + A8 I H 21 − I H 22 + A9 RH 1 − RH 2
+ A10 V H 1 − V H 2
≤A SH 1 − SH 2 + SV 1 − SV 2 + EV 1 − EV 2
+ I V 1 − I V 2 + EH 1 − EH2 + AH 1 − AH 2
+ I H 11 − I H 21 + I H 21 − I H 22 + RH 1 − RH 2
+ V H1 − V H2
It follows that A is a nonnegative constant which is independent of the state variable such that
A = max A1 , A2 , A3 , ⋯, A10
where
A1 = 2 η1 + η2 + ϱ1 + ϕ1 1 − u1 αβ1
21
φ′3 = φ3 − φ5 ϕa + φ3 − φ7 δ1 + μδ φ3
d
+ φ8 − φ9 1 − u1 αβ2 2 SV − H 2
NH
d
φ4′ = φ4 − φ3 ψa + μδ φ4 + φ8 − φ9 1 − u1 αβ2 1 SV − H 1
NH
φ5′ = φ5 − φ6 σa + φ5 − φ7 δ2 + φ5 − φ7 u2
d
+ μδ φ5 + φ8 − φ9 1 − u1 αβ2 3 SV − H 3
NH
φ6′ = φ6 − φ7 γa + φ6 − φ7 u2 + σ1 + μδ φ6
d
+ φ8 − φ9 1 − u1 αβ2 4 SV − H 4 ,
NH
φ7′ = μδ φ7
φ8′ = μd φ8 + u3 φ8 + φ8 − φ9 1 − u1 αβ2
d1 EH + d 2 AH + d3 I H1 + d4 I H2
− H5
NH
φ′9 = μd φ9 + φ9 − φ10 qm + u3 φ9
η1
S − H5
+ φ1 − φ4 1 − u1 αβ2
NH V
′ = μd φ10 + u3 φ10 + φ1 − φ4 1 − u1 αβ2
φ10
η2
S − H5
NH V
A2 = 2 d1 + d2 + d 3 + d 4 + μd + u3 1 − u1 αβ2
38
A3 = 2 1 − u1 ϱ7 + qm + u3 αβ1 η1
A4 = 2 1 − u1 μd + u3 αβ1 η1
with transversality conditions
A5 = 2 1 − u1 ϱ3 + ψa αβ2 d1 η1
φ j T = 0, j ∈ 1, 2, ⋯, 10
A6 = 2 1 − u1 ϱ4 + ϕa + δ1 αβ2 d 2 η1
39
A7 = 2 1 − u1 ϱ6 + γa + 2u2 αβ2 d 3 η1
A8 = 2 1 − u1 ϱ5 + σa + δ2 + 2u2 αβ2 d4 η1
u1′ t = min 1, max 0, ϑ1
A9 = μδ
u2′ t = min 1, max 0, ϑ2
A10 = ϱ2 + ϕ2
u3′ t = min 1, max 0, ϑ3
Furthermore, K X 1 − K X 2 ≤ Γ K 1 − K 2 , with Γ =
max J, L < ∞. Therefore, A K is uniformly Lipschitz
continuous. Hence, the malaria model (1) has a solution that
exists.
Now the method [51] is applied which converts the
malaria model (1) and (33) into the minimization of H m
subject to u1 , u2 , u3 .
where
η1 EV + η2 I V
SV
N H h1
d E + d 2 AH + d3 I H1 + d4 I H2
SV
+ φ9 − φ8 αβ2 1 H
N H h1
ϑ1 = φ4 − φ1 αβ1
Theorem 8. Suppose condition (34) is satisfied by the control
triplets ui , for, i = 1, ⋯, 3, then the adjoint variable φ j exists
which satisfy the underlying adjoint system;
φ1′ = φ1 − φ4 1 − u1 αβ1
+ φ1 − φ2 ϕ 1 + μ δ φ1
φ′2 = φ2 − φ7 ϕ2 + μδ φ2
40
η1 E V + η2 I V
NH
ϑ 2 = φ5 − φ 7
ϑ3 =
I H1
I
+ φ6 − φ7 H2
h2
h2
φ8
φ
φ
SV + 9 EV + 10 I V
h3
h3
h3
41
22
Journal of Applied Mathematics
The controls ui , for, i = 1, ⋯, 3 are characterised as presented in the given equation:
Proof 6. The Hamiltonian (42) given by
H m = H 1 EH + H 2 AH + H 3 I H1 + H 4 I H2 + H 5 SV + EV + I V
∂H m
=0
∂u1
+ φ1 Λa − ϕ1 SH − μδ SH − 1 − u1 αβ1
∂H m
=0
∂u2
η1 E V + η2 I V
SH + φ2 ϕ1 SH − ϕ2 VH − μδ VH
NH
+ φ3 ψa EH − ϕa AH − δ1 AH − μδ AH
η E + η2 I V
SH − ψa + μδ EH
+ φ4 1 − u1 αβ1 1 V
NH
+ φ5 ϕa AH − δ2 I H1 − σa I H1 − μδ I H1 − u2 I H1
+ φ6 σa I H1 − γa I H2 − σ1 I H2 − μδ I H2 − u2 I H2
+ φ7 γa I H2 + δ1 AH + δ2 I H1 + ϕ2 V H + u2 I H1 + u2 I H2
− μδ RH + φ8 Λb − μd SV − 1 − u1 αβ2
d1 EH + d2 AH + d3 I H1 + d4 I H2
SV − u3 SV
NH
d E + d2 AH + d3 I H1 + d4 I H2
SV
+ φ9 1 − u1 αβ2 1 H
NH
− qm EV − μd EV − u3 EV
+ φ10 qm EV − μd I V − u3 I V
42
is explored to deduce the adjoint system (38). We partially
differentiate (42) subject to the state variables given by
d
∂H m
φ =−
∂SH
dt 1
d
∂H m
φ2 = −
∂V H
dt
d
∂H m
φ =−
∂EH
dt 3
d
∂H m
φ =−
∂AH
dt 4
d
∂H m
φ5 = −
∂I H1
dt
d
∂H m
φ =−
∂I H2
dt 6
d
∂H m
φ7 = −
∂RH
dt
d
∂H m
φ =−
∂SV
dt 8
d
∂H m
φ9 = −
∂EV
dt
d
∂H m
φ =−
∂I V
dt 10
43
∂H m
=0
∂u3
Using standard argument, the characterization is done
by employing bounds such as
u∗i =
0
if ϑ∗i ≤ 0
ϑ∗i
if 0 ≤ ϑ∗i ≤ 1
1
if ϑ∗i ≥ 1
44
4.1. Numerical Simulation. With the aid of numerical examples to complement the analytic analysis, the method,
forward-backwards sweep, propounded by [52] is applied
to generate simulated graphs of the malaria model (32).
The optimality control simulated graphs are compared to
the noncontrol graph for comparative analysis to be made.
To understand the dynamics of the control strategies on
the model, we explored the scenario of pairing different
strategies to assess their impact on mitigating the spread of
the epidemic. The simulated graphs were generated with
the widely known scheme, and the fourth order Runge–
Kutta was employed to solve the numerical model’s (32)
optimality system. The scheme solves the adjoint system
(38) backwards in time and system (32) forward in time.
With a specified interval t ∈ 0,100 , chosen initial conditions
and the parameter values of Table 2, the iterative scheme
solves the optimality system, considering the associated
transversality condition. Further, the coefficients of the state
variable targeted for minimizing in the objective functional
(33) are chosen arbitrarily as H 1 = H 2 = H 3 = H 4 = H 5 = 10,
h1 = 5, h2 = 10, and h3 = 8. h2 which represents the treatment
intervention is considered to be greater than h1 and h3 since
it involves diagnosis and treatment cost. Further, h3 is
assumed to be greater than h1 since it may involve the cost
of insecticide and spraying.
4.2. A: The Intervention of u1 and u2 . To advance our understanding of the dynamics of the malaria model (1), the interventions of u1 and u2 are considered to generate graphical
solutions of the control model (32) and a comparative analysis is made with the noncontrol model (1). From the analysis, we noted that the exposed human’s noncontrol graph of
Figure 7(a) quickly rises in the early days to 8 8 × 105 in 3
days and gradually decreased and wiped out of the population in 90 days. The noncontrol asymptomatic human’s
graphs of Figure 7(b) rocketed to 6 9 × 104 in 5 days and
swiftly lowered as it degenerated from the population in 90
days. The mildly infected human’s graph of Figure 7(c)
Journal of Applied Mathematics
23
Table 2: Model 1: parameters and source with days as time.
Parameter
Values
Source
Parameter
Values
Source
a
0 0696,0 5
[20, 39, 53]
ϕa
0 31
Assumed
ϕ1
0 01,0 1
[39]
σa
0 03
Assumed
d1 , d2 , d3 , d4
0 1,1
[37, 54]
γa
0 4071
Assumed
η1 , η2
0 01,0 1
[20, 54]
σ1
0 0004,0 2
[39, 55]
Λa
100, 1000
[20]
δ 1 , δ2
0 01,1
[20]
β1
0 021,0 6
[24, 54, 56]
μδ
0 0000548
[37]
β2
0 1,0 5
[24, 54, 56]
Λb
50
[57]
ψa
0 06
[57]
qm
0 27
[58]
ϕ2
0 15
Assumed
μd
0 05,0 3
[22, 39, 58]
was picked from zero and gently moved to 17 6 × 104 in 18
days. It then changed its course and began to decline
smoothly to 0 8 × 104 in 100 days. The severely infected
noncontrol human’s graph of Figure 7(d) gently moved
from zero to 8 7 × 103 in 20 days and smoothly lowered
afterwards to 0 8 × 103 in 100 days. The susceptible mosquito graph of Figure 7(e) declined speedily to zero in 20
days. The noncontrol exposed mosquito graph of
Figure 7(f) swiftly increased from 1 0 × 1053 to 2 5 × 103 in
2 days and then quickly dropped to zero in 20 days. The
infected graph of Figure 7(g) steadily picked from 1 0 × 103
to 1 36 × 103 in 5 days and sharply decreased to zero in 30
days. The control-generated graphs of the intervention of
u1 and u2 produced the desired results. The exposed human
graph slowly decreased from 8 0 × 105 to zero in 100 days
but stayed below the noncontrol graph. The asymptomatic
depicted a similar pattern but lay below the noncontrol
throughout the simulated time. The mildly infected control
human’s graph took the same pattern but could only raise
a little about 2 0 × 104 . The graph slowly lowered from that
level and decreased to zero in 60 days. The severely infected
control human’s graph depicted the noncontrol graph and
slowly raised to 0 8 × 103 in the early days but became
greatly minimized afterwards. The control susceptible mosquito graph reached equilibrium with the noncontrol and
lowered in the same pattern to zero in 20 days. The exposed
mosquito control graph mimicked the noncontrol and raised
gently to 1 3 × 103 in 2 days but gradually lowered to zero
after day 2. The control infected mosquito graph dropped
gently from 1 0 × 103 to 9 30 × 102 in 2 days and gradually
decreased to zero in 30 days depicting the noncontrol graph
as observed in graphs of Figure 7. Figure 7(h) is the control
profile plot generated using intervention u1 and u2 . From the
graph, we observed that both u1 and u2 stayed on the top
bound till 100 days when they dropped to the lower bound.
4.3. B: The Intervention of u1 and u3 . The intervention of u1
and u3 is considered to provide detailed information on the
dynamics of the control model (32) generated solution to the
noncontrol solution. We observed from the simulated plots
that the noncontrol exposed human’s graph of Figure 8(a)
rose quickly to 8 8 × 105 in the first 3 days and fell smoothly
afterwards to zero in 90 days. The asymptomatic human’s
graph of Figure 8(b) soured from 0 3 × 104 to 6 9 × 104 in
5 days and slowly declined to zero in 90 days. The noncontrol mildly infected human’s graph of Figure 8(c) smoothly
picked from zero and soured to 17 6 × 104 in 18 days. The
dynamic behaviour of the graph changed after 18 days,
and it began to decrease gradually to 0 8 × 104 at the final
time. The severely infected human’s graph of Figure 8(d)
moved from zero to 8500 in 20 days and fell through to
450 at the final simulated time. The susceptible noncontrol
mosquito graph of Figure 8(e) continuously declined from
12 × 106 to zero in 10 days and remained there for the rest
of the simulated time. The exposed noncontrol graph of
Figure 8(f) steadily rose early to plummet to 2500 in 3 days.
The graph then decreased speedily to zero in 25 days and
stayed there for the rest of the simulated time. The infected
mosquito graph of Figure 8(g) rose to 1350 in 5 days and
then fell to zero in 30 days. However, with the intervention
of u1 and u3 , different dynamic behaviour was obtained.
The intervention caused the control exposed graph to
decrease from 12 × 106 to zero in 80 days, but the graph
stayed below the noncontrol graph throughout the simulated
time. The asymptomatic control human’s graph mimicked
the noncontrol graph but stayed below it. The mildly infected
control human’s graph, taking the same pattern as the noncontrol moved from zero to 12 3 × 104 in 18 days. It then
decreased slowly to 0 8 × 104 at the final simulated time.
The severely infected control graph depicted the noncontrol
graph and moved from zero to 6000 in 20 days. The graph
proceeded after 20 days and dropped slowly to 300 at the
final time. The control susceptible graph dropped sharply
to hit zero in 8 days and stayed there for the remaining time.
Further, the control exposed graph quickly fell from 1000 to
zero in 8 days, staying below the noncontrol graph. The control infected graph swiftly dropped from 1000 to zero in 8
days, remaining below the noncontrol graph as seen in the
graphs of Figure 8. Finally, Figure 8(h) is the control profile
plot generated using intervention of u1 and u3 . From the
graph, u1 and u3 both stayed in the upper limit until 100 days
when it dropped to the lower limit.
24
Journal of Applied Mathematics
9
×105
8
7
6
EH
5
4
3
2
1
0
0
20
40
60
80
100
80
100
Time (days)
Without control
With control
(a) Substate evolution of the exposed humans
8
×10
4
7
6
AH
5
4
3
2
1
0
0
20
40
60
Time (days)
Without control
With control
(b) Substate evolution of the asymptomatic humans
Figure 7: Continued.
Journal of Applied Mathematics
25
×105
2
1.8
1.6
1.4
IH1
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
Time (days)
Without control
With control
(c) Substate evolution of the mildly infected humans
10000
9000
8000
7000
IH2
6000
5000
4000
3000
2000
1000
0
0
20
40
60
Time (days)
Without control
With control
(d) Substate evolution of the severely infected humans
Figure 7: Continued.
80
100
26
Journal of Applied Mathematics
×106
12
10
SV
8
6
4
2
0
0
20
40
60
80
100
80
100
Time (days)
Without control
With control
(e) Substate evolution of the susceptible mosquitoes
2500
2000
EV
1500
1000
500
0
0
20
40
60
Time (days)
Without control
With control
(f) Substate evolution of the exposed mosquitoes
Figure 7: Continued.
Journal of Applied Mathematics
27
1800
1600
1400
1200
IV
1000
800
600
400
200
0
0
20
40
60
80
100
Time (days)
Without control
With control
(g) Substate evolution of the infected mosquitoes
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0
20
40
60
80
Time (days)
u1
u2
(h) Control profile
Figure 7: Substate evolution generated with u1 ≠ 0 and u2 ≠ 0.
100
28
Journal of Applied Mathematics
9
×105
8
7
6
EH
5
4
3
2
1
0
0
20
40
60
80
100
80
100
Time (days)
Without control
With control
(a) Substate evolution of the exposed humans
8
×104
7
6
AH
5
4
3
2
1
0
0
20
40
60
Time (days)
Without control
With control
(b) Substate evolution of the asymptomatic humans
Figure 8: Continued.
Journal of Applied Mathematics
2
29
×105
1.8
1.6
1.4
IH1
1.2
1
0.8
0.6
0.4
0.2
0
0
40
20
60
80
100
Time (days)
Without control
With control
(c) Substate evolution of the mildly infected humans
10000
9000
8000
7000
IH2
6000
5000
4000
3000
2000
1000
0
0
40
20
60
Time (days)
Without control
With control
(d) Substate evolution of the severely infected humans
Figure 8: Continued.
80
100
30
Journal of Applied Mathematics
6
12 ×10
10
SV
8
6
4
2
0
0
40
20
60
80
100
80
100
Time (days)
Without control
With control
(e) Substate evolution of the susceptible mosquitoes
2500
2000
EV
1500
1000
500
0
0
20
40
60
Time (days)
Without control
With control
(f) Substate evolution of the exposed mosquitoes
Figure 8: Continued.
Journal of Applied Mathematics
31
1800
1600
1400
1200
IV
1000
800
600
400
200
0
0
40
20
60
80
100
Time (days)
Without control
With control
(g) Substate evolution of the infected mosquitoes
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0
20
40
60
80
Time (days)
u1
u3
(h) Control profile
Figure 8: Substate evolution generated with u1 ≠ 0 and u3 ≠ 0.
100
32
4.4. C: The Intervention of u2 and u3 . A comparative analysis
involving the intervention of the control u2 and u3 of the
control model (32) to the noncontrol model (1) is made to
elucidate the dynamic behaviour of the disease. The investigation indicated that the exposed noncontrol human’s graph
of Figure 9(a) rose steadily in the early days to 8 8 × 105 and
decreased gradually until it entirely seemed to have been
wiped out of the population at the final time. The asymptomatic noncontrol graph of Figure 9(b) swiftly raised to
6 9 × 104 in 5 days and changed its direction to fall gradually
to zero in 100 days. The mildly infected noncontrol graph of
Figure 9(c) moved from zero to 17 5 × 104 in 18 days and
then decreased gradually to 0 9 × 104 at the final time. The
severely infected noncontrol graph of Figure 9(d) picked
from zero and steadily increased to 8500 in 20 days. The
graph then decreased gradually to 400 at the final simulated
time. The susceptible noncontrol mosquito graph of
Figure 9(e) dropped from 12 × 106 to zero in 20 days and
remained there for the rest of the simulated time. The
exposed noncontrol mosquito graph of Figure 9(f) sharply
rose from 1000 to 2500 in 3 days and suddenly fell to zero
in 20 days. The infected noncontrol mosquito graph of
Figure 9(g) picked from 1000 to 1350 in 5 days and then
lowered gradually till it hit zero in 30 days. The execution
of the intervention of the control u2 and u3 generated graphs
with different dynamic behaviour. The control exposed
human graph decreased right from 8 0 × 105 and progressed
steadily to zero at the final time. The control asymptomatic
human graph took a similar pattern as the noncontrol and
increased from 0 3 × 104 to 5 3 × 104 in 5 days. The graph
changed its dynamic behaviour and lowered gradually after
5 days to zero but stayed below the noncontrol graph for
the simulated time. The mildly controlled infected human
graph slightly increased similarly to the noncontrol graph
but was greatly minimized. In addition, the severely controlled infected human graph depicted the noncontrol and
increased for the 5 days and declined afterwards. The control
susceptible mosquito graph dropped sharply in the first 20 to
meet the noncontrol and maintained that level for the simulated time. The control exposed mosquito graph took the
same pattern as the noncontrol but declined sharply to hit
zero at 6. The control infected mosquito graph dropped
swiftly from 1000 to zero in 10 days as noticed in the graphs
of Figure 9. Figure 9(h) is the control profile of the model
using the intervention of u2 and u3 . The result of the graph
and the intervention of u2 and u3 should not be relaxed for
the entire simulated time.
4.5. D: The Intervention of u1 , u2 , and u3 . Finally, to further
boost management decision-makers with an advanced
understanding of the malaria 1 disease’s dynamics, we
employed all three interventions of the control model (32)
and circumspectly experimented with them to generate simulated plots to support the analytic investigation. From the
simulation, we observe that the noncontrol exposed human’s
graph of Figure 10(a) slightly increased from 8 0 × 105 to
8 8 × 105 in 3 days and gradually lowered after 3 days to zero
at the final time. The asymptomatic noncontrol human’s
Journal of Applied Mathematics
graph of Figure 10(b) steadily raised from 0 4 × 104 to 6 9
× 104 in 5 days and slowly declined to zero at the end of
the simulated time. The mildly infected noncontrol graph
of Figure 10(c) increased from zero to 17 5 × 104 in 18 days
and then reduced gradually to 0 8 × 104 at the final time. The
severely infected noncontrol human’s graph of Figure 10(d)
continuously increased from zero to 8500 in 20 days and
dropped slowly to 400 at the final time. The susceptible noncontrol mosquito graph of Figure 10(e) dropped speedily to
zero in 20 days from 12 × 106 and maintained that level for
the remaining time. The exposed noncontrol mosquito
graph of Figure 10(f) sharply increased from 1000 to 2500
in 3 days and then swiftly dropped to zero in 20 days. It then
stayed at zero for the remaining time. The noncontrol
infected mosquito graph of Figure 10(g) moved from 1000
to 1350 in 5 days and then changed the dynamics and began
to decrease. The graph progressed from 1350 to zero within
25 days. The intervention of u1 , u2 , and u3 reverted the
dynamic behaviour of the disease. We noted that the control
exposed human’s graph decreased right from day 1 and progressed steadfastly to zero in 100 days. The control asymptomatic graph mimicked the noncontrol graph and moved
from 0 4 × 104 to 4 4 × 104 in 3 days, and then gradually
decreased to zero in 100 days. The mildly infected control
human’s graph depicted the behaviour of the noncontrol
graph and slowly decreased to zero in 100 days. The severely
infected control human’s graph moves slightly to 500 in 5
days and gently decreases to zero in 100 days. The control
susceptible mosquito graph declined sharply in the first 5
days to zero and maintained the level for the simulated time.
The control exposed mosquito graph dropped from 1000 to
zero in 8 days and stayed there for the related time. The control infected mosquito graph swiftly dropped from 100 to
zero in 8 days and stayed at that level to meet the noncontrol
graph t = 30 as depicted by the graphs of Figure 10.
Figure 10(h) is the control profile diagram generated using
the three controls. From the graph, we observed that the
three controls remained at the upper limit before dropping
to the lower limit at the final time.
5. Cost-Effective Analysis
Cost-effective analysis is a method used to determine which
among the available interventions is the most cost-effective
in terms of achieving the desired outcomes. The analysis is
conducted to let decision-makers make informed decisions
about which interventions or treatments to prioritize based
on their budget and desired outcomes. By the cost-effective
analysis, we can determine the cost-effectiveness of each
considered intervention by comparing the costs and outcomes [59–62]. The section engages the method of ACER
(average cost-effectiveness ratio) and ICER (incremental
cost-effectiveness ratio) to make a judgment of the most
cost-effective intervention of the proposed interventions to
tackle the malaria epidemic by comparing the costs and performances of the interventions. The ACER gives the total
cost of an intervention to the prevented infections by the
intervention. Further, considering two interventions of p1
Journal of Applied Mathematics
9
33
×105
8
7
6
EH
5
4
3
2
1
0
0
20
40
60
80
100
80
100
Time (days)
Without control
With control
(a) Substate evolution of the exposed humans
4
8
×10
7
6
AH
5
4
3
2
1
0
0
20
40
60
Time (days)
Without control
With control
(b) Substate evolution of the asymptomatic humans
Figure 9: Continued.
34
Journal of Applied Mathematics
2
×105
1.8
1.6
1.4
IH1
1.2
1
0.8
0.6
0.4
0.2
0
0
40
20
60
80
100
Time (days)
Without control
With control
(c) Substate evolution of the mildly infected humans
10000
9000
8000
7000
IH2
6000
5000
4000
3000
2000
1000
0
0
40
20
60
Time (days)
Without control
With control
(d) Substate evolution of the severely infected humans
Figure 9: Continued.
80
100
Journal of Applied Mathematics
35
6
12 ×10
10
SV
8
6
4
2
0
0
40
20
60
80
100
80
100
Time (days)
Without control
With control
(e) Substate evolution of the susceptible mosquitoes
2500
2000
EV
1500
1000
500
0
0
40
20
60
Time (days)
Without control
With control
(f) Substate evolution of the exposed mosquitoes
Figure 9: Continued.
36
Journal of Applied Mathematics
1800
1600
1400
1200
IV
1000
800
600
400
200
0
0
40
20
60
80
100
Time (days)
Without control
With control
(g) Substate evolution of the infected mosquitoes
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0
20
40
60
80
Time (days)
u1
u2
u3
(h) Control profile
Figure 9: Substate evolution generated with u2 ≠ 0 and u3 ≠ 0.
100
Journal of Applied Mathematics
9
37
×105
8
7
6
EH
5
4
3
2
1
0
0
20
40
60
80
100
80
100
Time (days)
Without control
With control
(a) Substate evolution of the exposed humans
8
×104
7
6
AH
5
4
3
2
1
0
0
20
40
60
Time (days)
Without control
With control
(b) Substate evolution of the asymptomatic humans
Figure 10: Continued.
38
Journal of Applied Mathematics
2
×105
1.8
1.6
1.4
IH1
1.2
1
0.8
0.6
0.4
0.2
0
0
40
20
60
80
100
Time (days)
Without control
With control
(c) Substate evolution of the mildly infected humans
10000
9000
8000
7000
IH2
6000
5000
4000
3000
2000
1000
0
0
40
20
60
Time (days)
Without control
With control
(d) Substate evolution of the severely infected humans
Figure 10: Continued.
80
100
Journal of Applied Mathematics
12
39
×106
10
SV
8
6
4
2
0
0
40
20
60
80
100
80
100
Time (days)
Without control
With control
(e) Substate evolution of the susceptible mosquitoes
2500
2000
EV
1500
1000
500
0
0
40
20
60
Time (days)
Without control
With control
(f) Substate evolution of the exposed mosquitoes
Figure 10: Continued.
40
Journal of Applied Mathematics
1800
1600
1400
1200
IV
1000
800
600
400
200
0
0
40
20
60
80
100
Time (days)
Without control
With control
(g) Substate evolution of the infected mosquitoes
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0
20
40
60
80
Time (days)
u1
u2
u3
(h) Control profile
Figure 10: Substate evolution with u1 ≠ 0, u2 ≠ 0, and u3 ≠ 0.
100
Journal of Applied Mathematics
41
Table 3: Infections prevented in order of increasing magnitude.
Total infection averted ( × 107 )
Total cost ( × 103 )
ACER
ICER
B
35343
3.6579
1 035 × 10−8
1 035 × 10−8
A
40727
4.2199
1 036 × 10−8
1 044 × 10−8
C
40786
5.0625
1 241 × 10−8
1 428 × 10−6
D
40803
6.4702
1 586 × 10−8
8 280 × 10−6
Intervention
and p2 , the ICER is the difference in cost of p1 and p2 to the
difference in the infection prevented by the intervention.
Thus,
ICER =
Difference in costs of control strategies p1 and p2
Difference in infection averted by the strategies
45
Table 3 provides the infections prevented by the named
interventions ranked in increasing order of magnitude.
To find the most cost-effective intervention, a comparative analysis of the ACER’s values of the interventions must
be made. The intervention with the least ACER value is the
most cost-effective. By inspection, intervention B is the most
cost-effective intervention. However, intervention D has the
biggest ACER value, making it the least cost-effective intervention. But to decide on which intervention to discard, further detailed analysis needs to be carried out. Hence, we
compute the ICERs of the interventions for further probing.
From the ICER computations, we observe that intervention
D has the biggest ICER value. Hence, it is discarded from
the table and further analysis is made on the remaining
interventions. The underlying analyses are made in Table 4
Now, a further dive into the analysis from Table 4
shows that intervention A has the biggest ICER value than
B. Hence, intervention A is deleted from the entries of
Table 4. It follows that a new table is constructed in that
regard.
Following the computation in Table 5, we deduce that
intervention C has a bigger ICER value than B. Hence, we
delete intervention C from Table 5. B therefore is the most
cost-effective intervention, providing the optimum performance with the least cost.
6. Conclusion
The research studied a nonlinear differential equation
malaria model to assess the impact of vaccines on the control of the disease. The research analysed the model and
characterized its properties of positivity and boundedness.
The asymptotic dynamic behaviour of the disease at the
model’s equilibria was studied. Further, bifurcation analysis
was conducted to study the local dynamic behaviour of the
malaria model (1) at the malaria-free equilibrium. The
investigation indicated that malaria model (1) exhibits forward bifurcation. Further, with the model’s input and initial
conditions, sensitivity analysis was investigated to quantify
the amount of influence each parameter has on R0 . The
Table 4: Infections prevented in order of magnitude of
interventions B, A, and C.
Total infection
averted ( × 107 )
Total cost
( × 103 )
ICER
B
35343
3.6579
1 035 × 10−8
A
40727
4.2199
1 044 × 10−8
C
40786
5.0625
1 428 × 10−6
Intervention
Table 5: Infections prevented in order of magnitude of
interventions B and C.
Total infection
averted ( × 107 )
Total cost
( × 103 )
ICER
B
35343
3.6579
1 035 × 10−8
C
40786
5.0625
1 428 × 10−6
Intervention
influences of α, β1 , β2 , ψa and μδ on R0 are depicted in
Figure 5. We observe that the R0 changes with a change in
any of the given parameters. From Figure 5(a), we note that
when α is significantly small, the disease can be mitigated
even when μδ = 1. However, when α ≥ 1, the disease cannot
be mitigated with μδ ≤ 0 1. In Figure 5(b), we see that the
disease can be mitigated when β1 is small. Thus, when β1
is small, irrespective of the value of ψa , the disease could
be eliminated. In addition, we observe in Figure 5(c) that
the influence of β2 on R0 is great. Thus when ψa ≥ 1, the
disease can be eliminated when β2 is small. But when β2 is
big, the disease cannot be mitigated. Hence, α, β1 , and β2
have a greater influence on the R0 , and minimizing them
would be a prudent initiative by stakeholders. Furthermore,
the variability in R0 was assessed by the LHS-PRCC. With
the method, the uncertainty in R0 is computed with a 95%
confidence interval, with the mean, and 5th and 95th percentiles, respectively, simulated as 0 143788, 0 01545, and
0 41491. In addition, the bifurcation type invoked at the
disease-free state is analysed and the result revealed that
the convention that R0 < 1 is the condition for eradicating
the disease is not always sufficient when the system
undergoes backward bifurcation. The presence of backward
bifurcation means that the system is characterized by a coexistence of stable endemic and disease-free equilibria even
when R0 < 1. In such circumstances, the infection needs to
be dealt with at the early stages else it becomes difficult to
control once the transmission begins to get out of control.
42
Moreover, we formulated an intervention model to assess
the respective effects of the various pairings of interventions
on the dynamics of the disease. Lastly, an intervention cost
analysis was conducted to identify the most cost-effective
intervention regarding rewarding the desired outcome.
From the analysis, intervention B was found to be the most
cost-effective intervention. As recommended by the cost
analysis, public health, in their effort to mitigate malaria in
our communities, should target advocating and making
insecticide nets available to the communities for easy access.
In addition, areas in the community which serve as a hiding
place for the vector should be targeted for spraying to control the vector. Furthermore, as confirmed by the vaccine
analysis in Subsection 3.7, vaccines can assist in closing the
gap that has not been fixed by the existing interventions.
In addition, vaccines can help minimize the overall disease
burden and incidence, hospitalization, and death from the
disease especially in children. Hence, a vaccination program
with a proper awareness strategy should be carried out by
stakeholders to inform the public of the benefits of malaria
vaccination. Lastly, vaccine availability should be 100% at
all communities marked by the public health as areas of high
malaria incidence to help foster smooth vaccination activities in these areas.
6.1. Strengths and Limitations of the Study. The research
studied the asymptotic behaviour of malaria at the model
equilibria. The bifurcation type invoked at the disease-free
point D0f is analysed as the classical understanding that R0
< 1 is the condition for eradicating the disease is not always
sufficient when the system undergoes backward bifurcation.
The study analysed the global sensitivity to quantify the
influence each parameter related to the R0 has on it. The
model was redesigned into an intervention model for public
health benefit with an insight shed on the shift from the
classical understanding of nonpharmaceutical preventive
malaria control to pharmaceutical measures of vaccines. A
cost analysis of the intervention model was examined to
identify the most cost-effective intervention regarding
rewarding the desired outcome. Even though the proposed
model and interventions identified have successfully minimized the infectives of the disease, we believe that if the
model has been fitted to available data, it will help the public
health to take the necessary action in the eradication program as the estimate of the model parameters will bring a
better estimate of the R0 of the region of the dataset. Also,
using a different methodology to analyse the disease transmission will bring to light a better understanding of the disease dynamics at all compartments. In the future, the
authors propose considering calibrating the model to available data to estimate the model parameters and applying
fractional calculus to the work to account for the memory
effect in disease transmission.
Data Availability Statement
The authors confirm that the data supporting the findings of
this study are available within the article.
Journal of Applied Mathematics
Conflicts of Interest
The authors declare no conflicts of interest.
Author Contributions
Sacrifice Nana-Kyere: conceptualization, data curation, formal
analysis, investigation, methodology, software, validation,
writing-original draft, writing-review and editing. Baba Seidu:
supervision, writing-original draft, writing-review and editing. Kwara Nantomah: supervision, writing-original draft,
writing-review and editing.
Funding
The authors received no specific funding for this work.
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