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Modeling the Inflow of Exposed and Infected Migrants on the Dynamics of Malaria

Musah Konlan
Department of Mathematics and Statistics, University of Energy and Natural Resources, Sunyani, Ghana

Correspondence: musah.konlan@uenr.edu.gh

Abstract. Malaria is currently a life-threatening vector borne disease which is endemic in mostof the developing and underdeveloped countries associated with poor health care systems. In thisstudy, a host-vector mathematical model that takes into account the inflow of human migrants whohave been exposed or infected with malaria is formulated and analysed. The reproduction numberof the mosquito vector population is derived and used as a threshold quantity for determining theexistence of the model trivial and realistic steady states. The Routh-Hurwitz criterion and somestability theorems of Metzler matrices are used to show that the realistic disease free equilibriumis both locally and globally asymptotically stable whenever the disease reproductive number is lessthan one. We derived an equation for the model endemic condition and used Descartes Rule of SignChange to established the conditions for the model to admit one or three endemic equilibrium state(s).It is further shown that in the absence of inflow of exposed or infected migrants, the model admits aglobally asymptotically unique endemic equilibrium when R0 > 1 and two endemic equilibria when
R0 < 1. Our local sensitivity analysis revealed that the adults mosquito removal and biting rateswere respectively the most significant contributing parameters to the spread of malaria. The numericalsimulations results suggested that the exposed and infected immigrants have no significant impact onthe dynamical behaviour of the model population sub-classes.

1. Introduction
Malaria is currently a life-threatening vector borne disease which is endemic in most of thedeveloping and underdeveloped countries associated with challenging health care systems. Moreparticularly, malaria is highly endemic in sub-Saharan Africa characterized with poor hygienicconditions which serve as suitable breeding site for malaria vectors [1]. Plasmodium parasites andfemale Anopheles mosquitoes are respectively the causal agent and transmitting vectors of malaria.Among the most vulnerable groups to malaria are expectant mothers and infants under five yearsof age [1–3]. Common symptoms of malaria include: fever, chills, headache, pain, anaemia andvomiting [1,4]. The World Health Organization (WHO) reported that in 2022 alone, there were twohundred and forty nine million malaria cases recorded globally. Ninety four percent of theses cases
Received: 10 Jan 2024.
Key words and phrases. Malaria; immigrants; equilibrium states; stability analysis; local sensitivity analysis; nu-merical simulations. 1

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https://doi.org/10.28924/ada/ma.4.7


Eur. J. Math. Anal. 10.28924/ada/ma.4.7 2were recorded in Africa. For example, Ghana recorded within the same period five million threehundred and fifteen thousand five hundred and ninety three (5315593) and eleven thousand fivehundred and fifty seven (11557) estimated malaria cases and deaths respectively [3]. Currently,population migration caused by climate change induced factors and conflicts constitute a majorthreat to the malaria control programs.Mathematical modeling has become a significant tool box for understanding disease transmissiondynamics and evaluating the effectiveness of disease control strategies [5]. These models gener-ally explain the dynamics of infections, provide/estimate the thresholds indicators that determinewhether the disease will persist or die out [6–8].According to Mukhaktar et al. [9], mathematicallymodeling malaria can help better understand the disease dynamics and further unveil how cer-tain factors such as human migration influence the disease transmission process, In that regard,several modeling studies have been conducted concerning human migration and malaria. Authorsin [9] assessed how human mobility impact the malaria disease burden in South Sudan. Apriantiet al. [10] examined the effect of susceptible immigrants on the spread of malaria in Indonesia.Yiga et al. [11] analysed a malaria transmission model that takes into consideration the combinedeffect of infected immigrants and other variables that depend on temperature and rainfall. Ma-liki et al. [2] modelled the control of malaria in a population with infected immigrants. Witboi etal. [12] presented a malaria population dynamics model with human migrants. Yacheur et al. [13]studied the importation of malaria infections from sub-Saharan Africa to northern Africa and theabsorption effect of the immigrants. Researchers in [14, 15] formulated and analyzed mathematicalmodels for malaria disease dynamics that considered malaria vaccination campaigns and inflow ofinfective immigrants. Ahkrizal et al. [16] formulated a malaria dynamics model capturing the inflowof exposed and infected migrants and the recovery of exposed individuals.In the above mentioned literature, little attention is given to the aquatic phase of the malariavectors. Even though, the population of adults mosquitoes responsible for disseminating malariainfections is proportional to the density of the aquatic mosquitoes. It is therefore necessary totake into consideration the aquatic stages of the vector in a malaria model [11, 17]. Hence, in thisstudy, in order to explore the impact of exposed and infected individuals on the endemic conditionof malaria, we extend the malaria models formulated in [11] to include the exposed vectors and themodel in [16] to capture the aquatic stage of the Anopheles female mosquito without the relapsefactor of the recovered individuals. The rest of the organization of the paper is as follows: sectiontwo takes care of the model formulation and analysis, in section three, the sensitivity analysisresults is presented, population simulations is carried out in section four and the conclusion ispresented in section five.

https://doi.org/10.28924/ada/ma.4.7


Eur. J. Math. Anal. 10.28924/ada/ma.4.7 32. Malaria Model Development
The model considered the interactions of humans (hosts) and female Anopheles mosquitoes(vectors). Humans (hosts) are classified into susceptible (Sh), exposed (Eh), infected (Ih) andrecovered (Rh) sub-classes. As a result, the total human population at any given time t is:

Nh (t) = Sh (t) + Eh (t) + Ih (t) + Rh (t) (1)
The human/host population is sustained at a constant birth rate πh and immigration rate M.Hence, the susceptible human class is generated at a rate (πh + (1− p1 − p2)M), where p1 and

p2 are the immigration rate of exposed and infected migrants respectively. Recovered immigrantsare assumed to be susceptible to malaria. Susceptible humans become exposed to malaria infec-tions through effective contact with infected female Anopheles mosquitoes during blood meal ata rate λh. Exposed humans progress to infected class at rate γ. The size of exposed humans isaugmented as results of immigration of humans at a rate p1M . It is common to find people insettings with limited health facilities resorting to self medication after being bitten by mosquitoesor when a family member is suspected of suffering from malaria. Hence, in this model it is assumedthat exposed individuals recover from malaria at a rate ω. The density of the infected humans isreduced following treatment at rate τ or due to malaria induced mortality at a rate δ. The size ofthe infected humans is augmented due to migration of infected individuals at rate p2M . Recoveredindividuals lose their immunity and join the susceptible sub-class at a rate ϕ. The constant µh isthe human removal rate from each human compartment.Also, the vector (Anopheles mosquito) population is stratified into immature and adult mosquitosub-populations. The immature female Anopheles mosquito sub-population includes the mosquitoeggs, larvae and pupae stages.These aquatic stages are represented by a single compartment denoted by (Am). The aquaticvector (Am) is generated from the eggs laid by the matured mosquitoes (susceptible, exposed andinfected) at a rate πm (1− Am
K

)
(Sm + Em + Im).The population of aquatic vector is bounded above by the carrying capacity of the aquatic envi-ronment (K). The aquatic mosquito population declines due to natural death at a rate µa. Theaquatic mosquitoes mature into susceptible mosquitoes at a rate ψ. The matured mosquito is furtherstratified into susceptible (Sm), exposed (Em) and infected (Im) vectors. The susceptible vectorsbecome exposed to malaria parasites during blood meal from infectious (infected) humans at a rate

λm. Exposed vectors (Em) subsequently become infected at a rate σ. As the results of naturaldeath at a rate µm, the densities of adult mosquito populations ((Sm)), (Em), (Im) decrease. Thus,at any time t , the aquatic and adult malaria vector populations (Am and Nam) satisfy:
Am (t) ≤ K, Nam (t) = Sm (t) + Em (t) + Im (t) (2)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 4

λh = bβhIm
Nh

and λm = bβmIh
Nh

are respectively the forces of infection for the human and femaleAnopheles mosquitoes. The schematic diagram (figure 1) describes the transmission dynamics ofmalaria in an interacting human and mosquito populations. The model parameters are presentedin Table (1).
Table 1. Parameter description with their values and sources

Parameter Description Value[Range] Reference Unit

πh Human recruitment rate 0.03 [11] Day−1M Immigration rate of human 0.001 [11] Day−1

p1 Immigration rate of exposed humans 0.2 [11, 16] Day−1

p2 Immigration rate of infected humans 0.2 [11, 16] Day−1

µh Natural mortality rate of human 1/21900 [11] Day−1

βh Probability of transmission of infectionsfrom an infectious human to a 0.00021 [11] -susceptible mosquito (vector)
γ Progression rate from exposed humans 1/20 [11] Day−1to infected humans
ω Progression rate from exposed humans 0.055 [16] Day−1to recovered humans
τ Progression rate from infected humans 1/30 [11] Day−1to recovered humans
ϕ Progression rate from recovered humans 1/(20× 365) [11] Day−1to susceptible humans
δ Malaria induced death for humans 0.001 [11] Day−1

πm Anopheles mosquito egg deposition rate 6 [17,18] Day−1

K Carrying capacity for immature mosquitoes 40000 [18] Space
b Female Anopheles mosquito biting rate 0.94[0.1-1] [18] Day−1

βm Probability of transmission ofinfections from an infected 0.00021 [11] -Anopheles mosquito to a susceptible human
ψ Maturity rate of immature mosquitoes 0.08 [19] Day−1

σ Progression rate from exposed mosquitoesto infected mosquitoes 0.091 [18] Day−1

µm Natural mortality rate of adult mosquitoes 0.11346 [17] Day−1

µa Natural mortality rate of immature mosquito 0.1042 [18,19] Day−1

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 5

Figure 1. Schematic diagram for malaria transmission dynamics with human immi-grants
Based on figure 1, the following system of equations is derived:



dSh
dt

= πh + (1− p1 − p2)M + ϕRh − (λh + µh)Sh

dEh
dt

= p1M + λhSh − g0Eh

dIh
dt

= p2M + γEh − g1Ih

dRh
dt

= τIh + ωEh − g2Rh

dAm
dt

= πm
(

1− Am
K

)
(Sm + Em + Im)− g3Am

dSm
dt

= ψAm − (λm + µm)Sm

dEm
dt

= λmSm − g4Em

dIm
dt

= σEm − µmIm

(3)

where: g0 = (ω+γ+µh), g1 = (τ+δ+µh), g2 = (ϕ+µh), g3 = (ψ+µa) and g4 = (σ+µm)

https://doi.org/10.28924/ada/ma.4.7


Eur. J. Math. Anal. 10.28924/ada/ma.4.7 62.1. Boundedness of Solution.

Theorem 1. For non-negative initial values Sh(0), Eh(0), Ih(0), Th(0), Am(0), Sm(0) and Im(0)

of system (3), each element of the solution set {Sh(t), Eh(t), Ih(t), Th(t), Am(t), Sm(t) Im(t)}
is non-negative and bounded ∀ t ≥ 0.

Proof. Considering the first differential equation in system (3):
dSh
dt

= πh + (1− p1 − p2)M + ϕRh − (λh + µh)Sh (4)
=⇒

dSh
dt
≥ −(λh + µh)Sh

=⇒
∫

1

Sh
dSh ≥ −

∫
(λh + µh)dt

=⇒ Sh(t) ≥ Sh(0)e−(µht+
∫ t
0 λh(x)dx) ≥ 0Similarly:

dEh
dt

= p1M + λhSh − g0Eh =⇒ Eh(t) ≥ Eh(0)e−g0t ≥ 0

dIh
dt

= p2M + γEh − g1Ih =⇒ Ih(t) ≥ Ih(0)e−g1t ≥ 0

dRh
dt

= ωEh + τIh − g2)Rh =⇒ Rh(t) ≥ Rh(0)e−g2t ≥ 0

dAm
dt

= πm

(
1−

Am
K

)
(Sm + Em + Im)− g3Am =⇒ Am(t) ≥ Am(0)e−g3t ≥ 0

dSm
dt

= ψAm − (λm + µm)Sm =⇒ Sm(t) ≥ Sm(0)e−(µmt+
∫ t
0 λm(x)dx) ≥ 0

dEm
dt

= λmSm − g4Em =⇒ Sm(t) ≥ Sm(0)e−g4t ≥ 0

dIm
dt

= λmSm − µmIm =⇒ Im(t) ≥ Im(0)e−µmt ≥ 0

Therefore, for ∀ t ≥ 0, the state variables of the model have non-negative solutions.
2.2. Invariant Region. This section is dedicated to finding the region over which the solution setof our malaria model system of equations is well posed.
Theorem 2. The feasible region in which the solution set of the model system of equations make
biological sense is the set;

D = Dh ×Dm ⊂ R4+ × R4+ (5)
where

Dh =

{
(Sh, Eh, Ih, Rh) ∈ R4+ : Sh + Eh + Ih + Rh ≤

M + πh
µh

} (6)
and

Dm =

{
(Am, Sm, Em, Im) ∈ R4+ : Am ≤ K, Sm + Em + Im ≤

ψK

µm

} (7)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 7

Proof.Firstly, we determine the subset Dh.The human (host) population Nh at any given time t is:
Nh = Sh + Eh + Ih + Rh (8)

Taking the differential of both sides of equation (8) and simplifying gives:
dNh
dt

= M + πh − µhNh − δIh

=⇒
dNh
dt
≤ M + πh − µhNh (in the absence of malaria induced mortality)

=⇒
dNh

Nh − M+πh
µh

≤ −µhdt

(9)

Integrating the last inequality in (9) and taking the limit as t → +∞, yield: Nh → M+πh
µhConsequently, the following result is obtained

0 ≤ Nh ≤
M + πh
µh

(10)
Therefore:

Dh =

{
(Sh, Eh, Ih, Rh) ∈ R4+ : Sh + Eh + Ih + Rh ≤

M + πh
µh

} (11)
Secondly, the subset Dm is determined. At any point in time, the mosquito (vector) populationsatisfies:

Am ≤ K, Nam = Sm + Em + Im (12)
Now, Nam = Sm + Em + Im

=⇒
d

dt
(Nam) =

d

dt
(Sm + Em + Im)

=⇒
dNam
dt

=
dSm
dt

+
dEm
dt

+
dIm
dt

=⇒
dNam
dt

≤ ψK − µmNam

=⇒ Nam −
ψK

µm
≤
(
Nam(0)−

ψK

µm

)
e−µmt

=⇒ Nam ≤
ψK

µm
as t → +∞.

Therefore, Dm =

{
(Am, Sm, Em, Im) ∈ R4+ : Am ≤ K, Sm + Em + Im ≤

ψK

µm

} (13)
Thus, the feasible region for system (3) is the set:

D = Dh ×Dm ⊂ R4+ × R4+ (14)

https://doi.org/10.28924/ada/ma.4.7


Eur. J. Math. Anal. 10.28924/ada/ma.4.7 82.3. Model Equilibrium Points. To discuss the model equilibrium points, we consider two cases.The case where there is inflow of exposed and infected migrants (p1, p2 > 0). For this scenario,there is no disease free equilibrium and the model admits only the endemic equilibrium to bedetermine later. The second case is when there is no immigration of exposed and infected humans(p1 = p2 = 0). In this case, the computation of the model disease-free equilibria, is summarized inthe theorem below. This approach is adopted from [20–22].
Theorem 3. For convenience, we define the threshold parameter

N =
πmψ

µm(ψ + µa)
=
πmψ

g3µm
(15)

as the mosquito net reproduction or extinction number, then if:(1) N ≤ 1, system (3) admits a trivial disease-free equilibrium (TDFE) (which corresponds to
a population without mosquitoes) given by:

ξ0 = (S∗h, 0, 0, 0, 0, 0, 0, 0) (16)
(2) N > 1 (mosquitoes persist in the community), system (3) admits a realistic disease-free

equilibrium (RDFE) (since it corresponds to the existence of mosquitoes in the population)
given by:

ξ1 = (S∗h, 0, 0, 0, 0, A∗m, S
∗
m, 0) (17)

where: S∗h = M+πh
µh

, A∗m = K
(

1− 1
N
)
and S∗m = ψK

µm

(
1− 1

N
)

Proof.Suppose, (S∗h, E∗h , I
∗
h , R∗h, A∗m, S∗m, E∗m, I∗m

) is any arbitrary disease-free equilibrium point.Setting system (3) to zero with the condition that there are no infections at the disease-freeequilibrium, that is, p1 = p2 = E∗h = I∗h = R∗h = E∗m = I∗m = 0, gives: S∗h = M+πh
µh

for the firstequation.Also, it is not hard to see from system (3) that
S∗m + E∗m + I∗m =

ψA∗m
µm

(18)
Hence, from the sixth equation of system (3), we see that A∗m satisfies:

ψπm
µm

(
1−

A∗m
K

)
A∗m − g3A∗m = 0 (19)

=⇒ A∗m = 0 or A∗m = K

(
1−

1

N

) (20)
Now A∗m = 0 =⇒ S∗m = 0 and A∗m = K

(
1−

1

N

)
=⇒ S∗m =

ψK

µm

(
1−

1

N

) (21)
Hence, ξ0 and ξ1 are obtained respectively from A∗m = 0 and A∗m = K

(
1− 1

N
). Clearly, themagnitude of N dictates the existence of the model disease-free equilibrium points.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 9

N is a threshold quantity known as the vector offspring number or vector net reproduction number[18, 19, 23]. In general, N can be interpreted as a measure of the average number of new adultfemale Anopheles mosquitoes produced by one reproductive Anopheles mosquito during its entirereproductive life. It is expressed as a product of the egg deposition rate πm, the fraction of immaturemosquito that survive and develop into adult Anopheles mosquito ψ
ψ+µa

and the average life span ofadult Anopheles mosquito 1
µm

. Thus, if N > 1, the mosquito population persists in the community,otherwise if N ≤ 1, the malaria vector population becomes extinct and the local transmissionof malaria cannot take place. It is worth noting that the trivial disease-free equilibrium (TDFE)corresponds to the absence of female Anopheles mosquitoes in the community. Hence, the TDFEis biologically less meaningful.
2.4. The Basic Reproductive Number. In epidemiology, the basic reproductive number (Ro ) isa threshold quantity that is used to determine the extent of severity of the epidemics. In thisstudy, the method of next generating matrix is adopted to compute the model Ro . Expressing ourmodel differential equations in the form dX

dt = (F − V)XT where XT denotes the transpose of
X = (Eh, Ih, Em, Im), F and V are vectors denoting the rate of generation of new infections andtransfer rates respectively, gives:

F =


p1M + λhSh

p2M

λmSm

0

 and V =


g0Eh

−γEh + g1Ih

g4Em

−σEm + µmIm

 (22)
Evaluating the Jacobian matrices F and V of F and V at the RDFE gives respectively:

F =


0 0 0

bβhS
∗
h

N∗h

0 0 0 0

0
bβmS∗m
N∗h

0 0

0 0 0 0

 and V =


g0 0 0 0

−γ g1 0 0

0 0 g4 0

0 0 −σ µm

 (23)

From the expression of V, the inverse of V is:

V −1 =



1
g0

0 0 0

γ
g0g1

1
g1

0 0

0 0 1
g4

0

0 0 1
g4µm

1
µm


(24)

https://doi.org/10.28924/ada/ma.4.7


Eur. J. Math. Anal. 10.28924/ada/ma.4.7 10Hence, the next generation matrix FV −1 is given by:

FV −1 =



0 0
σbβhS

∗
h

g4N
∗
hµm

bβhS
∗
h

N∗hµm

0 0 0 0

γbβmS∗m
g0g1N

∗
h

bβmS∗m
g1N

∗
h

0 0

0 0 0 0


(25)

Solving for λ in the relation ∣∣FV −1 − λI∣∣ = 0 , where I is a unit matrix and λ an eigenvalue of
FV −1, we get the dominant eigenvalue as:

λmax = R0 =

√
σγb2βhβmS

∗
hS
∗
m

g0g1g4N
∗2
h µm

(26)
Taking N∗h = S∗h and simplifying the expression in (26), we obtain the reproductive number of themodel given by:

R0 =

√
σγb2βhβmµhKψ

g0g1g4(M + πh)µ2m

(
1−

1

N

)

=
√
R0h × R0m

(27)
where:

R0h =
γbβhµh

g0g1(M + πh)
and R0m =

σbβmKψ

g4µ2m

(
1−

1

N

)
The threshold quantities R0h and R0m characterized the contributions of malaria disease spreadfrom human to mosquito (host to vector) and from mosquito to human (vector to host) respectively.

R0h represents the number of secondary cases of Anopheles mosquitoes one infectious (infectedor treated) human will generate in a completely susceptible population of Anopheles mosquitoesduring its infectious phase. Similarly, R0m can be interpreted as the number of secondary humancases generated by an infected Anopheles mosquito in an entirely susceptible human populationover the course of its life time as infectious [2].
2.5. Stability of Malaria-Free Equilibrium.

2.5.1. Local Stability of Malaria-Free Equilibrium.

Theorem 4. The RDFE (ξ1) =
(
M+πh
µh

, 0, 0, 0, 0, K
(

1− 1
N
)
, Kψ
µm

(
1− 1

N
)
, 0
)

with N >

1 is locally asymptotically stable (LAS) if R0 < 1 and unstable if R0 > 1

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 11

Proof. Let matrix J0 be the Jacobian matrix of system (3) evaluated at the RDFE (ξ1). Thus,

J0 =



−µh 0 0 ϕ 0 0 0 −bβhS
∗
h

N∗h

0 −g0 0 0 0 0 0
bβhS

∗
h

N∗h

0 γ −g1 0 0 0 0 0

0 ω τ −g2 0 0 0 0

0 0 0 0 −
(
g3 +

πmS∗m
K

)
πm
N

πm
N

πm
N

0 0 −bβmS
∗
m

N∗h
0 ψ −µm 0 0

0 0
bβmS∗m
N∗h

0 0 0 −g4 0

0 0 0 0 0 0 σ −µm


(28)

It is not hard to see that the matrix in (28) admits two negative eigenvalues, namely λ1 = −µh and
λ2 = −g2. Using the matrix reduction method, the remaining eigenvalues can be obtained from thesub-matrix in (29) below:

J1 =



−g0 0 0 0 0
bβhS

∗
h

N∗h

γ −g1 0 0 0 0

0 0 −
(
g3 +

πmS∗m
K

)
πm
N

πm
N

πm
N

0 −bβmS
∗
m

N∗h
ψ −µm 0 0

0
bβmS∗m
N∗h

0 0 −g4 0

0 0 0 0 σ −µm


(29)

The characteristic equation of the sub-matrix in (29) is given by
(λ+ g0)(λ+ g1)(λ+ g4)(λ+ µm)

(
λ2 + Sλ+ P

)
= 0 (30)

where:
S =

πmS∗m
K + g3 + µm, and P =

πmµmS∗m
KIt can clearly be seen from (30) that four eigenvalues of the sub-matrix in (29) λ3 = −g0,

λ4 = −g1, λ5 = −g4, and λ6 = −µm are negative. Also, the nature of the remaining twoeigenvalues of the sub-matrix in (29) are determined from :
λ2 + Sλ+ P = 0 (31)

Since, S and P are positive whenever N > 1, it implies that the two remaining eigenvalues ofthe sub-matrix J1 are stricly negative. Consequently, all eigenvalues of the matrix J0 are real andnegative. Hence, according to the Routh-Hurwitz stability criterion, the malaria realistic disease-free equilibrium state ξ1 is locally asymptotically stable when N > 1 and R0 < 1 and unstableotherwise.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 122.5.2. Global Stability of Malaria-Free Equilibrium. Following [19, 24–27], the global stability ofa system equilibrium point can be established by first expressing the system in a triangular formas follows: dYs
dt = B1 (Ys − YRDFE) + B12Yi

d Yi
dt = B2Yi

(32)
Here, Ys and Yi denotes the compartments of non-transmitting and transmitting hosts and vec-tors respectively, with Ys = (Sh, Rh, Am, Sm)T Yi = (Eh, Ih, Em, Im)T and YRDFE =(
S∗h, R

∗
h, A

∗
m, S

∗
m

)
=
(
M+πh
µh

, K
(

1− 1
N
)
, ψKµm

(
1− 1

N
))

(Ys − YRDFE) =



Sh − M+πh
µh

Rh

Am −K
(

1− 1
N
)

Sm − Kψ
µm

(
1− 1

N
)

(33)

B1 =
∂Ys

∂(Sh, Rh, Am, Sm)
(34)

B12 =
∂Ys

∂(Eh, Ih, Em, Im)
(35)

B2 =
∂Yi

∂(Eh, Ih, Em, Im)
(36)Using our model system of equations system (3), we get:

B1 =


−µh ϕ 0 0

0 −g2 0 0

0 0 −
(
πmS∗m
K + ψ + µa

)
πm
N

0 0 ψ −µm

 (37)

B12 =


0 0 0 −bβhS

∗
h

N∗h

ω τ 0 0

0 0 πm
N

πm
N

0 −bβmS
∗
m

N∗h
0 0

 (38)

B2 =


−g2 0 0

bβhS
∗
h

N∗h

γ −g1 0 0

0
bβmS∗m
N∗h

−g4 0

0 0 σ −µm

 (39)
From the above we formulate the theorem as follows.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 13

Theorem 5. The system dYs
dt = B1 (Ys − YRDFE)+B12Yi is globally asymptotically stable (GAS) at the

RDFE when all eigenvalues of matrix B1 have negative real parts and B2 is a Metzler matrix.

Proof.Clearly, two eigenvalues of the matrix B1 are λ1 = −µh and λ2 = −g2. Thus, applying themethod of matrix reduction, B1 reduces to the sub-matrix:
A =

−(πmS∗mK + g3

)
πm
N

ψ −µm

 (40)
The nature of the remaining two eigenvalues of B1 are determined from characteristic equation:

λ2 +

(
g3 + µm +

πmS
∗
m

K

)
λ+ πmψ

(
1−

1

N

)
= 0 (41)

Since in equation (41), g3+µm +
πmS∗m
K > 0 and πmψ (1− 1

N
)
> 0 whenever N > 1, we concludeusing the Routh-Hurwitz stability condition, that the eigenvalues λ3 and λ4 have negative realparts. Hence, all the eigenvalues of the matrix B1 have negative real parts.Additionally, B2 is clearly a Metzler matrix (since all the off diagonal entries are non negative).Thus, we conclude that the system

d Ys
dt

= B1 (Ys − YRDFE) + B12Yi (42)
is GAS at the realistic disease free equilibrium [19,24–27].
2.6. Malaria Endemic Equilibrium. Let ξ2 = (S∗∗h , E

∗∗
h , I

∗∗
h , R

∗∗
h , A

∗∗
m , S

∗∗
m , E

∗∗
m , I

∗∗
m ) be theendemic equilibrium (EE) point for the malaria model, then setting system (3) to zero, the followingsystem of solutions is obtained

S∗∗h = Q0
(g0bϕτβmµh)I

∗∗2
h +[Q1bβmµh+µm(M+πh)]I

∗∗
h +Q1µm(M+πh)

[Q2σb2βhβmµhψA∗∗m+g4bβmµhµm(M+πh)]I
∗∗
h +g0g2g4µ

2
m(M+πh)

2

E∗∗h =
Q3I

∗∗3
h +Q4I

∗∗2
h +Q5I

∗∗
h +Q6

Q7I
∗∗2
h +Q8I

∗∗
h +Q9

R∗∗h =
τI∗∗h +ωE

∗∗
h

g2

A∗∗m = 0 or A∗∗m = K
(

1− 1
N
)

S∗∗m = 0 or S∗∗m =
(M+πh)ψA

∗∗
m

bβmµhI
∗∗
h +µm(M+πh)

E∗∗m = 0 or E∗∗m =
bβmµhS

∗∗
m I
∗∗
h

g4(M+πh)

I∗∗m = 0 or I∗∗m =
σbβmµhS

∗∗
m I
∗∗
h

g4µm(M+πh)here, I∗∗h satisfies : q3I
∗∗3
h + q2I

∗∗2
h + q1I

∗∗
h + q0 = 0

(43)

Where:
Q0 = g4µm(M+πh)

µh

Q1 = p1ϕωM + g0g2((1− p1 − p2)M + πh)

Q2 = g0g2 − ϕω

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 14

Q3 = 1
M+πh

g0g4ϕτσb
3βhβ

2
mµhµmψA

∗∗
m

Q4 = 1
M+πh

g4ϕτσb
2βhβmµhψA

∗∗
m

(
Q1bβmµm + M+πh

µh
g0ϕτ

)
+

g4p1Mbβmµm

(
Q2σb

2βhβmµhψA
∗∗
m

M+πh
+ g0g2g4bβmµhµm

)
Q5 = g4bβmµ

2
m (σbβhψA

∗∗
m (Q1 + p1MQ2) + 2g0g2g4p1Mµm(M + πh))

Q6 = g0g2p1M
µh

(
µ2mg4(M + πh)

)2
Q7 = g0g4bβmµhµm

M+πh

(
Q2σb

2βhβmψA
∗∗
m + g0g2g4bβmµm(M + πh)

)
Q8 = g0g4µ

2
m

(
Q2σb

2βhβmψA
∗∗
m + 2g0g2g4bβmµm(M + πh)

)
Q9 = g2

µh

(
g0g4µ

2
m(M + πh)

)2
q3 =g0g4bβmµm{

σb2βhβmµhA
∗∗
m

M + πh
(g0g1µh + ϕγ(µh + δ)) + g0g1g2g4bβmµhµm}

q2 =g0g1g4µ
2
m(M + πh)(

Q2σb
2βhβmψA

∗∗
m

M + πh
+ 2g0g2g4bβmµm−

γσb2βhβmA
∗∗
m

M + πh
(g4Q1bβmµhµm + g0g4ϕτµ

2
m(M + πh)−

g4Mbβmµhµm(p1γ + g0p2)(
ασb2βhβmA

∗∗
m

M + πh
+ g0g2g4bβmµm)

q1 =
g4µ

2
m(M + πh)

µh
{g20g1g2g4µ2m(M + πh)−

Q1γσb
2βhβmµhψA

∗∗
m

M + πh
−

g0p2M(
Q2σb

2βhβmµhψA
∗∗
m

M + πh
+ 2g20g2g4bβmµhµm}

q0 = −
g0g2
µh

M(p1γ + g0p2
(
g4µ

2
m(M + πh)

)2
To analyse the disease endemic condition, we consider the polynomial function:

f (I∗∗h ) = q3I
∗∗3
h + q2I

∗∗2
h + q1I

∗∗
h + q0 = 0 (44)

There is enough evidence that the polynomial in (44) admits a positive solution on the interval
[0,+∞) since: f (0) = q0 < 0 and lim

I∗∗h →+∞
f (I∗∗h ) = +∞. Next, we employ Descartes’ Rule ofSigns Change to explore more information on the roots of the polynomial f (I∗∗h ) (see table 2).

Table 2. Number (#) of Possible Positive Roots of f (I∗∗h )

Case q3 q2 q1 q0 # of sign change # of roots(i) + + + − 1 1(ii) + + − − 1 1(iii) + − + − 3 1, 3(iv) + − − − 1 1

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 15Based on the results in table 2, we claim that in the presence of importation of malaria infections,system (3) may admit one or three endemic equilibrium state(s). To better understand any possibleimpact of the inflow of humans who have been exposed or infected with malaria parasites on thedisease endemic condition, we now consider the endemic relation in the absence of exposed andinfected immigrants.To that effect, we set p1 = p2 = 0 into (44) and simplify to obtain:
I∗∗h
(
a2I
∗∗2
h + a1I

∗∗
h + a0

)
= 0 (45)

Equation (45) implies
I∗∗h = 0 or a2I

∗∗2
h + a1I

∗∗
h + a0 = 0 (46)where:

a2 =bβmµh{
σb2βhβmµhA

∗∗
m

M + πh
(g0g1µh + ϕγ(µh + δ)) + g0g1g4bβmµhµm} (47)

a1 =g1µhµm
(
Q2σb

2βhβmψA
∗∗
m + 2g0g2g4bβmµm(M + πh)

)
−

γσb2βhβmµhψA
∗∗
m (g2bβmµh + ϕτµm)

(48)
a0 = g0g1g2g4µ

3
m(M + πh)2

(
1− R20

) (49)With I∗∗h = 0 in (46), we retrieve the TDFE when A∗∗m = 0 and the RDFE when A∗∗m = K
(

1− 1
N
)

Furthermore, a solution to the quadratic equation in (46) can be obtained using the quadraticformula, that is :
I∗∗h =

−a1 ±
√
a21 − 4a0a2

2a2
(50)The expression in (50) leads to the following theorem:

Theorem 6. In the absence of inflow of exposed and infected human migrants, the malaria model
represented by system (3) admits:(i) one unique EE if a0 < 0, that is R0 > 1(ii) one unique EE if a1 < 0, and R0 = 1 or a21 − 4a0a2 = 0(iii) two EE if a1 < 0 and a0 > 0 that is R0 < 1 or a21 − 4a0a2 > 0(iv) no EE otherwise.

We deduce from case (i) of theorem 6 that for a specific case where the parameter accountingfor the importation of malaria infections is zero, system 3 admits a unique EE when R0 > 1. Thissuggests that even in the absence of importation of malaria infections from elsewhere, malariaepidemics can continue to propagate in the population. The occurrence of two endemic equilibriawhen R0 does not exceed one, case (iii) of theorem 6 shows that the model bifurcate backwardly.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 16That is, two stable equilibrium states coexist when the reproductive number of the model is lessthan one. In this case, R0 < 1 even though necessary is no more enough for the elimination ofmalaria. To obtain the value of R0 say, Rc0 at which backward bifurcation takes place in this case,we set the discriminant (∆) of equation 46 to zero and solve for Rc0 . That is
∆ = 0 =⇒ a21 − 4a0a2 = 0

=⇒ Rc0 =

√
1−

a21
4a2g0g1g2g4µ3m(M + πh)2

Hence, for values of R0 between Rc0 < R0 < 1, system 3 in the absence of importation of malariainfections experiences backward bifurcation.
2.7. Global Stability of the Malaria Endemic Equilibrium Point. In what follows, we explore thelong term behavior of the unique endemic equilibrium point whenever it exist.

Consider the Lyapunov candidate:
L
(
S∗∗h , E

∗∗
h , I

∗∗
h , R

∗∗
h , A

∗∗
m , S

∗∗
m , E

∗∗
m , I

∗∗
m

)
=

(
(Sh − S∗∗h )− S∗∗h ln

Sh
S∗∗h

)
+

(
(Eh − E∗∗h )− E∗∗h ln

Eh
E∗∗h

)
+

(
(Ih − I∗∗h )− I∗∗h ln

Ih
I∗∗h

)
+

(
(Rh − R∗∗h )− R∗∗h ln

Rh
R∗∗h

)
+

(
(Am − A∗∗m )− A∗∗m ln

Am
A∗∗m

)
+

(
(Sm − S∗∗m )− S∗∗m ln

Sm
S∗∗m

)
+

(
(Em − E∗∗m )− E∗∗m ln

Em
E∗∗m

)
+

(
(Im − I∗∗m )− I∗∗m ln

Im
I∗∗m

)
Taking the time derivative of L gives:
dL
dt

=

(
1−

S∗∗h
Sh

)
dSh
dt

+

(
1−

E∗∗h
Eh

)
dEh
dt

+

(
1−

I∗∗h
Ih

)
dIh
dt

+

(
1−

R∗∗h
Rh

)
dRh
dt

+

(
1−

A∗∗m
Am

)
dAm
dt

+

(
1−

S∗∗m
Sm

)
dSm
dt

+

(
1−

E∗∗m
Em

)
dEm
dt

+

(
1−

I∗∗m
Im

)
dIm
dt

=

(
Sh − S∗∗h
Sh

)
[πh + (1− p1 − p2)M + ϕRh − (λh + µh)Sh] +

(
Eh − E∗∗h
Eh

)
(p1M + λhSh − g0Eh)

+

(
Ih − I∗∗h
Ih

)
(p2M + γEh − g1Ih) +

(
Rh − R∗∗h
Rh

)
(τIh + ωEh − g2Rh)

+

(
Am − A∗∗m
Am

)
[πm

(
1−

Am
K

)
Nam − g3Am] +

(
Sm − S∗∗m
Sm

)
[ψAm − (λm + µm)Sm]

+

(
Em − E∗∗m
Em

)
(λmSm − g4Em) +

(
Im − I∗∗m
Im

)
(σEm − µmIm)

= πh + (1− p1 − p2)M + ϕRh + (λh + µh)S∗∗h − (πh + (1− p1 − p2)M + ϕRh)
S∗∗h
Sh (51)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 17

− (λh + µh)Sh + p1M + λhSh + g0E
∗∗
h − g0Eh − (p1M + λhSh)

E∗∗h
Eh

+ p2M + γEh + g1I
∗∗
h

− g1Ih − (p2M + γEh)
I∗∗h
Ih

+ τIh + ωEh + g2R
∗∗
h − g2Rh − (τIh + ωEh)

R∗∗h
Rh

+ Namπm + Namπm
A∗∗m
K

+ g3A
∗∗
m − Namπm

Am
K
− g3Am − Namπm

A∗∗m
Am

+ ψAm + λmSm + g4E
∗∗
m − g4Em − λmSm

E∗∗m
Em

S∗∗m
Sm

+ σEm + µmI
∗∗
m − µmIm − σEm

I∗∗m
Im

= L+ − L−where
L+ = πh +M + ϕRh + (λh + µh)S∗∗h + (p1 + p2)M

S∗∗h
Sh

+ λhSh + g0E
∗∗
h + γEh + g1I

∗∗
h + τIh

+ ωEh + g2R
∗∗
h + Namπm + Namπm

A∗∗m
K

+ g3A
∗∗
m + ψAm + (λm + µm)S∗∗m + λmSm + g4E

∗∗
m

+ σEm + µmI
∗∗
m

L− = (πh +M + ϕRh)
S∗∗h
Sh

+ (λh + µh)Sh + g0Eh + (p1M + λhSh)
E∗∗h
Eh

+ (p2M + γEh)
I∗∗h
Ih

+ g1Ih

+ g2Rh + (τIh + ωEh)
R∗∗h
Rh

+ Namπm
Am
K

+ g3Am + Namπm
A∗∗m
Am

+ λmSm
E∗∗m
Em

+ g4Em + µmIm

+ σEm
I∗∗m
Im

(52)

Since the model parameters and state variables are non-negative, it follows from (52) that
dL
dt ≤ 0 if L+ ≤ L−and dL

dt = 0 if and only if S∗∗h = Sh, E
∗∗
h = Eh, I

∗∗
h = Ih, R

∗∗
h =

Rh, A
∗∗
m = Am, S

∗∗
m = Sm, E

∗∗
m = Em, and I∗∗m = Im

Therefore, the largest compact invariant set within the model’s invariant region is the singleton
{S∗∗h , E∗∗h , I∗∗h , R∗∗h , A∗∗m , S∗∗m , E∗∗m , I∗∗m }. Hence, by the Lasalle’s invariant principle [28], theunique endemic equilibrium of system (3) is globally asymptotically stable whenever it exists.

3. Local Sensitivity Analysis
In this section, local sensitivity analysis is carried out to determine the parameters that mostlycontribute to disease spread or increase (R0). These parameters should be targeted during anyintervention aimed at combating the malaria infections. Using the normalized forward sensitivityindex relation:

Γw
x =

∂w
∂x
×
x

w (53)
and the model parameter values provided in Table 1 we compute the values for sensitivity indicesof the parameters of the model reproductive number, (R0) as presented in Table 3.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 18

Table 3. The values of the sensitivity indices
Parameter Sensitivity indexb +1.00

βh +0.50

βm +0.50

πh −0.4839M −0.0161

µh +0.4991

ω −0.2618

γ +0.2620

τ −0.4848

δ −0.0291

πm +1.98 ×10−4
k +0.50

ψ +0.4999
σ +0.2775

µa −1.12 ×10−4
µm −1.2779

If the sign of the sensitivity index of a given parameter of R0 is positive, it means R0 is directlyproportional to that parameter. That is, an increase (decrease) in the parameter value when otherparameters remain constant would result in an increase (decrease) in disease incidence. Conversely,if the sign of the sensitivity index of a given parameter is negative, then R0 is indirectly proportionalto that parameter [29]. From table 3, it is clear that an increase in the parameters: b, βh, βm,
K,ψ. γ, πm , and σ will lead to an increase in the disease spread (R0) while an increase in theparameters: µm, τ , ω and µa will result in a reduction of the disease spread R0 and vice versa.

4. Numerical Simulations
In order to explore the possible impact of the exposed and infected human immigrants on thedynamical behaviour of the malaria model sub-populations, system (3) is simulated using the fol-lowing assumed set of initial condition values of the state variables:

{Sh(0), Eh(0), Ih(0), Rh(0) Am(0), Sm(0), Em(0), Im(0)} = {700, 350, 100, 0, 5000, 1000, 300, 120}}and the parameter values provided in Table 1. The results (figures 10-13) suggest that the exposed andinfected human immigrants have no influence on the population density of the humans and mosquitoes inthe community. It can also be observed from the simulation results that the population of the immature andsusceptible Anopheles mosquitoes remain high in the community. This implies that efficient vector control

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 19

strategies for both the immature and mature Anopheles mosquitoes are urgently required if malaria is to beeradicated from the population.

Figure 2. SusceptibleHumans Figure 3. Exposed Hu-mans

Figure 4. Infected Hu-mans Figure 5. RecoveredHumans

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 20

Figure 6. AquaticMosquito Figure 7. SusceptibleMosquito

Figure 8. ExposedMosquitoes Figure 9. InfectedMosquitoes

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 21

Figure 10. Hu-man classes with
p1 = p2 = 0.2

Figure 11. Hu-man classes with
p1 = p2 = 0

Figure 12. Mosquitoclasses with
p1 = p2 = 0.2

Figure 13. Mosquitoclasses with
p1 = p2 = 0

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Eur. J. Math. Anal. 10.28924/ada/ma.4.7 22

5. Conclusion
In this study, a deterministic compartmental model for malaria dynamics that takes into consideration theinflow of exposed and infected migrants and the recovery of exposed humans is formulated and analysed.In the absence of inflow of exposed and infected humans from elsewhere, the model disease free states areobtained and the biologically desired infection-free equilibrium point (RDFE) is shown to be both locally andglobally asymptotically stable when the disease reproduction number (R0) is less than one and unstable if

R0 > 1. Furthermore, we derived the equation for the endemic condition and used the Descartes rule of signchange to establish the conditions for the model to admit one or three endemic equilibrium state(s). For aspecial case of no inflow of exposed or infected migrants, we proved that the model admits a global asymptoticstable unique endemic equilibrium if R0 > 1 and two endemic equilibria when R0 < 1. The results from ourlocal sensitivity analysis revealed that adult mosquito removal and biting rates (µm and b) are respectivelythe most sensitive parameters to the spread of malaria. This suggests that malaria vector control remainsa key factor for consideration in the elimination of malaria epidemics. Our numerical simulation graphicalresults indicate that the inflow of exposed and infected migrants has no significant impact on the dynamicalbehavior of the model population sub-classes. Thus, we recommend that real immigrants data is used to fitthe malaria model and explore more on the disease dynamics in the presence of exposed or infected humanimmigrants.
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https://doi.org/10.28924/ada/ma.4.7

	1. Introduction
	2. Malaria Model Development
	2.1. Boundedness of Solution
	2.2. Invariant Region
	2.3. Model Equilibrium Points
	2.4. The Basic Reproductive Number
	2.5. Stability of Malaria-Free Equilibrium
	2.6.  Malaria Endemic Equilibrium
	2.7. Global Stability of the Malaria Endemic Equilibrium Point

	3. Local Sensitivity Analysis
	4. Numerical Simulations
	5. Conclusion
	References

