Impact of Hygiene on Malaria Transmission Dynamics: A Mathematical Model

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by  Temidayo Oluwafemi, Temidayo Oluwafemi

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Journal of Multidisciplinary Applied Natural Science Vol. 2 No. 1 (2022) https://doi.org/10.47352/jmans.2774-3047.97
Research Article
Impact of Hygiene on Malaria Transmission Dynamics:
A Mathematical Model

Temidayo Oluwafemi*, Emmanuel Azuaba
Received : May 31, 2021 Revised : August 26, 2021 Accepted : August 26, 2021 Online : August 26, 2021
Abstract
Malaria continues to pose a major public health challenge, especially in developing countries, 219 million cases of malaria were
estimated in 89 countries. In this paper, a mathematical model using non-linear differential equations is formulated to describe the
impact of hygiene on Malaria transmission dynamics, the model is analyzed. The model is divided into seven compartments which
includes five human compartments namely; Unhygienic susceptible human population S, Hygienic Susceptible Human population
u
S , Unhygienic infected human population I , hygienic infected human population I and the Recovered Human population R and
n u n n
the mosquito population is subdivided into susceptible mosquitoes S and infected mosquitoes I. The positivity of the solution
v v
shows that there exists a domain where the model is biologically meaningful and mathematically well-posed. The Disease-Free
Equilibrium (DFE) point of the model is obtained, we compute the Basic Reproduction Number using the next generation method
and established the condition for Local stability of the disease-free equilibrium, and we thereafter obtained the global stability of the
disease-free equilibrium by constructing the Lyapunov function of the model system. Also, sensitivity analysis of the model system
was carried out to identify the influence of the parameters on the Basic Reproduction Number, the result shows that the natural
death rate of the mosquitoes is most sensitive to the basic reproduction number.
Keywords mathematical model, malaria, hygiene, stability analysis, basic reproduction number, lyapunov function, sensitivity
analysis

1. INTRODUCTION most especially for children under five years, after
pneumonia and diarrheal disease.
Malaria is one of the infectious diseases with an One of the factors that increase the transmission
adverse effect on the human population. The dynamics of malaria in poor environmental
malaria parasite that lives part of its life in humans conditions. These conditions include unsafe water
and the remaining in vectors is transmitted between supplies, poor personal hygiene, poor sanitary
human host and mosquito vector by the infected facilities, poor living standards, and unhygienic
female Anopheles mosquitoes through bites. In rare food. Poor personal hygiene may result in water-
cases, people may be infected via contaminated borne diseases [5]. According to Nkuo-Akenji et al.
blood, or a fetus may become infected by its mother [6] poor environmental sanitation (hygiene) and
during pregnancy or after delivery. Two of the five housing conditions might be significant risk factors
parasites species – Plasmodium Falciparum and for malaria burden. Also, Enebeli et al., [7] in their
Plasmodium vivax- pose the greatest public health study concluded that poor access to water,
challenges [1] – [3]. According to the World Health sanitation, and hygiene practices of caregivers
Organization [4], 219 million cases of malaria were directly relates to the prevalence of malaria among
estimated in 89 countries, the estimated deaths were their children. Olaniyi et al. [3] developed a
435,000 with the African region carrying a mathematical model of malaria dynamics with
disproportionately high share of the global malaria naturally acquired transient immunity in the
burden. Malaria is the third leading cause of death presence of protected travelers. Okuonghae and
Nwankwo [8] developed a non-autonomous model
Copyright Holder:
to assess the impact of different microclimate
© Oluwafemi, T., and Azuaba, E. (2022)
conditions on the transmission dynamics of malaria.
First Publication Right: Goni and Musa [9] proposed a mathematical model
Journal of Multidisciplinary Applied Natural Science
for the transmission dynamics of malaria by

incorporating change via education as a control
Publisher’s Note:
strategy. The human population follows the
Pandawa Institute stays neutral with regard to jurisdictional
claims in published maps and institutional affiliations. susceptible-protected-exposed-infectious-recovered
(SPEIR) pattern and the mosquito population
This Article is Licensed Under:
follows susceptible-exposed-infectious (SEI)
patterns. An analytical study was carried out to
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J. Multidiscip. Appl. Nat. Sci.
investigate the local stability of the system, the immunity after recovery given as , and reduced
basic reproduction number was obtained using the by the rate of progression to hygienic class , the
next-generation matrix method. The result shows force of infection for the unhygienic class and
that the disease-free equilibrium of the system is natural human death rate . The hygienic
locally asymptotically stable if . Bhuju et al. susceptible human compartment is increased by
[10] mathematically studied the impact of the , the rate at which hygienic human loss
temperature in malaria disease transmission immunity after recovery at , while the
dynamics. The SEIR model for the human compartment is reduced by natural human death
population and LSEI compartment model for rate and the force of infection for the hygienic
mosquito population. It was observed that class . The unhygienic infected human class
temperature affects the transmission dynamics of is increased by and reduced by natural
malaria significantly. Okosun and Makinde [11] human death rate , rate of progression from to
also modeled the impact of drug- resistance in given as , malaria induced death for unhygienic
malaria transmission. Many mathematical models human class and recovery for unhygienic
have been developed to study malaria dynamics but human . The hygienic infected class is
none has been discussed to study the impact of increased by and then reduced by the
hygiene on malaria transmission dynamics as recovery rate for a hygienic human class given
proposed in this work. as , malaria induced death for hygienic human
In this work, we propose a deterministic class and natural death rate . The Human
mathematical model for assessing the impact of recovery class is increased by and , then
hygiene on malaria transmission dynamics. We reduced by , and . The susceptible
compute the Basic Reproduction Number and mosquito class is increased by the Mosquito
establish the local and global stability of the disease recruitment rate given as , reduced by the
-free equilibrium and carry out the sensitivity mosquitoes death rate , and force of infection for
analysis of the parameters. mosquito given as . The infected mosquito class
is increased by and .
2. MATERIALS AND METHODS
Given the above description and definitions of
2.1. Model Formulation variables and parameters in Table 1 and 2, the
In this model, the total human population following are the model equations:
denoted by is subdivided into Unhygienic
susceptible human population , Hygienic (2.3)
Susceptible Human population , Unhygienic
infected human population , hygienic infected (2.4)
human population and the Recovered Human
population . The mosquito population denoted by (2.5)
is subdivided into susceptible mosquitoes and
infected mosquitoes . So (2.6)

(2.1) (2.7)

(2.2) (2.8)

Let be the recruitment rate of the human (2.9)
population. A fraction enters unhygienic
susceptible human class while the remaining “where”
fraction enters the hygienic susceptible human
class. The unhygienic susceptible class is increased
by the rate at which unhygienic human class lose (2.10)

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Figure 1. Model flow diagram.
2.2. Invariant Region i.e. the solution remains positive for all initial
The invariant region can be obtained by the values.
following theorem. Thus, the model is biologically meaningful and
mathematically well-posed in the domain .
Theorem 1
The solutions of the model are feasible for all 2.3. Disease Free Equilibrium (DFE)
if they enter the invariant region The DFE of the model equations can be found
by setting the right hand of the model (2.3) – (2.9)
(2.11) to zero, i.e.

Proof:

Let

which gives
(2.12)

(2.3)
be any solution of the system with non-negative

initial conditions. Hence, all feasible solution set of
(2.4)
the human population of the malaria model enters
the region
Table 1. Variables.
Symbols Description
(2.13) S Unhygienic Susceptible Human
u

S Hygienic Susceptible Human
n
Similarly, the feasible solution set of the vector
I Unhygienic Infected Human
population enter the region u
In Hygienic Infected Human
(2.14) R Recovered Human

S Susceptible Mosquitoes
v
Therefore, the region is positively invariant
I Infected Mosquitoes
v
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Table 2. Model Parameters.
Symbols Description
Recruitment rate of Human Population

Recruitment rate of mosquitoes

Progression from S to S
u n
Progression from I to I
u n
Disease—induced death for the unhygienic human class

Disease—induced death for the hygienic human class

Biting rate of mosquito for unhygienic human class

Biting rate of mosquito for hygienic human class

Transmission probability of infection from mosquito to human

Transmission probability of infection from human to mosquitoes

The force of infection for unhygienic human class

The force of infection for hygienic human class

Force of infection for mosquitoes

Biting rate of mosquitoes

Rate of reduction of infection for hygienic class

Modification parameter

Rate of recovery for unhygienic human class

Rate of recovery for hygienic human class

Rate at which recovered human become susceptible

Hygienic rate

Natural human death rate

Natural death rate of mosquitoes

Total human population

(2.5) At DFE,

(2.6) So we have,

(2.7)

(2.8)

(2.9)
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After computing simultaneously, we have The basic reproduction number is the largest
eigenvalue or spectral radius of FV-1. Hence,

(2.20)
Theorem 2
The disease-free equilibrium (DFE) for
model system is locally asymptotically stable if
Therefore, the Disease Free Equilibrium (DFE) and unstable otherwise.
point of the model is given by
Proof:
At DFE, the Jacobian matrix is given by
(2.15)

2.4. Reproduction Number (R )
0
The basic reproduction number (R ) is defined
0
as the number of secondary malaria infections
produced by one infected individual in a completely
susceptible community. The next-generation
method [12] will be employed to compute R . F(x)
0
Table 3. Indices of Sensitivity.
is the rate of new infection appearance while V(x)
is the rate of transfer of individuals into Symbols Sensitivity Index
compartments. So we have -1

1

(2.16) -0.00013

-0.000041

-0.000041

(2.17)
-0.29

0.00022

“where”
1

(2.18) 1

1

1

-0.087

1

Thus,
-0.000015

-0.71

-0.00011

(2.19) 1

-2

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2.5. Global stability of the Disease Free
Equilibrium (DFE)
Theorem 3
The DFE of the model system is globally
asymptotically stable if .

The eigenvalues are: Proof:
Consider the following Lyapunov function:

(2.21)
It is observed that all the eigenvalues are
negative, this implies that at the DFE point Differentiating we have
is locally asymptotically stable, this means that
Malaria infection can be eliminated from the
population.
(2.22)
Table 4. Parameter values of Model.
Symbols Values Source
100 [13]

1000 [14]

0.25 (Assumed)

0.5 (Assumed)

0.13 (Assumed)

0.06 (Assumed)

0.17 (Assumed)

0.1 (Assumed)

0.03 [2]

0.09 [2]

0.12 [15]

0.08 (Assumed)

0.5 (Assumed)

0.05 (Assumed)

0.15 (Assumed)

0.7902 [14]

0.46 (Assumed)

0.00004 [13]

0.0000569 [14]

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At DFE, we have mosquitoes ( ) is most sensitive to the Basic
Reproduction Number.

4. CONCLUSIONS

In this work, the mathematical model to
assess the impact of hygiene on malaria
transmission dynamics was proposed and analyzed.
(2.23) The model is divided into the Human Population
and Vector (Mosquito) Population, the Human
From the equation above, if (2.24) population is further subdivided into the Susceptible
Unhygienic human Population, Susceptible
Hence, the DFE is globally asymptotically hygienic Human population, infected unhygienic
stable. human population, infected hygienic human
population and Recovered human, while the vector
2.6. Sensitivity Analysis population is subdivided into the susceptible vector
In this section, sensitivity analysis is carried out and infected vector. We proved that the model
to identify the parameters that have a great equation is biologically meaningful and
influence on the basic reproduction number (R ). mathematically well-posed. The disease-free
0
The sensitivity index of R to a given parameter P is equilibrium (DFE) was established and it was
0
given by the relation observed that DFE is locally asymptotically stable
if and globally asymptotically stable if
(2.25) using Lyapunov function. Sensitivity
analysis of the model parameters was carried out
Table 3 shows the sensitivity indices of the basic and it shows that the natural death rate of
reproduction number to the parameters. The mosquitoes is most sensitive to the Basic
parameters with positive indices indicate that the reproduction Number.
basic reproduction number increases as their values This implies that individuals must continue to
increase. While the parameters with negative engage in activities that promote both personal
sensitivity indices indicate an increase in these hygiene and environmental hygiene so as reduce the
parameters will result in the decline of the basic growth of mosquito hence curbing the spread of
reproduction number and vice-versa. malaria also government and other Non-
Governmental Organizations (NGOs) must continue
3. RESULT AND DISCUSSIONS to intensify campaigns on hygienic practices at
individual and community levels. Future studies can
First, we show that this system of model is be carried out on the model this includes: establish
biologically meaningful and mathematically well- and prove the existence of the unique endemic
posed in the given domain . We compute the basic equilibrium point, Stability analysis (local and
reproduction number R of the model using the next global stability) of the endemic equilibrium point,
0
-generation method. We established the existence of and lastly solve the model equations using any
the disease-free equilibrium of the system and show analytic method available.
the condition for the local stability of the disease-
free equilibrium and global stability of the disease- AUTHOR INFORMATION
free equilibrium using the Lyapunov function. The
disease-free equilibrium is locally asymptotically Corresponding Author
stable if and globally asymptotically stable Temidayo Oluwafemi — General Studies
if . Sensitivity analysis of the model equation Department, Newgate College of Health
was carried out as illustrated in Table 3. From the Technology, Minna - 920211 (Nigeria);
table, it shows that the natural death rate of Email: [email protected]
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J. Multidiscip. Appl. Nat. Sci.
Author of Water, Sanitation and Hygiene Practices
Emmanuel Azuaba — Department of and the Occurrence of Childhood Malaria in
Mathematics, Bingham University Karu, Abia State, Nigeria”. Researchjournali’s
Nassarawa - 962101 (Nigeria) ; Journal of Public Health. 5 (6): 1–15.
[8] D. Okuonghae, and A. Nwankwo. (2019).
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