A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria

Sunday Nwokpoku Aloke; Patrick Agwu Okpara

doi:doi:10.11648/j.ajam.20241202.11

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A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria

Sunday Nwokpoku Aloke^*

, Patrick Agwu Okpara

Published in American Journal of Applied Mathematics (Volume 12, Issue 2)

Received: 2 March 2024 Accepted: 18 March 2024 Published: 2 April 2024

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Abstract

Lassa virus is transmitted from rodents to humans, but it is not known whether humans can transmit Lassa fever to rats. The virus is thought to spread to humans through contact with contaminated food or surfaces. Other forms of infection include handling rodents for food (people often get rodent blood and urine on their hands) and bites. It can also spread through the use of contaminated medical equipment, such as reusing needles. The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. As a result of measuring the spread of Lassa fever and determining the stability equilibrium, Lassa fever was found to be stable at an equilibrium point ε₀ for which the basic reproduction number R₀< 1. This paper optimized three control measures as a means to limit the spread of Lassa fever. The first two measures - regular hand washing and keeping homes and environment clean reduced the rate and impact of transmission between rodents and humans and the treatment of Lassa fever patients reduce transmission to human hosts, which were achieved by the operation of Pontryagin’s Maximum Principle. Therefore, the results of this study demonstrate that the joint control measures adopted in this paper are effective strategies to combat the spread of disease.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 2)
DOI	10.11648/j.ajam.20241202.11
Page(s)	24-36
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Lassa Fever, Scaling, Basic Reproduction Number, Stability Analysis, Controls

1. Introduction

Lassa fever which is an acute viral hemorrhagic disease belonging to several countries in West Africa, such as Benin, Ghana, Guinea, Liberia, Mali, Sierra Leone, and Nigeria

[18]

. The first case of this disease was reported in the 1950s, and the virus was identified in 1969 following the deaths of two missionary nurses in Lhasa, Nigeria. Lassa fever is endemic in Nigeria and outbreaks occur almost every year in different parts of the country

[1, 3]

. The Nigeria Center for Disease Control initiates regular report on Lassa fever supervision

[2]

. Lassa virus is present in wild rats that have multiple mammaries (udders) and excrete the virus in their urine and feces, They are common in rural areas of tropical Africa and often live in and around homes

[5, 6]

. The latent period for Lassa fever is 2-days to 3-weeks

[5]

. This infection is transmitted from rodents to humans and, to a lesser extent, from humans to humans. Infected rodents spread the virus throughout their lives and can spread the virus through urine, saliva, respiratory tract, and open blood vessels, even if they do not show clinical symptoms

[4, 5]

. Transmission from rodents to humans occurs through direct contact with the urine, feces, or saliva of infected rodents and discharge or secretion resulting from contact with infectious substances or consumption of food contaminated with feces

[8]

. In 2017, Innocent et al., formulated a Lassa fever model with measures to curtail it, studied the epidemiology of the disease, recommended avoidance of contact with virus-carrying rodents, and introducing human vaccines

[12]

. Martins et al., in their study developed a mathematical model to control the spread of lassa fever, analyzing the existence and stability of a lassa fever-free equilibrium

[10]

. Abdulkarim et al., discussed the objective factors and mortality rates lassa fevr in Bauchi State, Nigeria were data collected from 2015 to 2018 were used to show an increase in morbidity and mortality and the majority of deaths was shown to occurring within 7 days of symptom onset

[11]

Patrick et al.

[7]

proposed a mathematical model of Lassa fever transmission dynamics that included isolation and treatment as control strategies, their numerical simulations showed that the rate of spread of infection is an important parameter for the emergence of infectious disease. Therefore, efforts should be made to minimize transmission parameters to ensure eradication. Another model for the isolation of symptomatic Lassa fever was presented in

[9]

. This study extends studies

[7, 9]

by optimizing the prescribed control measures adopted by MEDECINS SANS FRONTERES

[15]

in the areas of regular hand washing and food hygiene, keeping homes and community environments clean, and treating patients infected with Lassa fever. This article experiments with the Forward-Backward Sweep approach, which uses the order four Runge-Kutta method to confirm the effectiveness of the control measures by the operation of the Pontryagin’s Maximum Principle to determine how the proliferate of the Lassa Fever can be limited. Additional, this article will also examines the incidence and recurrence rates of infection in Lassa fever survivors.

2. Assumptions of the Lassa Fever Model

a. All state variables and parameters are assumed to be positive.

b. The entire population is vulnerable to Lassa fever

c. There is believed to be an even mix infected and susceptible individuals.

d. Some recovered individuals may return to the susceptible class.

e. The entire population is at risk equal of Lassa Fever, regardless age or health condition.

2.1. Lassa Fever Model Equations

The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. The human population is divided into four classes; Susceptible

, Exposed

, Infected

, and Recovered

compartments and the rodent populations are divided into two; Susceptible

, and Infected

respectively. The susceptible people

go to the exposed section

to update population of exposed class to

. From the exposed population,

persons are transfer from

compartment to the infection ward

and, as a result of compliance with treatment and prevention measures,

persons move to the recovery group. Finally, the susceptible rodents

move to the infected compartment

and update the number of infected rodents to

(1)

(2)

Download: Download full-size image

Figure 1. Model Diagram.

Where

Table 1. Summary of Parameters and meaning.

Parameters	Meaning (Dimension: Time^-1)
	Recruitment rate for humans
	Recruitment rate for rodent
	Contact rate of humans
	Contact rate of rodents
	Progression rate to the infectious class
	Immunity lost rate
	Rate at which recovered individuals go back to the susceptible class
	Lassa Fever induced death rate
	Natural death rate of human
	Natural death rate of rodents

2.2. Scaling of the Model

To facilitate the analysis of equation 1 of the Lassa fever model, the ratio of the give populations is determined by scaling the population of each class based on the total number of species. Considering different populations,

and the ratio of every section within the species is given as:

it follows that

Rat population was estimated 7 billion in the world and this means that 1 rat for every human

[16]

i.e

. Let

and

Set

then

Then

After some simplification, we have

(3)

Where the dimensionless parameters are

2.3. Lassa Fever Model Properties

The Lassa Fever model (3) covers both the population of humans and rodents, the variables and parameters of the Lassa fever model are all non-negative for

Theorem 1: The Lassa fever model 1 of the initial condition in

and

are positively invariant in

and

Proof

The Lassa fever system (3) is split into two sections, i.e the class of humans

and the class of rodents

, defined by

(4)

(5)

Let

Then, equation (4), yields

By method of integrating factor

It follows as;

then

When

(6)

And equation (5), yield

Applying method of integrating factor

When

(7)

Therefore

and

then

and

respectively. This is an indication that the solutions of the Lassa Fever model (3) fall in the zone.

3. Existence of Lassa Fever Free Equilibrium and the Basic Reproduction Number

From the Lassa-Fever model equation 3, for the human population, the compartments

represent the disease-free states and

denote the infection class. The Lassa Fever-free equilibrium (LF-FE) point

by first setting

and

It follows that

and for the rodents population, the compartments

is only the disease-free states and the compartments

is the infection class,

then set

Therefore, the Lassa Fever-free equilibrium (DFE) is

To obtain the basic reproduction number,

of the model equation (3) at

, the application of the next-generation matrix is employed

[13]

. As the infected compartments are

then

formed the ongoing infection terms and the out sending terms shown below.

It follows that the measure of transmission potential of Lassa Fever, denoted by

, is obtained from the matrix

by calculating it’s spectral radius.

Theorem 2. When

, the Lassa Fever-free equilibrium

of the dynamical Lassa Fever equation 3 is locally asymptotically stable.

Proof

Simplifying the Jacobian matrix of the Lassa Fever equations at the Lassa Fever -free equilibrium point, the result is given by

It follows that

Where

and

, then

implies

. All eigenvalues will zeros or negatives, hence

is locally stable. If

implies

and

is locally unstable.

3.1. Global Stability of Lassa Fever - Free Equilibrium

The conditions for the global stability of the Lassa Fever model at

is obtained by applying the approach stated in

[23, 24]

which defines the human population

of the Lassa Fever model is defined by

Where

denotes the class of individuals free from Lassa Fever and

denotes the infected individuals. From the above notation, the Lassa Fever-free equilibrium is written as

. Then, these two conditions below clearly showed that the global stability of the Lassa Fever free equilibrium.

is globally asymptotically stable

where

for

Lemma 1: The equilibrium point

is globally asymptotically stable when

and the above assumptions on

are true.

Theorem 3: The Lassa Fever-free equilibrium point

of the Lassa Fever is globally asymptotically stable provided

Proof

Solving for the characteristic polynomial of

; we have

Therefore,

is globally asymptotically stable.

Then,

We have

Therefore, it follows that

satisfies all conditions stated in

3.2. Strategy for Prevention of Lassa Fever

The Preventive measures as adopted by MEDECINS SANS FRONTIERES, DOCTORS WITHOUT BORDERS

[15]

to curtail the spread of Lassa fever will be categorized as follows:

Regular Hand Washing with soap and clean water and food Hygiene; wash vegetables and fruits before eating, properly cover food, Food should be properly cooked, store food in containers with lids or covers, kitchen utensils should be kept clean and covered, avoid hunting and eating of rats will be set to reduce the spread of Lassa fever in the human population.

Maintaining a clean Environment at home and the community: keep a cat around, close holes in the house, use of door and window will be set to reduce the breeding of rats in the environment.

: Treatment of individuals infected with Lassa Fever.

3.3. Lassa Fever Model Equations with Controls

(8)

Download: Download full-size image

Figure 2. Model diagram with control.

4. Optimal Control Problem

The aim of the optimal control is to identify the control level which minimizes the number of exposed and infected classes. Therefore, we find the maximum value

of the controls

at time

so that the trajectories which are combined to the states

solve the above Lassa Fever model and

minimizes the function defined below:

(9)

Subject to model equation (8).

Equation 9 above consist of the exposed and infected cases with the severity of the side effects,

, of the control measures. Here, we employed

and

where

. The conditions required for these optimal controls was obtained by the application of Pontraygin’s Maximum Principle (PMP) is employed

[14]

. The Lagrangian is defined as following:

Where

are penalty multipliers, which satisfy

Theorem 4: Given optimal controls

and solution

of the corresponding state system 8, there exist adjoint variables

where

, transversality condition.

The optimality condition is given by

Moreso, the optimal controls are defined as follow:

Proof

The adjoint variables

were solved in the system in the Lagrangian.

Thus,

The optimal controls

were resolved from

, it follows that,

Therefore,

There are 3-cases for the optimal controls

respectively at time

Case I:

Case II:

Case III:

, since

Following a similar argument for

, it follows that

Therefore, it is summarized as follow in compact form.

Table 2. Parameters Values.

Parameters

Range

Reference

Scale Parameters

Values

1000*0.0003465

[17]

0.069-0.101

0.05

[20]

0.063-0.12

0.022-0.27

[17]

0.001

0.024-0.048

[17]

0.4329

0.333

Assumed

0.961

0.333-0.8

[19]

0.0095

0.00385

[19]

0.00056

0.00019231

[18]

0.12821

0.0003465

[18]

0.00641026

[19]

5. Numerical Simulations

The numerical simulation of the model is performed in other to examine the sequel of Lassa Fever parameters in the growth of the virus. The numerical values in Table 2 and the previous states

and

were used. The graph below shows the Model simulation for some period of time. A numerical approach known as the Forward-Backward Sweep method was used to enable numerical modeling of the state and adjoint equations, and MATLAB script was to iteratively update the control by implementing the fourth-order Runge-Kutta method. This state is repeated until successive iterations are sufficiently close to each other

[21, 22]

. In this paper, the proposed control measures for three numerical modeling strategies of Lassa Fever model are summarized as follow:

(i) Strategy A:

(ii) Strategy B:

(iii) Strategy C:

Where

Download: Download full-size image

Figure 3. Model Simulation for Strategy A.

Download: Download full-size image

Figure 4. Model Simulation for Strategy B.

Download: Download full-size image

Figure 5. Model Simulation for Strategy C.

Download: Download full-size image

Figure 6. The value of the objective function at a given maximum control level.

Figures 3-5 above shows at a strategic level the effectiveness of the control measures in terms of the number of susceptible human, exposed human, infected human, recovered human, susceptible rodent and infected rodents populations.

denote the objective function trajectory when

respectively.

6. Conclusion

This work conducts a theoretical study on the control of model of Lassa Fever dynamics in Ebonyi State, Nigeria. The infection free equilibrium solution appears to be both locally and globally stable. The introduction of favourable conditions will lead to the suppression of the Lassa Fever and reduce the prevalence of Lassa Feever in Nigeria if the proposed controls are implemented. Model simulation show that the level of control increases, the number infected individuals decreases and the population of the recovered individuals increases. It soon became clear that there was a significant increase from strategy A to strategy C. Therefore, the percentage of individuals complying with the degree and level of care recovered in this task should be interpreted and used with caution. In any case, since multiple routes of infection are likely to exist, intervention strategies must become more context-specific. A more holistic approach to rodent control is needed, using effective and targeted control methods while maintaining a clean environment in homes and communities as adopted in this paper. To reduce secondary transmission of Lassa fever, additional education on personal hygiene and access to health facilities during illness is needed.

Abbreviations

LFFE: Lassa Fever-free equilibrium

IVP: Initial Value Problem

Conflicts of Interest

The authors declare no conflicts of interest.

References

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APA Style

Aloke, S. N., Okpara, P. A. (2024). A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. American Journal of Applied Mathematics, 12(2), 24-36. https://doi.org/10.11648/j.ajam.20241202.11

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ACS Style

Aloke, S. N.; Okpara, P. A. A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. Am. J. Appl. Math. 2024, 12(2), 24-36. doi: 10.11648/j.ajam.20241202.11

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AMA Style

Aloke SN, Okpara PA. A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. Am J Appl Math. 2024;12(2):24-36. doi: 10.11648/j.ajam.20241202.11

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@article{10.11648/j.ajam.20241202.11,
  author = {Sunday Nwokpoku Aloke and Patrick Agwu Okpara},
  title = {A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {2},
  pages = {24-36},
  doi = {10.11648/j.ajam.20241202.11},
  url = {https://doi.org/10.11648/j.ajam.20241202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241202.11},
  abstract = {Lassa virus is transmitted from rodents to humans, but it is not known whether humans can transmit Lassa fever to rats. The virus is thought to spread to humans through contact with contaminated food or surfaces. Other forms of infection include handling rodents for food (people often get rodent blood and urine on their hands) and bites. It can also spread through the use of contaminated medical equipment, such as reusing needles. The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. As a result of measuring the spread of Lassa fever and determining the stability equilibrium, Lassa fever was found to be stable at an equilibrium point ε0 for which the basic reproduction number R0< 1. This paper optimized three control measures as a means to limit the spread of Lassa fever. The first two measures - regular hand washing and keeping homes and environment clean reduced the rate and impact of transmission between rodents and humans and the treatment of Lassa fever patients reduce transmission to human hosts, which were achieved by the operation of Pontryagin’s Maximum Principle. Therefore, the results of this study demonstrate that the joint control measures adopted in this paper are effective strategies to combat the spread of disease.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria

AU  - Sunday Nwokpoku Aloke
AU  - Patrick Agwu Okpara
Y1  - 2024/04/02
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241202.11
DO  - 10.11648/j.ajam.20241202.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 24
EP  - 36
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241202.11
AB  - Lassa virus is transmitted from rodents to humans, but it is not known whether humans can transmit Lassa fever to rats. The virus is thought to spread to humans through contact with contaminated food or surfaces. Other forms of infection include handling rodents for food (people often get rodent blood and urine on their hands) and bites. It can also spread through the use of contaminated medical equipment, such as reusing needles. The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. As a result of measuring the spread of Lassa fever and determining the stability equilibrium, Lassa fever was found to be stable at an equilibrium point ε0 for which the basic reproduction number R0< 1. This paper optimized three control measures as a means to limit the spread of Lassa fever. The first two measures - regular hand washing and keeping homes and environment clean reduced the rate and impact of transmission between rodents and humans and the treatment of Lassa fever patients reduce transmission to human hosts, which were achieved by the operation of Pontryagin’s Maximum Principle. Therefore, the results of this study demonstrate that the joint control measures adopted in this paper are effective strategies to combat the spread of disease.

VL  - 12
IS  - 2
ER  -

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Author Information

Sunday Nwokpoku Aloke

Department of Industrial Mathematics and Statistics, David Umahi Federal University of Health Sciences, Uburu, Nigeria

Contact Email

http://orcid.org/0009-0006-2366-6210
Patrick Agwu Okpara

Department of Industrial Mathematics and Statistics, David Umahi Federal University of Health Sciences, Uburu, Nigeria

Contact Email

http://orcid.org/0009-0003-8475-2874

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Aloke, S. N., Okpara, P. A. (2024). A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. American Journal of Applied Mathematics, 12(2), 24-36. https://doi.org/10.11648/j.ajam.20241202.11

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ACS Style

Aloke, S. N.; Okpara, P. A. A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. Am. J. Appl. Math. 2024, 12(2), 24-36. doi: 10.11648/j.ajam.20241202.11

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AMA Style

Aloke SN, Okpara PA. A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria. Am J Appl Math. 2024;12(2):24-36. doi: 10.11648/j.ajam.20241202.11

Copy | Download

@article{10.11648/j.ajam.20241202.11,
  author = {Sunday Nwokpoku Aloke and Patrick Agwu Okpara},
  title = {A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {2},
  pages = {24-36},
  doi = {10.11648/j.ajam.20241202.11},
  url = {https://doi.org/10.11648/j.ajam.20241202.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241202.11},
  abstract = {Lassa virus is transmitted from rodents to humans, but it is not known whether humans can transmit Lassa fever to rats. The virus is thought to spread to humans through contact with contaminated food or surfaces. Other forms of infection include handling rodents for food (people often get rodent blood and urine on their hands) and bites. It can also spread through the use of contaminated medical equipment, such as reusing needles. The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. As a result of measuring the spread of Lassa fever and determining the stability equilibrium, Lassa fever was found to be stable at an equilibrium point ε0 for which the basic reproduction number R0< 1. This paper optimized three control measures as a means to limit the spread of Lassa fever. The first two measures - regular hand washing and keeping homes and environment clean reduced the rate and impact of transmission between rodents and humans and the treatment of Lassa fever patients reduce transmission to human hosts, which were achieved by the operation of Pontryagin’s Maximum Principle. Therefore, the results of this study demonstrate that the joint control measures adopted in this paper are effective strategies to combat the spread of disease.
},
 year = {2024}
}

Copy | Download

TY  - JOUR
T1  - A Mathematical Model of Lassa Fever Transmission and Control in Ebonyi State, Nigeria

AU  - Sunday Nwokpoku Aloke
AU  - Patrick Agwu Okpara
Y1  - 2024/04/02
PY  - 2024
N1  - https://doi.org/10.11648/j.ajam.20241202.11
DO  - 10.11648/j.ajam.20241202.11
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 24
EP  - 36
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20241202.11
AB  - Lassa virus is transmitted from rodents to humans, but it is not known whether humans can transmit Lassa fever to rats. The virus is thought to spread to humans through contact with contaminated food or surfaces. Other forms of infection include handling rodents for food (people often get rodent blood and urine on their hands) and bites. It can also spread through the use of contaminated medical equipment, such as reusing needles. The state variables of the Lassa Fever model equation is expressed as nonlinear ordinary differential equations in the technique of an initial value problem (IVP) having 10 parameters. As a result of measuring the spread of Lassa fever and determining the stability equilibrium, Lassa fever was found to be stable at an equilibrium point ε0 for which the basic reproduction number R0< 1. This paper optimized three control measures as a means to limit the spread of Lassa fever. The first two measures - regular hand washing and keeping homes and environment clean reduced the rate and impact of transmission between rodents and humans and the treatment of Lassa fever patients reduce transmission to human hosts, which were achieved by the operation of Pontryagin’s Maximum Principle. Therefore, the results of this study demonstrate that the joint control measures adopted in this paper are effective strategies to combat the spread of disease.

VL  - 12
IS  - 2
ER  -

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