MATHEMATICAL MODELING FOR THE TRANSMISSION DYNAMICS FOR THE TRACHOMA INFECTION WITH TEMPORARY AND PERMANENT BLINDNESS

¹A. Alhassan, ²S. Musa and ¹A. A. Momoh

¹Department of Mathematics, Modibbo Adama University, Yola. Adamawa State.

²Department of Mathematics, Federal University of Agriculture, Mubi. Adamawa State.

​

Abstract

This study developed a non-linear deterministic model to analyse the transmission dynamics of trachoma. The positivity and boundedness of the model solutions were proven using integration methods alongside the comparison theorem.

The disease-free equilibrium (DFE) was determined, and the basic reproduction number was computed via the next generation matrix method. Local stability was assessed using the Hurwitz criterion and confirmed as being locally asymptotically stable (LAS), while global stability analysis showed the model to be globally asymptotically stable (GAS). ODE45 was used for numerical simulation to examine the effects of treatment, surgery, and vector population management. The model solution's findings show that efforts to treat reversible blindness, control vector populations, and treat and operate on these conditions could significantly aid in reducing the spread of trachoma disease.

INTRODUCTION

Trachoma

• Trachoma is a bacterial eye infection caused by the bacterium Chlamydia trachomatis.
• It is spread from person to person through contact with infected eye and nose secretions.
• Globally, 1.2 billion people live in endemic areas; 40.6 million people are suffering from active trachoma, and 48.5% of the global burden of active trachoma is concentrated in five countries: Ethiopia, India, Nigeria, Sudan and Guinea. Overall, Africa is the most affected continent, with 27.8 million (68.5% of the 40.6 million) cases of active trachoma.

• Trachoma accounts for about 3% of all cases of blindness worldwide. Africa is the most badly affected continent: 18.2 million cases of active trachoma (85.3% of all cases globally) and 3.2 million cases of trichiasis (44.1% of all cases globally) occur in 29/46 countries in WHO’s African Region (WHO, 2013).

​

Infection typically occurs in the eyelid and may be contracted from direct contact with discharge from the infected area on an infected individual or from contaminated surfaces. Re-infection is very common.

• If left untreated, it leads to the formation of irreversible opacities, with resulting visual impairment or blindness.

​

• Environmental factors associated with more intense transmission of C. trachomatis include: inadequate hygiene, crowded households, inadequate access to water, inadequate access to and use of sanitation. Elimination programmes in endemic countries are being implemented using the WHO-recommended SAFE strategy.

The SAFE consists of:

• Surgery to treat the blinding stage

(trachomatous trichiasis);

​

• Antibiotics to clear infection;

​

• Facial cleanliness; and

​

• Environmental improvement, particularly improving access to water and sanitation.

• we developed a model to study the dynamics of the disease transmission and its effects with respect to temporary blindness and permanent blindness. We considered applying some control measures which further help to curtail the transmission of the trachoma infection.

Some Assumptions of the Model

• It is assumed that individuals recovered with permanent blindness will not go back susceptible class because of non interaction with other people due to stigma.

​

• We assumed that those who developed permanent blindness can recover from the infection only but cannot recover from the blindness.

​

• Individuals with temporary blindness can recover from both the infection and the blindness.

Table 3.1(c): Variables for the trachoma model

Table 3.1 (d): Variables for the trachoma existing model

TRACHOMA DYNAMIC MODEL

Figure 1: Flow Diagram for the Trachoma Model

Trachoma model equations

Trachoma modified model equations… Continue

RESULTS

Trachoma Model Analysis

Basic properties of the model

The basic properties of the model are explored in order to prove the positivity and boundedness of the model solution.

​

​

Positivity of the solution

Theorem

Let the initial data be

​

Then the solution set

​

is positive for all time

Proof

Considering equation

We separate the variables, integrate both sides and took the antilog to obtain;

Considering equation;

We separate the variables, integrate both sides and took the antilog to obtain;

Applying similar approach gives;

4.1.1.2 Invariant Region (boundedness)

Theorem 4.2: The solutions to the system with initial conditions and the biological feasible regions

And

so that the regions are positively invariant and attractive.

Proof: we consider the human and the vector populations

​

Where,

if , then it follows that if

Employing the standard comparison theorem (Lakshmikantham, Leela & Martynyuk, 2000) which was employed using the method of separation of variables to obtain;

Using the method of integrating factor gives;

Particularly, if which implies that the region is positively invariant.

Thus, the model is mathematically well-posed in the domain as the variables used throughout the model, proves to be positive for t > 0. It is therefore, sufficient to consider the dynamics of the flow generated by the model in the domain .

For the vector’s population, we add equations to have;

Where

If then it follows that

Employing the standard comparison theorem (Lakshmikantham, Leela & Martynyuk, 2000) and the method of separation of variables to obtain;

Using the method of integrating factor

Particularly, if , it implies that the region is positively invariant.

Thus, the model is mathematically well-posed in the domain as the variables used throughout the model, proves to be positive for t > 0.

It is therefore, sufficient to consider the dynamics of the flow generated by the model in the domain .

​

4.1.2 Disease-free equilibrium point

We set the equations to be equal to zero;

​

​

And obtain;

​

In the absence of trachoma;

​

Therefore, the trachoma-free equilibrium point is given as;

​

Reproduction Number

The reproduction number is obtained as the most dominant eigenvalues

Local stability of trachoma disease-free

Linearization method has been adopted to analyze the stability of the disease-free equilibrium points of the human and vector’s population. Therefore, the Jacobian matrix of the system was evaluated.

Theorem 4.3

The DFE of the model is locally asymptotically stable (LAS) which has been determined based on the sign of the eigenvalues

Evaluating model equations at DFE and gives;

​

​

​

Finding the eigenvalues gives

​

​

The, we obtain the following eigenvalues;

and

Which are all negatives.

After we obtained some of the eigenvalues from (4.1.54), we now have a reduced matrix given as;

​

​

(4.1.55)

​

​

We carry out elementary row transformation on (4.1.55) in order to obtain the remaining eigenvalues and obatin the following;

​

​

​

The remaining eigenvalues obatined are

​

​

​

​

​

The eigenvalues obtained satisfied the condition for local stability.

4.4.4 Global stability of trachoma DFE

We adopt the method used by Castillo-Chavez, et al., (2002) with the following conditions;

Where denotes the uninfected populations and denotes the infected populations respectively.

At DFE, the two conditions that would be met for the global stability are;

is globally asymptotically stable (GAS)

Where A = D₂G(Y,0) represents an M-matrix and ∏ is the region where the model makes sense biologically. If the model satisfy the two conditions above then the following theorem holds, (Castillo-Chavez, et al., 2002).

Theorem 4.4

The equilibrium point M_T = F(Y^* ,0) is GAS.

Proof

We apply theorem 4.4 on the model using condition (i) above.

Condition (i)

​

​

It is solved using integrating factor gives;

The values of S_h(0) and S_v(0) where and as

Condition (ii)

​

Generating and evaluating at DFE gives;

​

​

While,

​

​

​

Therefore, going by we have;

​

​

​

​

Given that 0 < S_h < N_h and 0 < S_V < N_V are noticeable that and it is also evident that

is GAS. Therefore, by the above theorem M_T is GAS.

Figure 2: Graphical Demonstration of the Impact of Treatment

Numerical Simulations

Numerical simulation for the model was carried out using ODE45

Figure 3: Graphical Demonstration of the

Impact of Treatment and Surgery

Impact of Treatment and Surgery on the Transmission Dynamics

Figure 4: Graphical Demonstration of the Impact of Insecticides spray

Impact of Insecticide Spray on the Transmission Dynamics

Discussion of Results

• Basic Properties of the Model
• Trachoma Disease-Free Equilibrium Point
• Reproduction Number for Trachoma Model
• Local Stability for Trachoma Disease-Free Equilibrium
• Global Stability of Trachoma Disease-Free Equilibrium
• Numerical Simulations

References

Blake, I. M., Burton, M. J., Bailey, R. L., Solomon, A. W., West, S., et al., (2009) Estimating Household and Community Transmission of Ocular Chlamydia trachomatis. PLoS Negl Trop Dis 3(3): e401. Doi:10.1371/journal.pntd.0000401.

Borlase, A, Blumberg, S., Callahan, E. K., Deiner, M. S., Nash, S. D., Porco, T. C., Solomon, A. W., Lietman, T. M., Prada, J. M. and Hollingsworth, T. D. (2021) Modelling Trachoma Post-2020: Opportunities for Mitigating the Impact of COVID-19 and Accelerating Progress towards Elimination. Trans R Soc Trop Med Hyg, Vol. 115: pp 213–221. Doi:10.1093/trstmh/traa171.

References

Blake, I. M., Burton, M. J., Bailey, R. L., Solomon, A. W., West, S., et al., (2009) Estimating Household and Community Transmission of Ocular Chlamydia trachomatis. PLoS Negl Trop Dis 3(3): e401. Doi:10.1371/journal.pntd.0000401.

Borlase, A, Blumberg, S., Callahan, E. K., Deiner, M. S., Nash, S. D., Porco, T. C., Solomon, A. W., Lietman, T. M., Prada, J. M. and Hollingsworth, T. D. (2021) Modelling Trachoma Post-2020: Opportunities for Mitigating the Impact of COVID-19 and Accelerating Progress towards Elimination. Trans R Soc Trop Med Hyg, Vol. 115: pp 213–221. Doi:10.1093/trstmh/traa171.

THANK YOU

FOR

YOUR

TIME