Mathematical Model of Leptospirosis: Linearized Solutions and Stability Analysis

Abstract

In this paper the transmission of leptospirosis, an infectious disease caused by bacteria, is studied. Leptospirosis is currently spreading in Thailand and worldwide. A Susceptible-Infected-Removed sir model is used to study the stability analysis, analytical solution and global behavior of the spreading of the disease. The model was analysed using the techniques of non-linear dynamical systems. Two equilibrium points were found and the stability conditions for these equilibrium points were established. It will be shown that the linearised solutions of the sir equations are in good agreement with numerical solutions.

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Pimpunchat, B. , Wake, G. , Modchang, C. , Triampo, W. and Babylon, A. (2013) Mathematical Model of Leptospirosis: Linearized Solutions and Stability Analysis. Applied Mathematics, 4, 77-84. doi: 10.4236/am.2013.410A2008.

1. Introduction

In Thailand, leptospirosis occurs mainly in the rainy season, with an increase in cases beginning in August, reaching a peak in October, and beginning to fall in November [1]. Leptospirosis in contaminated material may infect a person or other mammal. The bacteria penetrates the blood vessels, multiplies and infects several organs. Acute disease may be observed in several species, especially humans and dogs. In livestock sub-acute or even endemic, leptospirosis is generally observed. Infection induces antibody production in the animal renal carriers, shedding leptospirosis in their urine several weeks or even months after infection [2]. In humans, symptoms are generally flu-like, but the disease can result in liver damage and renal failure.

In this study, a compartmental model, with human susceptible-infective-removed-susceptible (sirs) model and vector susceptible-infective (si) model, [3-5] was used to examine the dynamical behaviour of the spread of leptospirosis, a vector borne disease. The sir model has been used to describe the transmission dynamics of many diseases. J Holt et al. 2006 [6] used this model to understand the behavior of infection in an African rodent of Tanzania. In Thailand, W. Triampo 2007 [1] introduced a deterministic sir model for the transmission of the spread of leptospirosis for the Thai population. In this work a modification of the sir model, which is used to describe the general transmission dynamics of leptospirosis, is proposed, and thresholds for changes in stability are found.

2. Model

The sir model has both human and vector populations. The vector in this case are rats. The human population is divided into three subgroups; susceptible humans, infected humans, and recovered humans. The vector population is divided into two subgroups, susceptible vectors and infected vectors [1].

The following assumptions are considered in the model, as in Triampo [1], reproduced below:

• Newborns are not immunized and are therefore vulnerable instantly;

• All compartments of vector and human populations are well-mixed and therefore are spatially uniform;

• Humans can be infected by infected vectors but not by infected humans;

• Infected humans cannot infect susceptible vectors;

• Susceptible vectors become infected vectors instantly, with no incubation period needed for the infectious agents (leptospira) to develop.

• Perfect maternal transmission occurs, that is, new borns are susceptible if they are born to a susceptible mother and infective if they are born to an infective mother;

• Disease transmission is dependent on the product of the density of susceptibles and the infected (instead of the frequency of transmission: this difference disappears after normalisation);

• The total population remains constant.

Figure 1 shows the dynamics of the transmission of leptospirosis based on the assumptions above.

The law of mass action, where rates are proportional to the size (number) of each component, leads to the following set of coupled ordinary differential equations:

(1)

(2)

(3)

(4)

(5)

where * indicates a non-normalised variable.

The variables in (1)-(5) are normalised, giving proportions in each compartment, based on the assumption that the population remains constant, by letting

Then Equations (1)-(5) become

(6)

(7)

(8)

(9)

(10)

By using the constraints and the above system of equations can be simplified to:

(11)

(12)

(13)

Here we have omitted the equations involving, and used the constraints to make the system into only three compartments.

3. Equilibrium Points

Equilibrium points are found by setting the right hand sides of (11)-(13) equal to zero. This gives two equilibrium points in the feasible region. The disease-free equilibrium point and the endemic equilibrium point. After rearrangements this gives the endemic equlibrium point

(14)

and

(16)

When then and. In this work the spreading of infection between human and vector is assumed to depend on rainfall. This means that the basic reproduction number depends on. Equations (14)-(16) can be simplified to:

(15)

(17)

(18)

(19)

where and which is the basic reproduction number [1]. Infection in humans is described by an SIR model, but as animals do not recover, the disease in this vector is described by a SI process.

4. Local Stability Analysis

In this section the local stability analysis of (11)-(13) at steady-state is carried out. The Jacobian matrix of the system is given below:

(20)

The eigenvalues at are

(21)

The eigenvalues of the Jacobian matrix in (19) of the disease-free steady state have negative roots which are independent of any parameter. These eigenvalues tell us that there is only one case of solution, i.e., all eigenvalues are real and equal. The negative root condition is found by considering the eigenvalues (20). Now when so

. So the system is stable at the disease-free state since.

In the case of the endemic disease state, the eigenvalues at are

(22)

(23)

(24)

The following three theorems follow immediately from [7]:

Theorem 1 The disease-free state at is locally asymptotically stable if and unstable if.

Theorem 2 The endemic state at is locally asymptotically stable if and unstable if.

From theorem 2 and theorem 2 is a bifurcation point of the system where

. The expression above shows the relationships between and including bifurcations at and respectively. The parameters utilized in this calculation are

and. Letting and and using the above theorems the bifurcation point occurs when as displayed in Figures 2(a)-(c) In the proof of the following theorem, the Liapunov function is used to show that the disease free state at is globally asymptotically stable.

Theorem 3 If, the existence of local stability implies its global stability [4].

Proof. Define the Liapunov function as.

Then

(25)

This concludes the proof.

Then from theorem 2 the disease-free state at using is globally asymptotically stable since as shown in Figure 3.

5. Numerical Solutions

Numerical calculations were utilized for solving (11) and (12), however, an analytical solution was found for (13). The solutions were used to analyze and examine the behavior of the model and determine whether the results converged.

From (13)

(26)

Divide both sides by, then let and. Then

(27)

Multiply both sides by to get

This simplifies to

(28)

where is a constant determined by the value of at.

Let

(29)

Here (29) is the analytical solution of (13).

The negative root condition is found by examining (24) and (29).

As the eigenvalues and from (22) and (23) must be real, the following inequality is required:

(30)

Then from (30)

(31)

and

(32)

when. The cases are divided as will follow. The results correspond when using

.

5.1. Case 1

All eigenvalues are real and distinct at the steady state. When since , then the steady-state solution at is stable.

Figure 4(a) shows the stable steady state using a phase portrait. The graph of the numerical solution converges to the equilibrium point. The parameters

Figure 1. Flowchart of the transmission dynamics of leptospirosis.

used are and so. Figure 5(a) shows the solutions of and over time. The values of and are limited by.

At

(33)

(34)

(35)

When then so the equilibrium point is stable.

Figure 4(b) is the phase portrait of the solution. The graph of the numerical solution converges to an equilibrium point. The parameters are and so . So is stable. Figure 5(b) shows the solutions of and over time. The values for and are limited by . In the following cases, these is no because the values of (20) can not be found.

5.2. Case 2

All eigenvalues and from (21), (22) and (23) are real. Two eigenvalues are equal while the third is distinct.

Now

(a)(b)(c)

Figure 2. Equilibrium line between and (a), (b) (t), (c) that shows the bifurcation at. The heavy line represents the steady-state and the arrows represent the transient flow to the steady-states.

(36)

(37)

Figure 3. Graph shows global asymptotic stablity at.

(a)(b)

Figure 4. Phase portraits at (a); (b) at steady state.

(38)

Figure 6 shows the phase portrait for this case. The graph of the numerical solution converges to the equilibrium point. The parameters used are and so

(a)(b)

Figure 5. Numerical solutions for and at (a), (b).

Figure 6. Phase portrait at when.

so is stable. Figure 7 shows the solutions of and over time. The values of and are limited by .

5.3. Case 3

Two eigenvalues are complex while the third is real.

Figure 7. Numerical solution for and at when.

(39)

(40)

Figure 6 is the phase portrait of the solutions. The graph of the numerical solution converges to the equilibrium point where the parameters are and so so is stable. Figure 7 shows the solutions of and over time. The values of and approach the value of .

6. Discussion and Conclusion

The important factors of the spreading of leptospirosis are the rate of transmission of leptospirosis from an infected vector to a susceptible human and the rate of transmission of leptospirosis from an infected vector to a susceptible vector [1]. These two factors depend on rainfall. A summary of the two factors is included here:

1) The spreading of leptospirosis has two states: the disease-free state and the endemic state. The occurence of a state depends on. If, then the disease-free state will occur but if then the endemic state will occur as shown in Figures 2(a)-(c).

2) For the disease-free state, a Liapunov function was used to prove that it is globally asymptotically stable. So for any initial population, over a long time, the population will converge to the disease-free steady state, as shown in Figure 3.

3) Two factors impact on the endemic state case: and. Before the convergence point

Figure 8. Numerical solutions for and at when.

(41)

when then and which can be classified as below:

I. For the number of infected humans initially increases before decreasing to the endemic state while the number of recovered humans increases to the endemic state. This is shown in Figures 5(b) and 7.

II. For the number of infected humans initially increases before decreasing to the endemic state while the number of recovered human initially increases before decreasing to the endemic state. This is shown in Figure 8.

4) The death rate of the infected vector population is equal to or greater than the natural birth rate of the infected vector population.

5) The time to convergence to the disease free state is longer than the time to convergence to the endemic state because the amount of time before the disease disappears is longer than that of the endemic state. This reflects what would happen in the real world.

7. Acknowledgements

This research is supported by The Center of Excellence in Mathematics, CHE, Thailand.

Nomenclature

: number of humans susceptible (non-normalised).

: number of humans infected (non-normalised).

: number of humans recovered (non-normalised).

: number of vectors susceptible (non-normalised).

: number of vectors infected (non-normalised).

: time measured in days.

: natural birth rate of human population.

: human population.

: natural death rate of susceptible and recovered humans.

: rate of transmission (non-normalised) of leptospirosis from an infected vector to a susceptible human, varying with rain fall.

: rate immune individuals become susceptible again.

: natural death rate of infected humans plus the death rate due to the infection.

: rate infected humans can be cured by antibiotic medicines and become immune.

: natural birth rate of susceptible vectors.

: natural death rate of susceptible vectors.

: constant natural birth rate of infected vectors.

: natural death rate of infected vectors plus the death rate due to the infection.

: rate of transmission (non-normalised) of leptospirosis from an infected vector to a susceptible vector, varying with rain fall.

: proportion of humans susceptible (normalised).

: proportion of humans infected (normalised).

: proportion of humans recovered (normalised).

: proportion of vectors susceptible (normalised).

: vector population.

: proportion of vectors infected (normalised).

: rate of transmission of leptospirosis from an infected vector to a susceptible human, varying with rain fall.

: rate of transmission of leptospirosis from an infected vector to a susceptible vector, varying with rain fall.

: disease free equilibrium point.

: proportion of humans infected at equilibrium point (normalised).

: proportion of humans recovered at equilibrium point (normalised).

: proportion of vectors infected at equilibrium point (normalised).

: endemic equilibrium point.

: basic reproduction number.

: eigenvalues.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] W. Triampo, et al., “A Simple Deterministic Model for the Spread of Leptospirosis in Thailand,” International Journal of Biological and Medical Sciences, Vol. 2, No. 1, 2007, p. 22. [2] V. Michel, C. Branger and G. Andre-Fontaine, “Epidemiology of Leptospirosis,” Revista Cubana de Medicina Tropical, Vol. 54, No. 1, 2002, pp. 7-10. [3] J. D. Murray, “Mathematical Biology,” 2nd Edition, SpringerVerlag, Berlin, 2004. http://dx.doi.org/10.1007/b98868 [4] P. K. Tapaswi and J. Chattopadhyay, “Global Stability Results of a Susceptible-Infective-Immune-Susceptible (SIRS) Epidemic Model,” Ecological Modelling, Vol. 87, No. 1-3, 1996, pp. 223-226. [5] L. Cai and X.-Z. Li, “Global Analysis of a Vector-Host Epidemic Model with Nonlinear Incidences,” Applied Mathematics and Computation, Vol. 217, No. 7, 2010, pp. 3531-3541. [6] J. Holt, S. Davis and H. Leirs, “A Model of Leptospirosis Infection in an African Rodent to Determine Risk to Human: Seasonal Fluctuations and the Impact of Rodent Control,” Acta Tropica, Vol. 99, No. 2-3, 2006, pp. 218225. [7] S. M. O’Regan, T. C. Kelly, A. Korobeinikov, M. J. A. O’Callaghan and A. V. Pokrovskii, “Lyapunov Functions for Sir and Sirs Epidemic Models,” Applied Mathematics Letters, Vol. 23, No. 4, 2010, pp. 446-448.