Global Stability and Backward Bifurcation for a Lymphatic filariasis model


 An ODE model of Lymphatic filariasis is proposed with eight mutually disjoint compartments. The model is proven to be mathematically and epidemiologically well posed. Epidemiological interpretation of the effective reproduction number is presented. A special case is considered where the death rate due to the disease is negligible. An endemic equilibrium under this special scenario is explicitly computed and the presence of backward bifurcation under this condition is suggested. Bifurcation analysis is performed using the Castillo-Chavez and Song theorem in the special case where death due to disease is zero. When the re-infection rate is zero, backward bifurcation is shown not to be present. In such a situation, global asymptotic stability of the endemic equilibrium is established.


Introduction
Infectious diseases has always been a great concern of humans. The outbreak and spread of infectious diseases has been questioned and studied for many years, they are still the major causes of death in the developing countries. Lymphatic filariasis, a neglected tropical disease, is still a major public health problem in Africa, South America and Asia [1]. Despite existing knowledge of the disease pathology and global treatment campaign, nearly 1.4 billion people in 73 countries worldwide are threatened by the disease, of which over 120 million people are currently infected [2].
There are more than 500 filarial parasites that are known to infect mammals, birds, reptiles and amphibians [3]. Wuchereria bancrofti, brugia malayi, onchocerca vovulus, loa loa, mansonella perstans, mansonella streptocerca, mansonella ozzardi and brugia timori are the common parasites of man. The adult of brugia malayi, brugia timori and wuchereria bancrofti inhabit the lymphatic system, hence, the disease they cause is called lymphatic filariasis [3]. The worms have an estimated active reproductive span of 4-6 years, producing millions of small miniature microfilariae, which circulates in the peripheral blood [2]. They are transmitted from person to person by several species of mosquitoes [2]. Some of the symptoms of lymphatic filariasis include recurrent episodes of chills and fever during the day and at night, especially when the microfilariae circulates accross the bloodstream, paroxysmal coughing at night, swellings of legs and arms known as elephantiasis etc. Treatment include use of diethylcarbamazine (DEC). This drug focuses on killing the microfilariae which are active and circulating through the bloodstream. It is also effective against the adult wuchereria bancrofti worms and can effectively treat lymphatic filariasis. In cases where the lymph nodes are severely cligged and limbs are inflame, surgical procedures maybe prescribed to reduce the swellings and relieve the obstruction.
Several mathematical models have been formulated to understand the transmission dynamics of lymphatic filariasis under various circumstances. [4] developed and analysed alymphatic filariasis disease transmission model to determine the impact of multi-interventions campaign via health education and sterile insect technique (SIT) on the spread of lymphatic filariasis. [5] formulated and analysed a nonlinear differential equation model to study the effect of chemoprophylaxis on the exposed individuals. [6] developed a mathematical model to investigate the impact that vector genus specific dependent processes may have on overall lymphatic filariasis transmission. [7] used lymphatic filariasis Simulation Model (LYMFASIM)to estimate the duration of Mass Drug Administration (MDA) required for elimination and residual levels of microfilaemia (Mf) and antigenaemia (Ag) prevalence reached after that duration in different transmission settings varying from low to high. The result indicated that the duration of annual MDA increased with higher baseline endemicity and lower coverage. [8] used a model-based assessment to develop different plausible scale-up scenarios to reach global elimination and eradication of lymphatic filariasis. They predicted the duration of MDA to reach local elimination for different transmission archetypes and estimated the required number of treatments and the implication of rapid scale-up. [9] used mathematical models to assess the feasibility and strategic value of including vector control in the GPELF initiative to achieve the global elimination of lymphatic filariasis. [10] developed and analysed a mathematical model to quantify the potential effect that heterogeneous infection processes occurring in the major mosquito vector genera may have on parasite transmission and control. [11] formulated and analysed a deterministic differential equation model with two key control measures; quarantine and treatment. They assumed that no infection exists at the initial stage and that there is no vertical transmission in both human and mosquito population. In all the models considered, none has created a separate class for the symptomatic and asymptomatic individuals. In this work, we proposed eight non-linear mathematical models to study the transmission dynamics of lymphatic filariasis with treatment in the symptomatic class and those re-infected after treatment enters the exposed class.

Methods
The pathogenesis of lymphatic filariasis (LF), signs and symptoms, diagnosis and treatment as well as current epidemiological data were presented. Thereafter, an extensive review of lymphatic filariasis (LF) were carried out. Mathematical tools and techniques employed in the development and analysis of the formulated model were identified and studied. These methods and techniques include; use of Lyapunov functionals, Metzler matrix theory, bifurcation analysis, and other methods used for the study of dynamical systems. ( [12], [13], [14], [15], [16], [17]). Based on the biology and natural history of lymphatic filariasis, we formulated novel deterministic mathematical model. The model was rigorously analyzed. We formulated the following system of 8 nonlinear ordinary differential equations. The model is given below.
The total human population at time t, denoted by N h (t), is subdivided into five mutually-disjoint compartments of susceptible humans (S h (t)), latent individual (E h (t)), Asymptomatic individual (A h (t)), Symptomatic individual (I h (t)), and Treated individual (T h (t)). The total vector population at time t, denoted by N v (t) is subdivided into three mutually-disjoint compartments of susceptible mosquitoes (S v (t)), latent mosquitoes (E v (t)) and infectious mosquitoes (I v (t)), so that Based on the assumptions above, the model is given by the following system of nonlinear ordinary differential equations in (2.3) where are the forces of infection.
Population of susceptible mosquitoes E v (t) Population of exposed mosquitoes.
Population of infectious mosquitoes.

Explanation of Terms of the Equations in (2.3)
The equations in (2.3) represent the derivatives, with respect to time, of the different human and mosquito compartments. The first equation in (2.3) represent the derivatives of the susceptible human population, S h . The first term is Progression rates for mosquitoes δ L Disease-induced death rate for individuals with LF the rate of recruitment Λ h . The second term is the interaction between the susceptible human, S h and the infectious mosquitoes that leads to new cases of lymphatic filariasis in humans. This leaves the susceptible human class S h and moves into the exposed or latent class E h . The last term is the death rate for the susceptible human population, S h .
The second equation in (2.3) represent the derivatives of the latent or exposed class. The first term represent the number of new latent lymphatic filariasis cases. The second term represent those that got reinfected after treatment. The third term represent those that developed the lymphatic filariasis disease, are infective but does not show symptoms of the disease. They are regarded as the asymptomatic lymphatic filariasis cases. This leaves the latent class E h and move into the asymptomatic class, A h . The last term is the death rate for the latent humans.
The third equation in (2.3) represent the derivatives of the asymptomatic humans A h class (i.e. those with lymphatic filariasis disease but does not show symptoms). The first term is the number of asymptomatic individuals. The second term represent those that have developed the disease, are infective and shows symptoms of the disease. They are regarded as the symptomatic lymphatic filariasis cases. This leaves the asymptomatic class, A h and moves into the symptomatic class,I h . The last term is the natural mortality rate of the asymptomatic class.
The fourth equation in (2.3)represent the derivatives of the symptomatic class, I h (i.e. those showing symptoms of the disease). The first term represent the number of symptomatic individuals. The second term represent those undergoing treatment, they leaves the symptomatic class and moves into the recovered or treated class. The third term represent those that died due to the disease and the last term is the death rate.
The fifth equation in (2.3) represent the derivatives of the treated or recovered class. The first term on the right hand side represent the number of treated individuals. The second term of the equation represent those that are re-infected after they have been treated of the disease. These leaves the treated class and moves into the latent class. The last term is the death rate for the treated class.
The sixth equation in (2.3) represent the derivatives of the susceptible class, S v for mosquito population. The first term on the right hand side of the sixth equation is the recruitment term for mosquitoes, Λ v . The second term of the sixth equation is the interaction between the susceptible mosquito, I v , the exposed human,E h asymptomatic, A h and symptomatic humans, I h , leading to new cases of latent or exposed mosquitoes. These leaves the susceptible class S v and moves into the expose class, E v . The last term is the death rate for susceptible mosquitoes.
The seventh equation in (2.3) represent the derivatives of the latent or exposed class for mosquitoes. The first term represent the number of new cases of latent mosquitoes. The second term represent those mosquitoes that have fully develop the filarial parasite and can transmit the disease. The last term is the death rate for latent mosquitoes.
The eight equation in (2.3) represent the derivatives of the infectious class for mosquitoes. The first term on the right hand side is the number of new infectious mosquitoes. The last term is the death rate for infectious mosquitoes.

Theorem 1: Let the initial data for the LF model (2.3) be given as
of the model with positive initial conditions, will remain positive for all time t > 0.

Proof:
Consider the first equation of model (2.3), given below as Using similar approach, it can be shown that the other state variables of the model (2.3) remain positive for all t > 0.

Proof:
Adding up the first five equations of the right hand side of (2.3), yields [16] can be used to show that Similarly, it can also be shown for the total vector population.
,then either the orbits enters the domain D in finite time or N h (t) asymptotically approaches Λ h µ h as t → ∞. and N v (t) asymptotically approaches Λv µv as t → ∞. Thus, the domain D attracts all trajectories in R 8 + . Since the domain D is positively-invariant, it is enough to study the dynamics of the flows generated by the system (2.3) in D. We conclude, therefore, that the model (2.3) is both mathematically and epidemiologically well posed.

Local Asymptotic Stability (LAS) of the Disease Free Equilibrium (DFE)
The DFE of the model (2.3) is given by Applying the method in [17], we investigate the LAS of the DFE: The basic reproduction number is , R 1 and R 2 are highly dependent on the biting rate Using Theorem 2 in [17], we state the following: Lemma 1: The DFE of the model (4.5) is LAS in D if R L < 1, and unstable if R L > 1.

Implication Of Lemma 1
• LF can be eradicated from a population where there is diagnosis of both asymptomatic and symptomatic LF cases, when R L < 1.
• This implies that if a small number of infectious LF patients enters a population, no outbreak of the disease in the population where LF control programmes are already being implemented.
The threshold quantity R 0 is the effective reproduction number for lymphatic filariasis. It represent the average number of secondary cases that one infectious individual (or mosquito) would generate over the duration of the infectious period if introduced into a susceptible human (or mosquito) population.

Epidemiological Interpretation of the terms of R 0
The epidemiological interpretation of the terms in the effective reproduction number R L is as follows. Susceptible humans S h acquire lymphatic filariasis infection following effective contact with an infected mosquito I v . The number of human infection generated by an infected mosquito is the product of the infectious rate of infected mosquito, the probability that an exposed mosquito survives the exposed stage and moves to the infectious stage, and the average life expectancy of the infected mosquito.
The average number of new human infection is given as Equation (3.6) represents the total number of secondary lymphatic filariasis infection in human caused by one infected mosquito.
Similarly, susceptible mosquito S v acquire lymphatic filariasis parasite infection following effective contact with an exposed human E h , asymptomatic human A h and infectious human I h . The number of mosquito infections generated by an exposed human is given by the product of the infection rate of exposed humans and the average duration in the exposed E h class.

Thus,
Number of mosquito infection generated by exposed humans is given by Using the fact that N 0 The number of mosquito infection generated by an asymptomatic human A h is given by the product of the infection rate of asymptomatic humans, the probability that an exposed human survives the exposed stage and moves to the asymptomatic stage and the average duration in the asymptomatic class.
Thus, the average number of mosquito infections generated by the asymptomatic humans is given by Using the fact that N 0 Finally, the number of mosquito infections generated by infectious human I h is given by the infection rate of infectious humans, the probability that an asymptomatic humans survives the asymptomatic stage and moves to the infectious class and the average duration in the infectious class.
Thus, the average number of mosquito infection generated by the infectious humans is given by Using the fact that N 0 The average number of new mosquito infections generated by infected humans (exposed, asymptomatic and infectious) is given by the addition of equations (3.8), (3.10) and (3.12).
That is Simplifying, we have The expression in equation (3.14) represents the total number of lymphatic filariasis infected mosquitoes caused by (exposed, asymptomatic and infectious humans).
The geometric mean of equations (3.6) and (3.14) gives the effective reproduction number R L and this is equal to R 0 .

Existence Of Endemic Equilibrium Point
We consider a special case of the model (2.3) where δ L = 0. We now have where and are the forces of infection at endemic steady state.
Adding up the first five equations and the last three equations in (4.1), we have be the EEP for the mass-action model.
The effective reproduction number now denoted by R L1 , is given as: The EEP of the mass-action system is given as: and are the forces of infections at the EEP, respectively Substituting the expressions for the EEP in (4.7) into the forces of infection in (4.8), it follows (after several algebraic manipulations) that the EEP of the special case satisfies the following polynomial at the steady: The components of the EEP are then obtained by solving for λ v * from the polynomial (4.10).
The number of positive roots of (4.10) depends on the sign of B and C, since A > 0 The following results can be deduced. ii. unique endemic equilibrium if B > 0 and R L > 1.
iii. no endemic equilibrium otherwise, when R L < 1.
Item 1 of Theorem 3 suggests the possibility of backward bifurcation (BB) in the model (4.1) with δ L = 0. This is a significant result because models of vector-borne diseases have been established in literature that disease-induced death is responsible for backward bifurcation.( [18], [19], [20])

Bifurcation Analysis
Here, we investigate the possibility of the co-existence of a stable DFE with a stable EEP when R L1 < 1. We claim the following Theorem 4 The model (4.1) with δ L = 0 exhibit backward bifurcation at R L1 = 1 whenever a bifurcation coefficient, denoted by a is positive.

Proof:
Consider the case with β v = β * v , a bifurcation parameter. Solving for .
It follows, that the model (4.1) with δ L = 0 can be re-written asẋ The Jacobian of the transformed system at the DFE with β v = β * v , is given by: The matrix J β * v has a simple zero eigenvalue and all other eigenvalues have negative real part.
The Jacobian J β * v has a right eigenvector w = (w 1 , w 2 , . . . , w 8 ) T , where Applying the Center Manifold Theory in [14], we compute the associated non-zero partial derivatives of the right hand sides of the transformed system (5.2), (evaluated at the DFE with β v = β v * ) that the associated bifurcation coefficients, a and b, are given by Computation of a For the system (5.2), the associated non-zero partial derivatives are given by: , , , , , , , , , It follows from the above expressions, (after several algebraic manipulations), that Computation of b using β v = β * v as a bifurcation parameter gives It follows also from the above expressions that (5.10) Obviously b > 0 for all biologically feasible parameter values.
Hence, backward bifurcation occurs if and only if a > 0.

Non-existence of backward bifurcation
Consider the special case of the model (2.3) with δ L = 0 and negligible re-infection rate (i.e., ν = 0). Then the backward bifurcation coefficient, a, given by equation (5.8) reduces to: Hence, this study has confirmed that the presence of re-infection induces backward bifurcation in the transmission dynamics of LF in a system where there are negligible disease-induced deaths. Hence, as was previously postulated in literature, the absence of disease-induced death does not necessarily rule out the possibility of a backward bifurcation as seen from this analysis.

Global Asymptotic Stability (GAS) OF EEP
In this subsection, we consider the global asymptotic stability (GAS) of the endemic equilibrium point (EEP) of model (2.3). We consider a special case where ν = δ L = 0, so that equation (2.3) becomes Since T h does not appear in the other equations in system (6.1), we only need to consider the reduced system given below are the forces of infection at endemic steady state.

Proof:
In proving the GAS of the EEP of system (6.2), we consider the following non-linear Lyapunov function F has Lyapunov derivatives, given as: Substituting the right hand side of the equations of model (6.2) into (6.7), we have: .
Using equation (6.6) and simplifying, we have Finally, using the arithmetic mean geometric mean relation, the following inequalities hold: (6.10) So thatḞ ≤ 0 for R L1 > 1. Considering the system (6.2) at EEP, the necessary state variables can be substituted into the equation for T h , hence T h (t) approaches T * h as t → ∞. Hence,Ḟ is a Lyapunov function of system (6.2) with ν = δ L = 0 on D/D O .
Thus, by the Lyapunov function F and Lasalle's Invariance Principle [21], every solution to the equations in the reduced model (6.2), with ν = δ L = 0 approaches E 2 as t → ∞ for R L1 > 1.

Results and Discussion
We developed a new compartmental model to describe the transmission dynamics of lymphatic filariasis. We found out that, the disease-free state was globally asymptotically stable when the reproduction number is less than unity; indicating that lymphatic filariasis cannot successfully invade the community. When the reproduction number was greater than one, the endemic state was globally stable. We established that the model exhibit backward bifurcation even when the disease-induced death was negligible. We confirmed that the presence of re-infection induces backward bifurcation in the transmission dynamics of lymphatic filariasis in a system where there are negligible disease-induced deaths. This is a significant result because models of vector-borne diseases (Malaria, Dengue, West Nile Virus etc.) have been established in literature that disease-induced death is responsible for backward bifurcation ( [18], [19], [20]). We conclude that the absence of disease-induced death does not necessarily rule out the possibility of a backward bifurcation.

Conclusion
In this paper, we confirmed that the presence of re-infection induces backward bifurcation in the transmission dynamics of Lymphatic filariasis in a system where there are negligible disease-induced deaths. Hence, as was previously postulated in literature, the absence of disease-induced death does not necessarily rule out the possibility of a backward bifurcation. We also showed that the unique endemic equilibrium point (EEP) with negligible re-infection and disease-induced death is globally asymptotically stable (GAS) whenever R L1 > 1.

Abbreviations
DFE: Disease-free equilibrium EEP: Endemic equilibrium point GAS: Global asymptotic stability GPELE: Global program for the elimination of LF LF: Lymphatic filariasis LYMFASIM: Lymphatic filariasis simulation MDA: Mass drug administration ODE: Ordinary differential equation WHO: World health organization 10 Declaration

Availability of Data and Material
All data used during this study are included in this article.

Competing Interest
The authors declared that they have no competing interests.

Funding
This research is partially supported by the European Mathematical Society (EMS-Simons for Africa) collaborative research visit grant (Type 2) to CRM, Barcelona, Spain. The funder partially supported my accommodation and feeding during my stay at the Centre for Research in Mathematics (CRM).

Authors' contributions
EI formulated the model equations, analysed the existence of the endemic equilibrium point, bifurcation analysis and was a major contributor in writing the manuscripts. DO and FO jointly analysed the global stability of the endemic equilibrium point. All authors read and approve the final manuscript.

Not applicable
Substituting the new value of I * h in (11.5), we have Substituting the value of S * h and the new value of T * h in (11.2) and after some simplification, we have (11.8) where g 1 = (k 1 + µ h ), g 2 = (k 2 + µ h ), g 3 = (τ h1 + µ h ), g 4 = (k v + µ v ) Substituting the new value of E * h in (11.3), we have (11.9) Substituting the new value of A * h in (11.4 ), we have (11.10) Substituting the new value of I * h in (11.5), we have (11.11) From the sixth equation in (4.1), setting the right hand side to zero, we have From the seventh equation in (4.1), setting the right hand side to zero and substituting the value of S * v , we have From the eighth equation in (4.1), setting the right hand side to zero and substituting the value of E * v , we have