Global stability analysis and control of leptospirosis

1 Introduction

Leptospirosis is caused by numerous distinct serovars of a spiral-shaped bacterium known as Leptospira interrogans and it is a disease of animals and humans. These serovars are harboured by a wide range of animals, and all of them are capable of causing illness in humans. Leptospira serovars Pomona and hardjo are particularly important in livestock, however the number of other serovars of concern, detected in domestic animals and in humans, is fast growing (Alabama Cooperative Extension Sytems (ANR-0858)). Leptospirosis is a cause of economic losses in the farming of animals. Many infected animals do not show signs of clinical disease.

Leptospirosis is commonly spread by the urine of infected animals and with moisture acting as an important factor of the survival of the bacteria in the environment. Livestock pick up infection by contact with pasture or water contaminated by the urine of infected livestock or wild animals. In warm, moist conditions the organisms may survive in the environment and cause infection for several weeks, so that under suitable climate conditions, many livestock are almost continually exposed for long periods [1].

Infections can range from asymptomatic or sub-clinical to acute and fatal. Symptoms of acute leptospirosis in animals include sudden agalactia in the lactating female, icterus and haemoglobinuria in the young, nephritis and hepatitis in dogs, and meningitis. Chronic leptospirosis can cause abortion, stillbirth, high mortality among young calves, decreased milk production, runting, and infertility. Often chronically infected animals remain as asymptomatic carriers for life with the organism localized in the kidneys and in the reproductive organs and while horses can develop periodic ophthalmia as a result of leptospirosis [2]. In humans, leptospirosis is capable of causing headaches, fever, chills, sweats and myalgia. Also other symptoms may include lethargy, aching joints, and long periods of sickness. Some highly pathogenic serovars may cause pulmonary haemorrhaging and death. While mild type leptospirosis is probably the most common form of infection, they can sometimes be chronic in nature and have a mental component to their clinical manifestations.

The disease can either be transmitted directly between animals or indirectly through the environment. Leptospirosis is of increasing importance as an occupational disease as intensive farming practices become more widely adopted. For instance, during 1999, those working in agricultural industries in Australia accounted for 35.3% of notifications while those working in livestock industries accounted for 22.9% of notifications [2].

There have been applications of optimal control methods to epidemiological models, but most of these studies focused on HIV and TB diseases dynamics. The authors in [3–6] studied the optimal chemotherapy treatment in controlling the virus reproduction in an HIV patient. In [7–10], optimal control was used to minimize the costs of both diseases and treatment. In [1112] the authors used optimal control to investigate the best strategy for educational campaigns during the outbreak of an epidemic and at the same time minimizing the number of infective humans. The authors in [13] also used Optimal control to study a nonlinear mathematical SIR epidemic model with a vaccination program. Optimal control was applied to study the impact of chemo-therapy on malaria disease with infective immigrants and the impact of basic amenities [1415], while [16] studied the effects of prevention and treatment on malaria, using an SEIR model. It was also used in a malaria model with genetically modified mosquitoes but without human population [17]. For other applications of optimal control to modelling of infectious diseases [18–21].

Very little has been done in the area of applying optimal control theory to study and analyse the dynamics of leptospirosis. Recently, the authors in [22] studied the dynamical interactions between leptospirosis infected vector and human population. While [23] considered a leptospirosis epidemic model to implement optimal campaign using multiple control variables. However, none of these studies carried out cost-effectiveness analysis of the control strategies.

In this paper, an extension of the SIR Leptospirosis model presented in [22] is considered by incorporating both human and vector populations (livestocks) and also incorporates vector vaccination, treatments and prevention strategies. The aim is to gain some insights into the best intervention for minimizing the transmission of the disease within the population and to explore the impacts of various intervention scenarios, namely, prevention, vaccination and treatment. We analyse the stability and bifurcation of the model, then we incorporate into the model appropriate cost functions in order to study and determine the possible impacts of these strategies in controlling the disease. We further carried out detailed qualitative optimal control analysis of the resulting model and give the necessary conditions for optimal control of the disease using Pontryagin’s Maximum Principle, in order to determine optimal strategies for controlling the spread of the disease. The cost-effectiveness analysis of the control strategies is further considered, in order to ascertain the most cost-effective out of the strategies.

The organization of the paper is as follows, in Section 2, we derive a model consisting of ordinary differential equations that describes the interactions between humans and livestocks populations and the underlying assumptions. Section 3 is devoted to the mathematical analysis of the leptospirosis model. In Section 4, the optimal control analysis of the disease is presented. In Sections 5, the simulation results are shown to illustrate the effects of preventions, vaccination and treatment. The cost-effectiveness analysis is presented in Section 6 while the conclusions are in Section 7.

2 Model formulation

The model sub-divides the total human population, denoted by N_h, into sub-populations of susceptible individuals (S_h), individuals with leptospirosis symptoms (I_h), recovered human (R_h). So that N_h = S_h + I_h + R_h.

The total vector (livestock) population, denoted by N_v, is sub-divided into susceptible vector (S_v), infectious vector (I_v), recovered vector (R_v) and vaccinated vector (V_v). Thus, N_v(t) = S_v + I_v + R_v + V_v.

The model is given by the following system of ordinary differential equations:

where βm∗=Iv+Ih.

Susceptible individuals are recruited at a rate Λ_h. Susceptible individuals acquire leptospirosis through contact with infectious vectors and infectious humans at a rate (I_v + I_h)β. Infected individuals recovered from the disease at a rate γ. Individuals with the disease are treated under control, at a rate u₂(t), while u₁(t) is the control efforts on prevention. Non treated infected individuals die at a rate δ_h. Recovered individual loose immunity at a rate σ_h and become susceptible again. The term μ_h is the natural death rate.

Susceptible vector (S_v) are generated at a rate Λ_v, where a proportion u₃ ∈ [0, 1] is successfully vaccinated individual vector. Vectors with the disease are treated under control, at a rate u₄(t). Leptospirosis is acquired through contacts with infected humans and infectious vectors at a rate (I_v + Ih)λ. Leptospirosis infected livestocks are assumed to suffer death due to natural causes and disease induced death rates, μ_v and δ_v respectively. The vectors recovery rate is α and due to wanning effect some vaccinated vectors will move to the infected class at a rate bβm∗λ, where (1 — b) ∈ [0, 1] is the efficacy of the vaccine or they loose their immunity completely and move to the susceptible class at a rate τ.

3 Mathematical analysis of the Leptospirosis model

4 Optimal control analysis of the Leptospirosis model

We seek here to minimize the number of infective individuals and the cost of applying prevention, treatment and vaccination controls. The objective functional that we consider is given by

subject to differential equations system (1).

Here w₁I_v and w₂I_h are the cost associated with a number I_v of infected vectors and It of infected individuals. The term w5mu32 is the cost associated with vaccination, where m is the number of vectors vaccinated and w4u22,w6u42 are the costs associated with human and vector treatments respectively. The cost associated with preventive measure is w3u12, while t_f is the time period of the intervention and the coefficients, w₁, w₂, w₃, w₄, w₅, w₆ are thebalancing cost factors due to scales and importance of the ten parts of the objective function. In line with [35, 15, 28], a linear function for the cost on infection, w₁I_v, w₂I_h, and quadratic forms for the cost on the controls w3u12,w4u22,w5mu32 and w6u42.

We seek an optimal control u1#,u2#,u3#,u4# such that

where U = {u: u is measurable and 0 ≤ u_t, (t) ≤ 1 for t ∈ [0, t_f], i = 1, 2, 3,4} is the control set.

The necessary conditions that an optimal control must satisfy come from the Pontryagin’s Maximum Principle [29]. This principle converts (1) and (20) into a problem of minimizing pointwise a Hamiltonian H, with respect to (u₁, u₂, u₃, u₄)

where MSh,MIh,MRh,MSv,MIv,MRv and MVv are the adjoint variables or co-state variables solutions of the following adjoint system: