Stability analysis of a mathematical model of Crimean Congo haemorrhagic fever disease

Abstract

Today, ticks are harmful parasitic creatures feared by humans. Ticks do not always carry dangerous diseases. However, we should not ignore the pathogens and viruses that may be carried because these creatures can carry various viruses and seriously threaten human health. If it is not diagnosed early, it can result in fatal consequences. Ticks can get viruses from their hosts at various stages of their lives. Ticks can transmit these viruses to humans in the adult tick stage. Here we can say that the animals that ticks use as hosts are only vectors. Cattle, bovine or chickens do not show symptoms of diseases which are caused by ticks. In this thesis, the spread of Crimean-Congo haemorrhagic fever disease is investigated by considering the problem as an epidemic model. Before stating the problem, in first chapter, some information about dynamic systems is given. The definition of systems of differential equations and their stability analysis are mentioned. Besides, the autonomous systems of equations are briefly explained. And how their stability can be analysed is mentioned. Then, to guide our own problem, information about the well-known SI, SIR, SIS epidemic models and Prey-Predator model and their stability is given in the second chapter. And finally in the third chapter the original problem of the thesis is examined. The system of equation of these models is non-linear. After writing system of equation we found the equilibrium points first. Then, we do linearisation by substituting the equilibrium point in to the Jacobian matrix. We investigated sign of the eigenvalues of these Jacobian matrices which are evaluated by equilibrium points of epidemic models. If all eigenvalues are negative the equilibrium point is stable. If at least one eigenvalue is positive, then the equilibrium point is called unstable. It is not always possible to determine the sign of eigenvalues. In such a case, we could talk about basic reproduction number. Basic reproduction number is represented by R0. If R0 < 1, all eigenvalues are negative and the equilibrium point is a stable equilibrium point. The disease disappear over time. Otherwise, if R0 > 1, at least one of the eigenvalues is positive. Also, the endemic equilibrium point exist when R0 > 1. In addition to, when R0 > 1 disease free equilibrium point is unstable and endemic equilibrium point is stable. And the disease becomes endemic. The problem is expressed as the combination of the variation of population dynamics of human, tick and birds(chicken). In all dynamics of human and tick we considered the in and outs to the compartments, outs as both in the meaning of transfers between compartments and removals such as death. The inputs to the system are either taken constants or logistic growth effects. In this thesis, we investigate the problem in three different ways. • The model which takes logistic growth both in tick and chicken populations, • The model which takes logistic growth only in chicken population, • The model which takes logistic growth only in tick population. We use a system of five ODEs to represent the interaction between chicken population, susceptible and infected populations of humans and ticks. It can be said that there is SI model between infected tick and susceptible tick, SIS model between infected human and susceptible human, and Prey-Predator model between tick and chicken. We have determined the equilibrium points for each model and investigate the stability of the equilibrium points. During the studies the reproduction numbers were found and the stability is investigated with respect to the reproduction numbers. The bifurcation analysis has also been done for tick logistic - chicken logistic model and tick logistic -chicken constant model. According to the results of the first and second models, it was observed that there was a decrease in the number of ticks when the chicken population in the environment was increased. In addition, if the frequency of unleashing of chickens into the environment is increased, then ticks can be more likely to increase among chickens is. Therefore, the number of ticks in the environment may decrease. Due to this decrease, it has mathematically shown that the Crimean Congo Haemorrhagic Fever disease decreases over time.