The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

Dengue disease is the most common vector borne infectious disease transmitted to humans by infected adult female Aedes mosquitoes. Over the past several years the disease has been increasing remarkably and it has become a major public health concern. Dengue viruses have increased their geographic range into new human population due to travel of humans from one place to the other. In the present paper, we have proposed a multi patch SIR-SI model to study the host-vector dynamics of dengue disease in different patches including the travel of human population among the patches. We have considered different disease prevalences in different patches and different travel rates of humans. The dimensionless number, basic reproduction number R0 which shows that the disease dies out if R 0 < 1 and the disease takes hold if R 0 ≥ 1, is calculated. Local and global stability of the disease free equilibrium are analyzed. Simulations are observed considering the two patches only. The results show that controlling the travel of infectious hosts from high disease dominant patch to low disease dominant patch can help in controlling the disease in low disease dominant patch while high disease dominant becomes even more disease dominant. The understanding of the effect of travel of humans on the spatial spread of the disease among the patches can be helpful in improving disease control and prevention measures. In the present study, a patch may represent a city, a village or some biological habitat.

Dengue fever is a mosquito-borne infectious disease threatening more than a hundred tropical countries of the world. The heterogeneity of mosquito bites of human during the spread of dengue virus is an important factor that should be considered while modeling the dynamics of the disease. However, traditional models assumed homogeneous transmission between host and vectors which is inconsistent with reality. Mathematically, we can describe the heterogeneity and uncertainty of the transmission of the disease by introducing fuzzy theory. In the present work, we study transmission dynamics of dengue with the fuzzy SEIR-SEI compartmental model. The transmission rate and recovery rate of the disease are considered as fuzzy numbers. The dynamical behavior of the system is discussed with different amounts of dengue viruses. Also, the fuzzy basic reproduction number for a group of infected individuals with different virus loads is calculated using Sugeno integral. Simulations are made to illustrate the mathematical results graphically.

Malaria is one of the most common mosquito borne diseases. Temperature is an important factor which affects the life cycle of the mosquitoes and transmission dynamics of the malaria disease. In the present work, we use SEIR compartmental model for the human population and LSEI compartmental model for mosquito population taking temperature dependent parameters. Basic reproduction number, R 0 of the model is computed using Next Generation Matrix Method. Stability of the disease free equilibrium and the existence of the endemic equilibrium point are discussed by basic reproduction number, R 0. Numerical results are carried out with different temperature levels. It is observed that temperature affects the transmission dynamics of malaria disease significantly.

Dengue is a vector-borne infectious disease endemic in many parts of the world. The disease is spreading in new places due to human movement into the dengue disease supporting areas. Temperature is the major climatic factor which affects the biological processes of the mosquitoes and their interaction with the viruses. In the present work, we propose a multipatch model to assess the impact of temperature and human movement in the transmission dynamics of dengue disease. The work consists of system of ordinary differential equations that describe the transmission dynamics of dengue disease between humans and mosquitoes. Human population is divided into four classes: susceptible, exposed, infectious, and recovered. Mosquito population is divided into three classes: susceptible, exposed, and infectious. Basic reproduction number ℛ0 of the model is obtained using Next-Generation Matrix method. The qualitative analysis of the model is made in terms of the basic reproduction number. Parameters used in the model are considered temperature dependent. Dynamics of vector and host populations are investigated with different human movement rates and different temperature levels. Numerical results show that proper management of human movement between patches helps reducing the burden of dengue disease. It is also seen that the temperature affects the transmission dynamics of the disease significantly.