Nursanti Anggriani

A Critical Protection Level Derived from Dengue Infection Mathematical Model Considering Asymptomatic and Symptomatic Classes

In this paper we formulate a model of dengue fever transmission by considering the presence of asymptomatic and symptomatic compartments. The model takes the form as a system of differential equations representing a host-vector SIR (Susceptible – Infective -Recovered) disease transmission. It is assumed that both host and vector populations are constant. It is also assumed that reinfection of recovered hosts by the disease is possible due to a wanning immunity in human body. We analyze the model to determine the qualitative behavior of the model solution and use the concept of effective basic reproduction number (p) as a control criteria of the disease transmission. The effect of mosquito biting protection (e.g. by using insect repellent) is also considered. We compute the long-term ratio of the asymptomatic and symptomatic classes and show a condition for which the iceberg phenomenon could appear.

Lymphatic Filariasis Transmission and Control: A Mathematical Modelling Approach

Lymphatic filariasis has an effect on almost 120 million individuals all over the world. The disease may cause a chronic morbidity if the persons who are infected are left untreated. It is endemic in many parts of tropical countries. To prevent worldwide parasite transmission, the World Health Organization initiated the Global Programme to Eliminate Lymphatic Filariasis (GPELF) by eliminating filarial parasites from their human hosts (Molyneux & Zagaria, 2002). Various GPELF implementations are done in many participating countries. In 2004 alone there were more than thirty countries have started elimination program and this number is still rising. Various degrees of success have emerged as a result of the implementation of this program. Although it was reported that in some places the program has interrupted the transmission, in many other places the program could not stop the transmission of the disease (WHO, 2005). It has been argued that strategic choices and operational or biological factors contribute to the success or failure of the program. In general, it is difficult to evaluate the success or the failure of a health program, especially in the beginning of the program. A mathematical model provides useful tools for planning and evaluation of control program in disease elimination (Goodman, 1994). In our earlier work (Supriatna et al., 2009) we develop a mathematical model for the transmission of Lymphatic Filariasis disease in Jati Sampurna, Indonesia. In Indonesia, the disease is already alarming. For example, the incidence of filariasis in Jati Sampurna (a district in the West Java province) is more than 1%. Within less than five years since the date of the publication confirming that Jati Sampurna is an endemic area, almost all regions nearby Jati Sampurna, and other relatively far distance areas are affected by the disease, and some of them are also categorized as endemic areas. Other cases of filarial prevalence are reported outside Java island, such as in Alor islands (the province of Nusa Tenggara Timur). On Alor islands, both B. timori and W. bancrofti are circulated, with a prevalence of up to 20% (Supali et al., 2002). Indonesia joined the GPELF since 2001 and implemented administration of a single dose regimen of diethylcarbamazine (DEC) and albendazole in endemic areas (Krentel et al., 2006). Our previous model tries to capture the effectiveness of this scenario in the attempt of controlling the spread of the disease, inspired by the transmission of the disease in Jati Sampurna. The model assumes that acute infected humans are infectious and treatment is given to a certain number of acute infected humans found from screening process. The screening is

Application of differential transformation method for solving dengue transmission mathematical model

The differential transformation method (DTM) is a semi-analytical numerical technique which depends on Taylor series and has application in many areas including Biomathematics. The aim of this paper is to employ the differential transformation method (DTM) to solve system of non-linear differential equations for dengue transmission mathematical model. Analytical and numerical solutions are determined and the results are compared to that of Runge-Kutta method. We found a good agreement between DTM and Runge-Kutta method.The differential transformation method (DTM) is a semi-analytical numerical technique which depends on Taylor series and has application in many areas including Biomathematics. The aim of this paper is to employ the differential transformation method (DTM) to solve system of non-linear differential equations for dengue transmission mathematical model. Analytical and numerical solutions are determined and the results are compared to that of Runge-Kutta method. We found a good agreement between DTM and Runge-Kutta method.

A Gompertz population model with Allee effect and fuzzy initial values

Growth and population dynamics models are important tools used in preparing a good management for society to predict the future of population or species. This has been done by various known methods, one among them is by developing a mathematical model that describes population growth. Models are usually formed into differential equations or systems of differential equations, depending on the complexity of the underlying properties of the population. One example of biological complexity is Allee effect. It is a phenomenon showing a high correlation between very small population size and the mean individual fitness of the population. In this paper the population growth model used is the Gompertz equation model by considering the Allee effect on the population. We explore the properties of the solution to the model numerically using the Runge-Kutta method. Further exploration is done via fuzzy theoretical approach to accommodate uncertainty of the initial values of the model. It is known that an initial valu...

The optimal strategy of wolbachia-infected mosquitoes release program: An application of control theory in controlling dengue disease

In this paper we discuss an application of control theory in a bio-system. We develop a mathematical model of dengue transmission in human and mosquito populations. We assume that the growth of human population is governed by a logistic equation. Two interventions are then introduced to control the disease, i.e. human vaccination and wolbachia-infected mosquitoes discharging into the wild mosquitoes population. We study the basic reproduction number for the system and also present the optimal control for the interventions via the Pontryagin Maximum Principle. Some numerical examples are explored, and the result indicates that the effect of the optimal control into the reduction of infected human population is critically influenced by both epidemiological parameters, such as the level of the contagiousness of the wolbachia infection, and economics factors, such as the cost of the implementation of the intervention program.