ABSTRACT
Pneumonia is a dangerous disease that can lead to death without proper treatment. It is caused by a bacterial infection that leads to the inflammation of the air sacs in human lungs and potentially results in a lung abscess if not properly untreated. Here in this article we introduced a novel mathematical model to investigate the potential impact of Pneumonia treatments on disease transmission dynamics. The model is then validated against data from Jakarta City, Indonesia. In the model, the infection stage in infected individuals is categorized into three stages: the Exposed, Congestion and Hepatization, and the Resolution stage. Mathematical analysis shows that the disease-free equilibrium is always locally asymptotically stable when the basic reproduction number is less than one and unstable when larger than one. The endemic equilibrium only exists when the basic reproduction number is larger than one. Our proposed model always exhibits a forward bifurcation when the basic reproduction number is equal to one, which indicates local stability of the endemic equilibrium when the basic reproduction number is larger than one but close to one. A global sensitivity analysis shows that the infection parameter is the most influential parameter in determining the size of the total infected individual in the endemic equilibrium point. Furthermore, we also found that the hospitalization and the acceleration of the treatment duration can be used to control the level of endemic size. An optimal control problem was constructed from the earlier model and analyzed using the Pontryagin Maximum Principle. We find that the implementation of treatment in the earlier stage of infected individuals is needed to avoid a more significant outbreak of Pneumonia in a long-term intervention.
ABSTRACT
Public health education is pivotal in the management and control of infectious and non-infectious diseases. This manuscript presents and analyses a nonlinear fractional model of tungiasis dynamics with the impact of public health education for the first time. The human population is split into five classes depending on their disease status. The infected population is split into two subgroups; infected but unaware and infected but aware. The model focuses on the impacts of public health education, contact and treatment contact on tungiasis transmission dynamics. Notably, public health education is important for containing as well as reducing disease outbreaks in communities. The Caputo fractional derivative is utilised in defining the model governing equations. Model equilibrium points existence and stability are investigated using simple matrix algebra. Model analysis shows that tungiasis is contained when the reproduction number is less than unity. Otherwise, if it is greater than unity, the disease persists and spread in the population. The generalised Adams-Bashforth-Moulton approach is utilised in solving the derived tungiasis model numerically. The impacts of public health education, treatment and contact rate on overall disease dynamics are discussed through numerical simulations. From the simulations, we see that for given fractional order, public health education and treatment increase the quality of life plus reduce equilibrium numbers of tungiasis-infected individuals. We observe that population classes converge quicker to their steady states when α is increased. Thus, we can conclude that the derivative order α captures the role of experience or knowledge that individuals have on the disease's history.
Subject(s)
Tungiasis , Humans , Quality of Life , Health Education , Disease Outbreaks , ReproductionABSTRACT
In the present study, we propose and analyze an epidemic mathematical model for malaria dynamics, considering multiple recurrent phenomena: relapse, reinfection, and recrudescence. A limitation in hospital bed capacity, which can affect the treatment rate, is modeled using a saturated treatment function. The qualitative behavior of the model, covering the existence and stability criteria of the endemic equilibrium, is investigated rigorously. The concept of the basic reproduction number of the proposed model is obtained using the concept of the next-generation matrix. We find that the malaria-free equilibrium point is locally asymptotically stable if the basic reproduction number is less than one and unstable if it is larger than one. Our observation on the malaria-endemic equilibrium of the proposed model shows possible multiple endemic equilibria when the basic reproduction number is larger or smaller than one. Hence, we conclude that a condition of a basic reproduction number less than one is not sufficient to guarantee the extinction of malaria from the population. To test our model in a real-life situation, we fit our model parameters using the monthly incidence data from districts in Central Sumba, Indonesia called Wee Luri, which were collected from the Wee Luri Health Center. Using the first twenty months' data from Wee Luri district, we show that our model can fit the data with a confidence interval of 95%. Both analytical and numerical experiments show that a limitation in hospital bed capacity and reinfection can trigger a more substantial possibility of the appearance of backward bifurcation. On the other hand, we find that an increase in relapse can reduce the chance of the appearance of backward bifurcation. A non-trivial result appears in that a higher probability of recrudescence (treatment failure) does not always result in the appearance of backward bifurcation. From the global sensitivity analysis using a combination of Latin hypercube sampling and partial rank correlation coefficient, we found that the initial infection rate in humans and the mosquito infection rate are the most influential parameters in determining the increase in total new human infections. We expand our model as an optimal control problem by including three types of malaria interventions, namely the use of bed net, hospitalization, and fumigation as a time-dependent variable. Using the Pontryagin maximum principle, we characterize our optimal control problem. Results from our cost-effectiveness analysis suggest that hospitalization only is the most cost-effective strategy required to control malaria disease.
ABSTRACT
Malaria is one of the major causes of a high death rate due to infectious diseases every year. Despite attempts to eradicate the disease, results have not been very successful. New vaccines and other treatments are being constantly developed to seek optimal ways to prevent malaria outbreaks. In this article, we formulate and analyze an optimal control model of malaria incorporating the new pre-erythrocytic vaccine and transmission-blocking treatment. Sufficient conditions to guarantee local stability of the malaria-free equilibrium were derived based on the controlled reproduction number condition. Using the non-linear least square fitting method, we fitted the incidence data from the province of Papua and West Papua in Indonesia to estimate the model parameter values. The optimal control characterization and optimality conditions were derived by applying the Pontryagin Maximum Principle, and numerical simulations were also presented. Simulation results show that both the pre-erythrocytic vaccine and transmission-blocking treatment significantly reduce the spread of malaria. Accordingly, a high doses of pre-erythrocytic vaccine is needed if the number of infected individuals is relatively small, while transmission blocking is required if the number of infected individuals is relatively large. These results suggest that a large-scale implementation of both strategies is vital as the world continues with the effort to eradicate malaria, especially in endemic regions across the globe.