Optimal control of a discrete age-structured model for tuberculosis transmission

Optimal control and cost-effectiveness analysis for leptospirosis epidemic

This paper aims to apply an optimal control theory for the autonomous model of the leptospirosis epidemic to examine the effect of four time-dependent control measures on the model dynamics with cost-effectiveness. Pontryagin's Maximum Principle was used to derive the optimality system associated with the optimal control problem. Numerical simulations of the optimality system were performed for different control strategies and the results were presented graphically with and without controls. The optimality system was simulated using the Forward-Backward Sweep method in the Matlab programme. The numerical results revealed that the combination of all optimal control measures is the most effective strategy for minimizing the spread and impact of disease in the community. Furthermore, a cost-effectiveness analysis was performed to determine the most cost-effective strategy using the incremental cost-effectiveness ratio approach and we observed that the rodenticide control-only strategy is most effective to combat the spread of disease when available resources are limited.

OPTIMAL CONTROL ANALYSIS OF A TUBERCULOSIS MODEL

In this paper, an optimal control theory was applied to the tuberculosis (TB) model governed by system of nonlinear ordinary differential equations. The aim is to investigate the impact of treatment failure on the TB epidemic. An optimal control strategy is proposed to minimize the disease effect and cost incurred due to treatment failure. The existence and uniqueness of optimal controls are proved. The characterization of optimal paths is analytically derived using Pontryagin’s Minimum Principle. The control-induced model is then fitted using TB infected cases reported from the year 2010–2019 in East Shewa zone Oromia regional state, Ethiopia. Different simulation cases were performed to compare with analytical results. The simulation results show that the combined effect of awareness via various mass media and continuous supervision during the treatment period helps to reduce treatment failure and hence reduced the TB epidemic in the community.

Intensive care unit occupancy predictions in the COVID-19 pandemic based on age-structured modelling and differential flatness

The COVID-19 pandemic confronts governments and their health systems with great challenges for disease management. In many countries, hospitalization and in particular ICU occupancy is the primary measure for policy makers to decide on possible non-pharmaceutical interventions. In this paper a combined methodology for the prediction of COVID-19 case numbers, case-specific hospitalization and ICU admission rates as well as hospital and ICU occupancies is proposed. To this end, we employ differential flatness to provide estimates of the states of an epidemiological compartmental model and estimates of the unknown exogenous inputs driving its nonlinear dynamics. A main advantage of this method is that it requires the reported infection cases as the only data source. As vaccination rates and case-specific ICU rates are both strongly age-dependent, specifically an age-structured compartmental model is proposed to estimate and predict the spread of the epidemic across different age groups. By utilizing these predictions, case-specific hospitalization and case-specific ICU rates are subsequently estimated using deconvolution techniques. In an analysis of various countries we demonstrate how the methodology is able to produce real-time state estimates and hospital/ICU occupancy predictions for several weeks thus providing a sound basis for policy makers.

Mathematical assessment of the role of denial on COVID-19 transmission with non-linear incidence and treatment functions

A mathematical model describing the dynamics of Corona virus disease 2019 (COVID-19) is constructed and studied. The model assessed the role of denial on the spread of the pandemic in the world. Dynamic stability analyzes show that the equilibria, disease-free equilibrium (DFE) and endemic equilibrium point (EEP) of the model are globally asymptotically stable for R0<1 and R0>1, respectively. Again, the model is shown via numerical simulations to possess the backward bifurcation, where a stable DFE co-exists with one or more stable endemic equilibria when the control reproduction number, R0 is less than unity and the rate of denial of COVID-19 is above its upper bound. We then apply the optimal control strategy for controlling the spread of the disease using the controllable variables such as COVID-19 prevention, hospitalization and maximum treatment efforts. Using the Pontryagin maximum principle, we derive analytically the optimal controls of the model. The aforementioned control strategies are performed numerically in the presence of denial and without denial rate. Among such experiments, results without denial have shown to be more productive in ending the pandemic than others where the denial of the disease invalidates the effectiveness of the controls causing the disease to continue ravaging the globe.

Dynamic behaviors of a modified SIR model with nonlinear incidence and recovery rates

A complex SIR epidemic dynamical model using nonlinear incidence rate and nonlinear recovery rate is established to consider the impact of available hospital beds and interventions reduction on the spread of infectious disease. Rigorous mathematical results have been established for the model from the point of view of stability and bifurcation. The model has two equilibrium points when the basic reproduction number R0>1; a disease-free equilibrium E0 and a disease endemic equilibrium E1. We use LaSalle’s invariance principle and Lyapunov’s direct method to prove that E0 is globally asymptotically stable if the basic reproduction number R0<1, and E1 is globally asymptotically stable if R0>1, under some conditions on the model parameters. The existence and nonexistence of limit cycles are investigated under certain conditions on model parameters. The model exhibits Hopf bifurcation near the disease endemic equilibrium. We further show the occurring of backward bifurcation for the model when there is limited number of hospital beds. Finally, some numerical results are represented to validate the analytical results.

Inhalable solid lipid nanoparticles for intracellular tuberculosis infection therapy: macrophage-targeting and pH-sensitive properties

Mycobacterium tuberculosis (MTB) is one of the most threatening pathogens for its latent infection in macrophages. The intracellular MTB isolated itself from drugs and could spread via macrophages. Therefore, a mannose-modified macrophage-targeting solid lipid nanoparticle, MAN-IC-SLN, loading the pH-sensitive prodrug of isoniazid (INH), was designed to treat the latent tuberculosis infection. The surface of SLNs was modified by a synthesized 6-octadecylimino-hexane-1,2,3,4,5-pentanol (MAN-SA) to target macrophages, and the modified SLNs showed a higher cell uptake in macrophages (97.2%) than unmodified SLNs (42.4%). The prodrug, isonicotinic acid octylidene-hydrazide (INH-CHO), was synthesized to achieve the pH-sensitive release of INH in macrophages. The INH-CHO-loaded SLNs exhibited a pH-sensitive release profile and accomplished a higher accumulated release in pH 5.5 media (82.63 ± 2.12%) compared with the release in pH 7.4 media (58.83 ± 3.84%). Mycobacterium smegmatis was used as a substitute for MTB, and the MAN-IC-SLNs showed a fourfold increase of intracellular antibiotic efficacy and enhanced macrophage uptake because of the pH-sensitive degradation of INH-CHO and MAN-SA in SLNs, respectively. For the in vivo antibiotic efficacy test, the SLNs group displayed an 83% decrease of the colony-forming unit while the free INH group only showed a 60% decrease. The study demonstrates that macrophage targeting and pH-sensitive SLNs can be used as a promising platform for the latent tuberculosis infection. Graphical Abstract Table of contents: Macrophage-targeting and pH-sensitive solid lipid nanoparticles (SLN) were administrated to the lung via nebulization. Macrophage targeting was achieved by appropriate particle size and surface mannose modification with synthesized MAN-SA. After being swallowed by macrophages, the prodrug, Isonicotinic acid octylidene-hydrazide (INH-CHO), quickly released isoniazid, which was triggered by the intracellular acid environment. The SLNs exhibited higher intracellular antibiotic efficacy due to their macrophage-targeting and pH-sensitive properties.

Optimal control problem and backward bifurcation on malaria transmission with vector bias

This article aims to apply a mathematical model to investigate the spread of malaria by considering vector bias, saturated treatment, and an optimal control approach. A mathematical analysis of the equilibrium points and an investigation of the basic reproduction number show that if the basic reproduction number (R0) is less than one, the disease-free equilibrium is locally asymptotically stable. Furthermore, the center-manifold theory is applied to analyze the stability of the endemic equilibrium when R0=1. We find that our model performs a backward bifurcation phenomenon when the saturated treatment or vector bias parameter is larger than the threshold. Interestingly, we found that uncontrolled fumigation could increase the chance of the appearance of backward bifurcation. From the sensitivity analysis of R0, we find that the fumigation and vector bias are the most influential parameters for determining the magnitude of R0. Using the Pontryagin maximum principle, the optimal control problem is constructed by treating fumigation and medical treatment parameters as the time-dependent variable. Our numerical results on the optimal control simulation suggest that time-dependent fumigation and medical treatment could suppress the spread of malaria more efficiently at minimum cost.

Dengue infection modeling and its optimal control analysis in East Java, Indonesia

In this study, we present a mathematical model of dengue fever transmission with hospitalization to describe the dynamics of the infection. We estimated the basic reproduction number for the infected cases in East Java Province for the year 2018 is R0≈1.1138. The parameters of the dengue model are estimated by using the confirmed notified cases of East Java province, Indonesia for the year 2018. We formulated the model for dengue with hospitalization and present its dynamics in details. Initially, we present the basic mathematical results and then show briefly the stability results for the model. Further, we formulate an optimal control problem with control functions and obtain the optimal control characterization. The optimal control problem is solved numerically and the results comprised of controls system for different strategies. The controls such as prevention and insecticide could use the best role in the disease eradication from the community. Our results suggest that the prevention of humans from the mosquitoes and the insecticide spray on mosquitoes can significantly reduce the infection of dengue fever and may reduce further spread of infection in the community.

Optimal vaccination strategy for dengue transmission in Kupang city, Indonesia

Dengue is a public health problem with around 390 million cases annually and is caused by four distinct serotypes. Infection by one of the serotypes provides lifelong immunity to that serotype but have a higher chance of attracting the more dangerous forms of dengue in subsequent infections. Therefore, a perfect strategy against dengue is required. Dengue vaccine with 42-80% efficacy level has been licensed for the use in reducing disease transmission. However, this may increase the likelihood of obtaining the dangerous forms of dengue. In this paper, we have developed single and two-serotype dengue mathematical models to investigate the effects of vaccination on dengue transmission dynamics. The model is validated against dengue data from Kupang city, Indonesia. We investigate the effects of vaccination on seronegative and seropositive individuals and perform a global sensitivity analysis to determine the most influential parameters of the model. A sensitivity analysis suggests that the vaccination rate, the transmission probability and the biting rate have greater effects on the reduction of the proportion of dengue cases. Interestingly, with vaccine implementation, the mosquito-related parameters do not have significant impact on the reduction in the proportion of dengue cases. If the vaccination is implemented on seronegative individuals only, it may increase the likelihood of obtaining the severe dengue. To reduce the proportion of severe dengue cases, it is better to vaccinate seropositive individuals. In the context of Kupang City where the majority of individuals have been infected by at least one dengue serotype, the implementation of vaccination strategy is possible. However, understanding the serotype-specific differences is required to optimise the delivery of the intervention.

Detailed epidemiological analysis as a strategy for evaluating the actual behavior of tuberculosis in an apparently low-incidence region

Tuberculosis control in developing regions with apparent low incidence, like the low-income Mexican state of Michoacán, with mean annual incidence rates below 10/100,000 inhabitants, requires knowledge of the actual behavior of the disease. This can be determined using an epidemiological profile at sub-regional level, allowing disclosure of the clinical and social factors that may be hampering efforts to control tuberculosis. In this work, a detailed epidemiological profile was outlined using data of all new monthly cases registered in the National System of Epidemiological Surveillance Database for Michoacán municipalities from 2000 to 2012. Cases were grouped by gender and age, and sociodemographic data were obtained both from the National Institute of Statistics and Geography and from the United Nations Development Programme. Correlations were calculated by Chi-square, Mann-Whitney U, and Kruskal-Wallis H tests. We observed no statistically significant differences between notification rates for the years 2000, 2005 and 2010 (χ² = 0.222, p = 0.895). The percentage of cases is similar between all age groups older than 15, while some regions had low notification rates but high proportions of pediatric cases. Higher proportions of cases of extrapulmonary tuberculosis were observed in municipalities in northern Michoacán. No correlation was found between municipal Human Development Index values and municipal notification rates. Michoacán is undergoing an epidemiological transition with three regions having different epidemiological profiles and particular needs for effective prevention and containment of tuberculosis. Our work shows the importance of the spatial scale of epidemiological profiles for determining specific regional needs of surveillance and containment.