A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde

Transmission dynamics of symptom-dependent HIV/AIDS models

In this study, we proposed two, symptom-dependent, HIV/AIDS models to investigate the dynamical properties of HIV/AIDS in the Fujian Province. The basic reproduction number was obtained, and the local and global stabilities of the disease-free and endemic equilibrium points were verified to the deterministic HIV/AIDS model. Moreover, the indicators $ R_0^s $ and $ R_0^e $ were derived for the stochastic HIV/AIDS model, and the conditions for stationary distribution and stochastic extinction were investigated. By using the surveillance data from the Fujian Provincial Center for Disease Control and Prevention, some numerical simulations and future predictions on the scale of HIV/AIDS infections in the Fujian Province were conducted.

A potential transition from a concentrated to a generalized HIV epidemic: the case of Madagascar

BACKGROUND

HIV expansion is controlled by a range of interrelated factors, including the natural history of HIV infection and socio-economical and structural factors. However, how they dynamically interact in particular contexts to drive a transition from concentrated HIV epidemics in vulnerable groups to generalized epidemics is poorly understood. We aim to explore these mechanisms, using Madagascar as a case-study.

METHODS

We developed a compartmental dynamic model using available data from Madagascar, a country with a contrasting concentrated epidemic, to explore the interaction between these factors with special consideration of commercial and transactional sex as HIV-infection drivers.

RESULTS

The model predicts sigmoidal-like prevalence curves with turning points within years 2020-2022, and prevalence reaching stabilization by 2033 within 9 to 24% in the studied (10 out of 11) cities, similar to high-prevalence regions in Southern Africa. The late/slow introduction of HIV and  circumcision, a widespread traditional practice in Madagascar, could have slowed down HIV propagation, but, given the key interplay between risky behaviors associated to young women and acute infections prevalence, mediated by transactional sex, the protective effect of circumcision is currently insufficient to contain the expansion of the disease in Madagascar.

CONCLUSIONS

These results suggest that Madagascar may be experiencing a silent transition from a concentrated to a generalized HIV epidemic. This case-study model could help to understand how this HIV epidemic transition occurs.

Deterministic-stochastic analysis of fractional differential equations malnutrition model with random perturbations and crossover effects

To boost the handful of nutrient-dense individuals in the societal structure, adequate health care documentation and comprehension are permitted. This will strengthen and optimize the well-being of the community, particularly the girls and women of the community that are welcoming the new generation. In this article, we extensively explored a deterministic-stochastic malnutrition model involving nonlinear perturbation via piecewise fractional operators techniques. This novel concept leads us to analyze and predict the process from the beginning to the end of the well-being growth, as it offers the possibility to observe many behaviors from cross over to stochastic processes. Moreover, the piecewise differential operators, which can be constructed with operators such as classical, Caputo, Caputo-Fabrizio, Atangana-Baleanu and stochastic derivative. The threshold parameter is developed and the role of malnutrition in society is examined. Through a rigorous analysis, we first demonstrated that the stochastic model's solution is positive and global. Then, using appropriate stochastic Lyapunov candidates, we examined whether the stochastic system acknowledges a unique ergodic stationary distribution. The objective of this investigation is to design a nutritional deficiency in pregnant women using a piecewise fractional differential equation scheme. We examined multiple options and outlined numerical methods of coping with problems. To exemplify the effectiveness of the suggested concept, graphical conclusions, including chaotic and random perturbation patterns, are supplied. Consequently, fractional calculus' innovative aspects provide more powerful and flexible layouts, enabling us to more effectively adapt to the system dynamics tendencies of real-world representations. This has opened new doors to readers in different disciplines and enabled them to capture different behaviors at different time intervals.

Optimal Control and Bifurcation Analysis of HIV Model

In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R ₀ is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R ₀ ≤ 1, the disease-free equilibrium is stable both locally and globally whereas if R ₀ > 1, based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R ₀ = 1, the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin's maximum principle is applied to form an optimality system. Further, forward fourth-order Runge-Kutta's method is applied to obtain the solution of state variables whereas Runge-Kutta's fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population's dynamic behavior.

Incorporating social determinants of health into the mathematical modeling of HIV/AIDS

Currently, it is estimated that 37.6 million people are living with the HIV/AIDS virus worldwide, placing HIV/AIDS among the ten leading causes of death, mostly among low- and lower-middle-income countries. Despite the effective intervention in the prevention and treatment, this reduction did not occur equally among populations, subpopulations and geographic regions. This difference in the occurrence of the disease is associated with the social determinants of health (SDH), which could affect the transmission and maintenance of HIV. With the recognition of the importance of SDH in HIV transmission, the development of mathematical models that incorporate these determinants could increase the accuracy and robustness of the modeling. This article aims to propose a theoretical and conceptual way of including SDH in the mathematical modeling of HIV/AIDS. The theoretical mathematical model with the Social Determinants of Health has been developed in stages. For the selection of SDH that were incorporated into the model, a narrative literature review was conducted. Secondly, we proposed an extended model in which the population (N) is divided into Susceptible (S), HIV-positive (I), Individual with AIDS (A) and individual under treatment (T). Each SDH had a different approach to embedding in the model. We performed a calibration and validation of the model. A total of 31 SDH were obtained in the review, divided into four groups: Individual Factors, Socioeconomic Factors, Social Participation, and Health Services. In the end, four determinants were selected for incorporation into the model: Education, Poverty, Use of Drugs and Alcohol abuse, and Condoms Use. the section "Numerical simulation" to simulate the influence of the poverty rate on the AIDS incidence and mortality rates. We used a Brazilian dataset of new AIDS cases and deaths, which is publicly available. We calibrated the model using a multiobjective genetic algorithm for the years 2003 to 2019. To forecast from 2020 to 2035, we assumed two lines of poverty rate representing (i) a scenario of increasing and (ii) a scenario of decreasing. To avoid overfitting, we fixed some parameters and estimated the remaining. The equations presented with the chosen SDH exemplify some approaches that we can adopt when thinking about modeling social effects on the occurrence of HIV. The model was able to capture the influence of the employment/poverty on the HIV/AIDS incidence and mortality rates, evidencing the importance of SDOH in the occurrence of diseases. The recognition of the importance of including the SDH in the modeling and studies on HIV/AIDS is evident, due to its complexity and multicausality. Models that do not take into account in their structure, will probably miss a great part of the real trends, especially in periods, as the current on, of economic crisis and strong socioeconomic changes.

Stochastic optimal control of pre-exposure prophylaxis for HIV infection

The aim of the paper is to apply the stochastic optimal control problem in order to optimize the number of individual which will have the pre-exposure prophylaxis (PReP) treatment in the stochastic model for HIV/AIDS with PReP. By using the stochastic maximum principle, we derive the stochastic optimal control of PReP for the unconstrained control problem. Furthermore, by combining the stochastic maximum principle with a version of the Lagrange multiplier method, we solve the PReP problem for two different types of budget constrains with a given constrain for the costs (possible of different kind, transportation, price of the treatment, etc.). Obtained results for the different percentage of the individuals who got the vaccine, as well as results for unconstrained and constrained problems, are illustrated by a numerical example.

Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia

The newest infection is a novel coronavirus named COVID-19, that initially appeared in December 2019, in Wuhan, China, and is still challenging to control. The main focus of this paper is to investigate a novel fractional-order mathematical model that explains the behavior of COVID-19 in Ethiopia. Within the proposed model, the entire population is divided into nine groups, each with its own set of parameters and initial values. A nonlinear system of fractional differential equations for the model is represented using Caputo fractional derivative. Legendre spectral collocation method is used to convert this system into an algebraic system of equations. An inexact Newton iterative method is used to solve the model system. The effective reproduction number (R0) is computed by the next-generation matrix approach. Positivity and boundedness, as well as the existence and uniqueness of solution, are all investigated. Both endemic and disease-free equilibrium points, as well as their stability, are carefully studied. We calculated the parameters and starting conditions (ICs) provided for our model using data from the Ethiopian Public Health Institute (EPHI) and the Ethiopian Ministry of Health from 22 June 2020 to 28 February 2021. The model parameters are determined using least squares curve fitting and MATLAB R2020a is used to run numerical results. The basic reproduction number is R0=1.4575. For this value, disease free equilibrium point is asymptotically unstable and endemic equilibrium point is asymptotically stable, both locally and globally.

Optimal control of an HIV model with a trilinear antibody growth function

<p style='text-indent:20px;'>We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.</p>

Model-free based control of a HIV/AIDS prevention model

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.

A dynamically-consistent nonstandard finite difference scheme for the SICA model

In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible-Infected-Chronic-AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.

Novel fractional order SIDARTHE mathematical model of COVID-19 pandemic

Nowadays, COVID-19 has put a significant responsibility on all of us around the world from its detection to its remediation. The globe suffer from lockdown due to COVID-19 pandemic. The researchers are doing their best to discover the nature of this pandemic and try to produce the possible plans to control it. One of the most effective method to understand and control the evolution of this pandemic is to model it via an efficient mathematical model. In this paper, we propose to model COVID-19 pandemic by fractional order SIDARTHE model which did not appear in the literature before. The existence of a stable solution of the fractional order COVID-19 SIDARTHE model is proved and the fractional order necessary conditions of four proposed control strategies are produced. The sensitivity of the fractional order COVID-19 SIDARTHE model to the fractional order and the infection rate parameters are displayed. All studies are numerically simulated using MATLAB software via fractional order differential equation solver.

The Impact of Pre-exposure Prophylaxis for Human Immunodeficiency Virus on Gonorrhea Prevalence

Pre-exposure prophylaxis (PrEP) has been shown to be highly effective in reducing the risk of HIV infection in gay and bisexual men who have sex with men (GbMSM). However, PrEP does not protect against other sexually transmitted infections (STIs). In some populations, PrEP has also led to riskier behavior such as reduced condom usage, with the result that the prevalence of bacterial STIs like gonorrhea has increased. Here, we develop a compartmental model of the transmission of HIV and gonorrhea and the impacts of PrEP, condom usage, STI testing frequency and potential changes in sexual risk behavior stemming from the introduction of PrEP in a population of GbMSM. We find that introducing PrEP causes an increase in gonorrhea prevalence for a wide range of parameter values, including at the currently recommended frequency of STI testing once every three months for individuals on PrEP. Moreover, the model predicts that a higher STI testing frequency alone is not enough to prevent a rise in gonorrhea prevalence, unless the testing frequency is increased to impractical levels. However, testing every 2 months in combination with a 10-25 % reduction in risky behavior by individuals on PrEP would maintain gonorrhea prevalence at pre-PrEP levels. The results emphasize that programs making PrEP more available should be accompanied by efforts to support condom usage and frequent STI testing, in order to avoid an increase in the prevalence of gonorrhea and other bacterial STIs.

Numerical Optimal Control of HIV Transmission in Octave/MATLAB

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin’s maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva, C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations. An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three standard methods implemented by us in Octave/MATLAB: Euler method and second-order and fourth-order Runge–Kutta methods. Afterwards, a control function is introduced into the normalized HIV model and an optimal control problem is formulated, where the goal is to find the optimal HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least HIV new infections and cost associated with the control measures. The optimal control problem is characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed numerically by implementing a forward-backward fourth-order Runge–Kutta method. Complete algorithms, for both uncontrolled initial value and optimal control problems, developed under the free GNU Octave software and compatible with MATLAB are provided along the article.