A mathematical model of COVID-19 transmission

Dynamics of HIV transmission among men who have sex with men in Taiwan: a mathematical modeling study

BACKGROUND

In Taiwan, the human immunodeficiency virus (HIV) is primarily transmitted among men who have sex with men (MSM). We believe that the primary source of heterogeneity influencing transmission is from whether the infected individuals engage in behaviors that promote transmission. Therefore, we differentiated and modeled the statuses of infected individuals, estimated parameters, and predicted future infection trends to better understand HIV transmission dynamics among MSM.

METHODS

We devised a computational approach to accurately estimate the number of infectious individuals during early surveillance years in Taiwan using monthly data of newly diagnosed patients with HIV from January 2005 to December 2006. Subsequently, using annual incident cases from 2005 to 2019, we estimated the susceptibility rate, transmission rates, and duration of risk and conducted short-term forecasts of HIV infection among MSM through mathematical modeling based on the susceptible-infected-removed model. Parameters and their values, such as HIV and acquired immunodeficiency syndrome (AIDS) transmission rates and removal rates, were also estimated.

RESULTS

A total of 850 infectious individuals with HIV were identified in 2005. From 2005 to 2019, approximately 0.68% (95% confidence interval: 0.64-0.71%) of Taiwan's male population were estimated to be susceptible to HIV infection via MSM. The transmission rate did not significantly differ between individuals with HIV and AIDS (0.0110 and 0.107 per 10,000 individuals, respectively). The average infection duration between infection and AIDS onset was 4.5 years and between AIDS onset and death was 2.1 years. In 2019, the proportion of infectious infected individuals continuously decreased. By 2026, only 11.85% and 10.21% of individuals with HIV and AIDS, respectively, will be infectious.

CONCLUSIONS

Our study presents a new dynamic mathematical model for estimating infectious HIV cases among MSM in Taiwan. By accounting for the heterogeneity of infectious HIV/AIDS within the MSM population, we effectively estimated the number of infections during early surveillance years and implement concentrated measures, thereby reducing infectious HIV/AIDS cases, which contributes to the overall reduction in HIV/AIDS cases. With these estimations, identifying target populations and implementing effective intervention measures at appropriate times can be improved.

Analytical and Numerical Investigation of the SIR Mathematical Model

This is a theoretical study of the SIR model — a popular mathematical model of the propagation of infectious diseases. We construct a solution of the Cauchy problem for a system of two ordinary differential equations describing in integral form the concentration dynamics of infected and recovered individuals in an immune population. A qualitative analysis is carried out of the stationary system states using the Lyapunov function. An expression is obtained for the coordinates of the equilibrium points in terms of the Lambert W-function for arbitrary initial values. The application of the SIR model for the description of COVID-19 propagation dynamic is demonstrated.

A Coupled Mathematical Model of the Dissemination Route of Short-Term Fund-Raising Fraud

To effectively protect citizens’ property from the infringement of fund-raising fraud, it is necessary to investigate the dissemination, identification, and causation of fund-raising fraud. In this study, the Susceptible Infected Recovered (SIR) model, Back-Propagation (BP) neural network, Fault tree, and Bayesian network were used to analyze the dissemination, identification, and causation of fund-raising fraud. Firstly, relevant data about fund-raising fraud were collected from residents in the same area via a questionnaire survey. Secondly, the SIR model was used to simulate the dissemination of victims, susceptibles, alerts, and fraud amount; the BP neural network was used to identify the data of financial fraud and change the accuracy of the number analysis of neurons and hidden layers; the fault-tree model and the Bayesian network model were employed to analyze the causation and importance of basic events. Finally, the security measures of fund-raising fraud were simulated by changing the dissemination parameters. The results show that (1) for the spread of the scam, the scale of the victims expands sharply with the increase of the fraud cycle, and the victims of the final fraud cycle account for 12.5% of people in the region; (2) for the source of infection of the scam, the initial recognition rate of fraud by the BP neural network varies from 90.9% to 93.9%; (3) for the victims of the scam, reducing fraud publicity, improving risk awareness, and strengthening fraud supervision can effectively reduce the probability of fraud; and (4) reducing the fraud rate can reduce the number of victims and delay the outbreak time. Improving the alert rate can reduce victims on a large scale. Strengthening supervision can restrict the scale of victims and prolong the duration of fraud.