The human immunodeficiency virus (HIV) has infected many countries' populations currently. It has triggered above 16 million deaths throughout the world. For instance, about 35% of the population between the ages of 15 and 50 are infected by HIV in Africa. In the past few decades, mathematical models have been proposed to illustrate the immunological response to the HIV infection, which is developed only based on the one type of population growth called the Lotka-Volterra model. The Lotka-Volterra population growth model is governed by three ordinary differential equations, whose solution for the HIV model represents the evolution of the numbers or densities of uninfected host cells, infected cells, and virus particles. This population model does not consider age, memory, or hereditary of the process, especially for the virus fusion processes, which will not be instantaneous. In other words, this model assumes that the HIVs infect all the immune cells that they are encountered instantly and convert them immediately.

Volterra proposed a convolution integral with a special kernel and the usual terms used in the logistic equation to take the influence of aging and hereditary. However, the main burden of using Volterra's population growth model is determining the kernel of the integral properly. It is shown that the kernel of Volterra's integral can naturally be chosen by extending the definition of well-known functions, operators, and relations in classic calculus such as the factorial operator or Cauchy's n−fold integral. This leads to the emergence of a new branch of calculus called fractional calculus. This is a generalization of classic calculus wherein the derivative/integral operator's order accepts fractional or even complex numbers. Differential equations with fractional orders are called fractional-order differential equations (FDEs) that have been shown highly-efficient in capturing the memory effects or hereditary properties of dynamical systems. Specifically, FDEs have attracted considerable interest owing to their ability in the characterization of the diffusion process.

In this project, we propose Volterra's population model with a properly chosen kernel inspired by fractional calculus to model HIV infection. The proposed model is more efficient and realistic compared to Lotka's model. It also models the hereditary of population growth. Besides, we study the quantification of uncertainty in the proposed model by introducing parameter uncertainty into it. Finally, we will show the advantages of the proposed model by using experimental data.