Introduction: The first case of COVID-19 in Kenya was reported on March 13, 2020, prompting the collection of baseline data during the initial spread of the disease. Subsequently, the Kenyan government implemented non-pharmaceutical interventions (NPIs) on April 9, 2020, to mitigate disease transmission over a two-month period. These measures were later gradually relaxed starting from June 9, 2020.
Methods: We applied a deterministic mathematical model to simulate the dynamics of COVID-19 transmission in Kenya. Using baseline data, we estimated transmission and recovery rates and proposed a mathematical model of how NPIs affect disease transmission rates. The model extends to interventions that yield an increase in disease transmission, unlike previous models that were limited to a decrease in transmission. We computed the mitigation and relaxation fractions and hence deduced the impact of the interventions.
Results: The mitigation measures imposed from April 9, 2020, reduced the disease transmission by 43.7% from the baseline level, while the relaxation from June 9, 2020, increased the transmission by 32% over the mitigation level. Without intervention, the model predicts that infections would have peaked at 30% by late May 2020. However, due to the combined effect of mitigation and relaxation, the epidemic peaked at 13% infection in mid-July 2020.
Discussion: The model’s projections closely align with observed data, providing valuable insights for planning. Ongoing research aims to refine the model to capture sub-waves and spikes, as well as simulate multiple waves of infection. These efforts will enhance our understanding of COVID-19 dynamics and inform effective public health strategies. The estimated basic reproduction number , consistent with previous findings, underscores the validity of our model and its relevance in predicting disease transmission dynamics.
Tuberculosis (TB), a disease caused by bacteria Mycobacterium tuberculosis (Mtb), remains one of the major infectious diseases of humans with 10 million TB cases and 1.5 million deaths due to TB worldwide yearly. Upon exposure of a new host to Mtb, bacteria typically infect one local site in the lung, but over time, Mtb disseminates in the lung and in some cases to extrapulmonary sites. The contribution of various host components such as immune cells to Mtb dynamics in the lung, its dissemination in the lung and outside of the lung, remains incompletely understood. Here we overview different types of mathematical models used to gain insights in within-host dynamics of Mtb; these include models based on ordinary or partial differential equations (ODEs and PDEs), stochastic simulation models based on ODEs, agent-based models (ABMs), and hybrid models (ODE-based models linked to ABMs). We illustrate results from several of such models and identify areas for future resesarch.
Mathematical modeling is a powerful method to understand how biological systems work. By creating a mathematical model of a given phenomenon one can investigate which model assumptions are needed to explain the phenomenon and which assumptions can be omitted. Creating an appropriate mathematical model (or a set of models) for a given biological system is an art, and classical textbooks on mathematical modeling in biology go into great detail in discussing how mathematical models can be understood via analytical and numerical analyses. In the last few decades mathematical modeling in biology has grown in size and complexity, and along with this growth new tools for the analysis of mathematical models and/or comparing models to data have been proposed. Examples of tools include methods of sensitivity analyses, methods for comparing alternative models to data (based on AIC/BIC/etc.), and mixed-effect-based fitting of models to data. I argue that the use of many of these “toolbox” approaches for the analysis of mathematical models has negatively impacted the basic philosophical principle of the modeling—to understand what the model does and why it does what it does. I provide several examples of limitations of these toolbox-based approaches and how they hamper generation of insights about the system in question. I also argue that while we should learn new ways to automate mathematical modeling-based analyses of biological phenomena, we should aim beyond a mechanical use of such methods and bring back intuitive insights into model functioning, by remembering that after all, modeling is an art and not simply engineering.
“Getting something for nothing is impossible; there is always a price to pay.” Louis Gross.
“There is not such a thing as a free lunch.”
In this study, we present a nonlinear deterministic mathematical model for co-infection of pneumonia and COVID-19 transmission dynamics. To understand the dynamics of the co-infection of COVID-19 and pneumonia sickness, we developed and examined a compartmental based ordinary differential equation type mathematical model. Firstly, we showed the limited region and non-negativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Secondly, the Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the related reproduction numbers , , and are smaller than unity, then pneumonia, COVID-19, and their co-infection have disease-free equilibrium points that are both locally and globally asymptotically stable otherwise the endemic equilibrium points are stable. Sensitivity analysis is used to determine how each parameter affects the spread or control of the illnesses. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates four controls, namely, prevention of pneumonia, prevention of COVID-19, treatment of infected pneumonia and treatment of infected COVID-19. Then the Pontryagin's maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system reveals that the combination of treatment and prevention is the most optimal to minimize the diseases.