Dynamics between infectious diseases with two susceptibility conditions: A mathematical model (original) (raw)

Mathematical Modeling of Infectious Disease Transmission Dynamics in a Metapopulation

IOSR Journals , 2019

Epidemic modeling is an important theoretical approach for investigating the transmission dynamics of infectious diseases. It formulates mathematical models to describe the mechanisms of disease transmissions and dynamics of infectious agents and then informs the health control practitioners the likely impact of the control methods. In this paper we investigate the spread of an infectious disease in a human population structured into n-patches. The population is initially fully susceptible until an infectious individual is introduced in one of the patches. The interaction between patches is dominated by movement of individuals between patches and also the migration of individuals and therefore any infection occurring in one patch will have a force of infection on the susceptible individuals on the other patches. We build a mathematical model for a metapopulation consisting of í µí±› patches. The patches are connected by movement of individuals. For í µí±› = 2, we obtained the basic reproduction number and obtained the condition under which the disease free equilibrium will be asymptotically stable. We further described in terms of the model parameters how control methods could be applied to ensure that the epidemic does not occur and validated the results by the use of the numerical simulation. We showed that the global basic reproduction number cannot exceed one unless the local basic reproduction number is greater than one in at least one of the sub-populations. We further showed that the control of the epidemic in this case can be achieved by applying a control method that decreases the transmission parameters in patches where the local basic reproduction number is greater than one.

Epidemic models with differential susceptibility and staged progression and their dynamics

Mathematical Biosciences and Engineering, 2009

We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each stage. We consider the contact rates to be proportional to the total population or constant such that the infection rates have a bilinear or standard form, respectively. We derive explicit formulas for the reproductive number R 0 , and show that the infectionfree equilibrium is globally asymptotically stable if R 0 < 1 when the infection rate has a bilinear form. We investigate existence of the endemic equilibrium for the two cases and show that there exists a unique endemic equilibrium for the bilinear incidence, and at least one endemic equilibrium for the standard incidence when R 0 > 1.

Differential susceptibility epidemic models

Journal of mathematical biology, 2005

We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotical...

A New Susceptible-Infectious (SI) Model With Endemic Equilibrium

arXiv (Cornell University), 2020

The focus of this article is on the dynamics of a new susceptible-infected model which consists of a susceptible group (S) and two different infectious groups (I 1 and I 2). Once infected, an individual becomes a member of one of these infectious groups which have different clinical forms of infection. In addition, during the progress of the illness, an infected individual in group I 1 may pass to the infectious group I 2 which has a higher mortality rate. In this study, positiveness of the solutions for the model is proved. Stability analysis of species extinction, I 1-free equilibrium and endemic equilibrium as well as disease-free equilibrium is studied. Relation between the basic reproduction number of the disease and the basic reproduction number of each infectious stage is examined. The model is investigated under a specific condition and its exact solution is obtained.

Mathematical modeling of infectious disease

A B S T R A C T Human suffers from infectious disease since prehistoric time. Some times epidemic infectious disease causes mass death toll. So attempts were been taken to save human kind from such infectious diseases. With the advent of science branches of it are have been associate with this endeavour. In recent yearsmathematical modeling has become a valuable tool in the analysis of infectious disease dynamics and to support the development of control strategies.Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. Models use some basic assumptions and mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of possible interventions, like mass vaccination programmes. Here author attempt to discuss basic problems from mathematical view.

Mathematical Modeling and Role of Dynamics in Epidemiology

2013

This study aims at providing the Considerable role of correlation of mathematical modeling and dynamical aspects of some epidemic diseases. This study emphasizes an understanding of deterministic modelling applied to the population dynamics of infection diseases. Here we are mainly emphasizing the historical background of mathematical modelling and role of dynamics in different infection diseases such as measles, AIDS, Cholera, Plague, Malaria, T.B., and Dengue etc. Our investigation is focusing on historical aspects of bioepidemiological mathematical survey. Keyword: Mathematical modelling, Epidemic disease, Biomathematical aspects, Dynamics.

Stability analysis of epidemiological models incorporating heterogeneous infectivity

Computational and Applied Mathematics

In this paper we analyze general deterministic epidemiological models described by autonomous ordinary differential equations taking into account heterogeneity related to the infectivity and vital dynamics, in which the flow into the compartment of the susceptible individuals is given by a generic function. Our goal is to provide a new tool that facilitates the qualitative analysis of equilibrium points, which represent the disease free population, generalizing the result presented by Leite et al. (Math Med Biol J IMA 17:15-31, 2000) , and population extinction. The epidemiological models exposed are the type SEIRS (Susceptible-Exposed-Infectious-Recovered-Susceptible) and SEIR (Susceptible-Exposed-Infectious-Recovered) with vaccination. Moreover, we computed the basic reproduction number from the models by van den Driessche and Watmough (Math Biosci 180:29-48, 2002) and correlate this threshold parameter with the stability of the equilibrium point representing the disease free population.