Narrow Results

This study investigates the effects of vaccination and treatment on the spread of HIV/AIDS. The objectives are (i) to derive conditions for the success of vaccination and treatment programs and (ii) to derive threshold conditions for the existence and stability of equilibria in terms of the effective reproduction number R. It is found, firstly, that the success of a vaccination and treatment program is achieved when R_0t<R₀, R_0t<R_0v and γ_eR_VT(σ)<R_UT(α), where R_0t and R_0v are respectively the reproduction numbers for populations consisting entirely of treated and vaccinated individuals, R₀ is the basic reproduction number in the absence of any intervention, R_UT(α) and R_VT(σ) are respectively the reproduction numbers in the presence of a treatment (α) and a combination of vaccination and treatment (σ) strategies. Secondly, that if R<1, there exists a unique disease free equilibrium point which is locally asymptotically stable, while if R>1 there exists a unique locally asymptotically stable endemic equilibrium point, and that the two equilibrium points coalesce at R=1. Lastly, it is concluded heuristically that the stable disease free equilibrium point exists when the conditions R_0t<R₀, R_0t<R_0v and γ_eR_VT(σ)<R_UT(α) are satisfied.

The focus of this paper is on COVID-19, the December 2019 dated respiratory epidemic or infection that has ravaged most, if not all, of the world’s 253 “countries” (consisting of 194 independent nation-states, 55 dependent-states, Antarctica, and 3 other territories). Readers of this paper should be aware of at least three facts. First, as of January 2022, the submission date of this paper, COVID-19 continues to impact the world, although to a much lesser extent, given the protection afforded by the mRNA vaccines, the boosters, and the antiviral medications (e.g., oseltamivir, penciclovir, acyclovir, Paxlovid, etc.). Second, the paper should be regarded as being inconclusive in both its outlook and its list of references; it provides, at best, an intermediary account of the continuing COVID-19 pandemic. Third, like the Spanish Flu of 1918, it is speculated that the COVID-19 pandemic will not endure beyond three years and will conclude as an endemic problem by, hopefully, early 2023. Therefore, the paper is partitioned into three sections which, respectively, address the infection’s three phases: epidemic (initial infection), pandemic (worldwide infection) and endemic (pervasive but nonlife-threatening infection) phases. The lessons learned from the range of decisions made throughout the COVID-19 phases should help to inform and better prepare the world for future pathogens and deadly diseases.

A recent paper [W. D. Wang, Modeling adaptive behavior in influenza transmission, Math. Model. Nat. Phenom.7(3) (2012) 253–262] presented the local stability of the endemic equilibrium $E^{\ast }$ of an influenza transmission model incorporating human mobility behavior. In the present paper, we prove that $E^{\ast }$ is globally stable if the basic reproduction number ${{ℛ}}_0>1$.

In this paper, we study a system of differential equations that models the population dynamics of SEIR vector transmission of dengue fever. The model studied breeding value based on the number of reported cases of dengue fever in Selangor because the state had the highest case in Malaysia. The model explains that maximum level of human infection rate of dengue fever achieved in a very short period. It is also revealed that there existed suitability result between theoretical and empirical calculation using the model. The result of SEIR model will hopefully provide an insight into the spread of dengue fever in Selangor Malaysia and basic form for modeling this area.