The long and binding treatment of tuberculosis (TB) at least 6-8 months for the new cases, the partial immunity given by BCG vaccine, the loss of immunity after a few years doing that strategy of TB control via vaccination and treatment of infectious are not sufficient to eradicate TB. TB is an infectious disease caused by the bacillus Mycobacterium tuberculosis. Adults are principally attacked. In this work, we assess the impact of vaccination in the spread of TB via a deterministic epidemic model (SV ELI) (Susceptible, Vaccinated, Early latent, Late latent, Infectious). Using the Lyapunov-Lasalle method, we analyse the stability of epidemic system (SV ELI) around the equilibriums (disease-free and endemic). The global asymptotic stability of the unique endemic equilibrium whenever R 0 > 1 is proved, where R 0 is the reproduction number. We prove also that when R 0 is less than 1, TB can be eradicated. Numerical simulations, using some TB data found in the literature in relation with Cameroon, are conducted to approve analytic results, and to show that vaccination coverage is not sufficient to control TB, effective contact rate has a high impact in the spread of TB.

COVID-19 is an acute respiratory illness in humans caused by a coronavirus, capable of producing severe symptoms and, in some cases, death, especially in older people and those with underlying health conditions. It was originally identified in China in 2019 and became a pandemic in 2020. On 6 March 2020, Cameroon recorded its first cases of infection with COVID-19. The Government of Cameroon (GOC) took 13 barrier measures on 18 March 2020. On 1 May 2020, 19 new measures were adopted, easing restrictions and encouraging economic activity. On 1 June, schools and universities were reopened, after which massive screening began to take place throughout the country. In this study, we have modelled the COVID-19 epidemic in Cameroon in order to assess the governmental measures of response and predict the behaviour of epidemic As a result of these measures, the pandemic evolved in three phases. The first phase began on 18 March and ended on 15 May 2020. During this phase, the actual curve of cumulative positive cases based on field data closely fit the theoretical curve resulting from mathematical modelling. In the beginning of May, we predicted that nearly 3000 positive cases would be declared by mid-May 2020. The actual data confirmed these predictions: there were 2954 cases as of 15 May 2020. The second phase, beyond mid-May 2020, encompasses the period when the GOC’s relaxation of measures takes effect. This phase was marked by an acceleration of the cumulative number of positive cases starting in the third week of May, postponing the expected peak by two weeks. Under Phase 2 conditions, the onset of the peak will occur in early June and extend through the first two weeks of June. However, a third phase occurs in the first week of June, with the reopening of schools and universities combined with massive screening; the peak is therefore expected in the second week of June (around 15 June). The GOC should, at this stage, strengthen its response plan by tripling the current coverage capacity to regain the first phase convergence conditions associated with the first 13 measures. The pandemic will begin its descent in the month of august, but COVID-19 will remain endemic for at least one year.

This paper focuses on the study and control of a non-linear mathematical epidemic model (vih SS VELI) based on a system of ordinary differential equation modeling the spread of tuberculosis infectious with HIV/AIDS coinfection. Existence of both disease free equilibrium and endemic equilibrium is discussed. Reproduction number 0  is determined. Using Lyapunov-Lasalle methods, we analyze the stability of epidemic system around the equilibriums (disease free and endemic equilibrium). The global asymptotic stability of the disease free equilibrium whenever 1 vac <  is proved, where 0  is the reproduction number. We prove also that when 0  is less than one, tuberculosis can be eradicated. Numerical simulations are conducted to approve analytic results. To achieve control of the disease, seeking to reduce the infectious group by the minimum vaccine coverage, a control problem is formulated. The Pontryagin's maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically using the Runge Kutta fourth procedure.

COVID-19 is a highly contagious disease which has spread across the world. A deterministic model that considers an important component of individuals with vertically transmitted underlying diseases (high-risk susceptible individuals), rather than the general public, is formulated in this paper. We also consider key parameters that are concerned with the disease. An epidemiological threshold, R0, is computed using next-generation matrix approach. This is used to establish the existence and global stability of equilibria. We identify the most sensitive parameters which effectively contribute to change the disease dynamics with the help of sensitivity analysis. Our results reveal that increasing contact tracing of the exposed individuals who are tested for COVID-19 and hospitalizing them, largely has a negative impact on R0. Results further reveal that transmission rate between low-risk/high-risk susceptible individuals and symptomatic infectious individuals β and incubation rate of the exposed individuals σ have positive impact on R0. Numerical simulations show that there are fewer high-risk susceptible individuals than the general public when R0<1. This may be due to the fact that high-risk susceptible individuals may prove a bit more difficult to control than the low-risk susceptible individuals as a result of inherited underlying diseases present in them. We thus conclude that high level of tracing and hospitalizing the exposed individuals, as well as adherence to standard precautions and wearing appropriate Personal Protective Equipment (PPE) while handling emergency cases, are needed to flatten the epidemic curve.

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