Mathematical modeling of COVID-19 pandemic in the context of sub-Saharan Africa: a short-term forecasting in Cameroon and Gabon

In this paper, we propose and analyse a compartmental model of COVID-19 to predict and control the outbreak. We first formulate a comprehensive mathematical model for the dynamical transmission of COVID-19 in the context of sub-Saharan Africa. We provide the basic properties of the model and compute the basic reproduction number $\mathcal {R}_0$ when the parameter values are constant. After, assuming continuous measurement of the weekly number of newly COVID-19 detected cases, newly deceased individuals and newly recovered individuals, the Ensemble of Kalman filter (EnKf) approach is used to estimate the unmeasured variables and unknown parameters, which are assumed to be time-dependent using real data of COVID-19. We calibrated the proposed model to fit the weekly data in Cameroon and Gabon before, during and after the lockdown. We present the forecasts of the current pandemic in these countries using the estimated parameter values and the estimated variables as initial conditions. During the estimation period, our findings suggest that $\mathcal {R}_0 \approx 1.8377 $ in Cameroon, while $\mathcal {R}_0 \approx 1.0379$ in Gabon meaning that the disease will not die out without any control measures in theses countries. Also, the number of undetected cases remains high in both countries, which could be the source of the new wave of COVID-19 pandemic. Short-term predictions firstly show that one can use the EnKf to predict the COVID-19 in Sub-Saharan Africa and that the second vague of the COVID-19 pandemic will still increase in the future in Gabon and in Cameroon. A comparison between the basic reproduction number from human individuals $\mathcal {R}_{0h}$ and from the SARS-CoV-2 in the environment $\mathcal {R}_{0v}$ has been done in Cameroon and Gabon. A comparative study during the estimation period shows that the transmissions from the free SARS-CoV-2 in the environment is greater than that from the infected individuals in Cameroon with $\mathcal {R}_{0h}$ = 0.05721 and $\mathcal {R}_{0v}$ = 1.78051. This imply that Cameroonian apply distancing measures between individual more than with the free SARS-CoV-2 in the environment. But, the opposite is observed in Gabon with $\mathcal {R}_{0h}$ = 0.63899 and $\mathcal {R}_{0v}$ = 0.39894. So, it is important to increase the awareness campaigns to reduce contacts from individual to individual in Gabon. However, long-term predictions reveal that the COVID-19 detected cases will play an important role in the spread of the disease. Further, we found that there is a necessity to increase timely the surveillance by using an awareness program and a detection process, and the eradication of the pandemic is highly dependent on the control measures taken by each government.

Bifurcation analysis of a phage-bacteria interaction model with prophage induction

A predator-prey model is used to investigate the interactions between phages and bacteria by considering the lytic and lysogenic life cycles of phages and the prophage induction. We provide answers to the following conflictual research questions: (1) what are conditions under which the presence of phages can purify a bacterial infected environment? (2) Can the presence of phages triggers virulent bacterial outbreaks? We derive the basic offspring number $\mathcal N_0$ that serves as a threshold and the bifurcation parameter to study the dynamics and bifurcation of the system. The model exhibits three equilibria: an unstable environment-free equilibrium, a globally asymptotically stable (GAS) phage-free equilibrium (PFE) whenever $\mathcal N_0<1$, and a locally asymptotically stable environment-persistent equilibrium (EPE) when $\mathcal N_0>1$. The Lyapunov-LaSalle techniques are used to prove the GAS of the PFE and estimate the EPE basin of attraction. Through the center manifold approximation, topological types of the PFE are precised. Existence of transcritical and Hopf bifurcations are established. Precisely, when $\mathcal N_0>1$, the EPE loses its stability and periodic solutions arise. Furthermore, increasing $\mathcal N_0$ can purify an environment where bacteriophages are introduced. Purposely, we prove that for large values of $\mathcal N_0$, the overall bacterial population asymptotically approaches zero, while the phage population sustains. Ecologically, our results show that for small values of $\mathcal N_0$, the existence of periodic solutions could explain the occurrence of repetitive bacteria-borne disease outbreaks, while large value of $\mathcal N_0$ clears bacteria from the environment. Numerical simulations support our theoretical results.

A patchy model for the transmission dynamics of tuberculosis in sub-Saharan Africa

Tuberculosis (TB) spreads through contact between a susceptible person and smear positive pulmonary TB case (TPM+). The spread of TB is highly dependent on people migration between cities or regions that may have different contact rates and different environmental parameters, leading to different disease spread speed in the population. In this work, a metapopulation model, i.e., networks of populations connected by migratory flows, which overcomes the assumption of homogeneous mixing between different regions was constructed. The TB model was combined to a simple demographic structure for the population living in a multi-patch environment (cities, towns, regions or countries). The model consist of a system of differential equations coupling TB epidemic at different strength and mobility between the patches. Constant recruitment rate, slow and fast progression to the disease, effective chemoprophylaxis, diagnostic and treatment are taken into account to make the model including the reality of people in the sub-Saharan African countries. The basic reproduction number ( R 0 ) was computed and it was demonstrated that the disease-free equilibrium is globally asymptotically stable ifR 0 < 1 . WhenR 0 > 1 , the disease-free equilibrium is unstable and there exists one endemic equilibrium. Moreover, the impact of increasing migration rate between patches on the TB spread was quantified using numerical implementation of the model. Using an example on 15 inter-connected patches on the same road, we demonstrated that most people was most likely to get infected if the disease starts in a patch in the middle than in border patches.