A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan

Sex-structured disease transmission model and control mechanisms for visceral leishmaniasis (VL)

BACKGROUND

Leishmaniasis are a group of diseases caused by more than 20 species of the protozoan that are transmitted through the bite of female sand fly. The disease is endemic to 98 countries of the world. It affects most commonly the poorest of the poor and mainly males. Several research has been conducted to propose disease control strategies. Effective medical care, vector control, environmental hygiene, and personal protection are the mainstays of the current preventative and control methods. The mathematical models for the transmission dynamics of the disease studied so far did not consider the sex-biased burden of the disease into consideration.

METHODOLOGY

Unlike the previous VL works, this study introduces a new deterministic sex-structured model for understanding the transmission dynamics of visceral leishmaniasis. Basic properties of the model including basic reproduction number ([Formula: see text]), and conditions for the existence of backward bifurcation of the model are explored. Baseline parameter values were estimated after the model was fitted to Ethiopia's VL data. Sensitivity analysis of the model was performed to identify the parameters that significantly impact the disease threshold. Numerical simulations were performed using baseline parameter values, and scenario analysis is performed by changing some of these parameters as appropriate.

CONCLUSION

The analysis of the model shows that there is a possibility for a backward bifurcation for [Formula: see text], which means bringing [Formula: see text] to less than unity may not be enough to eradicate the disease. Our numerical result shows that the implementation of disease-preventive strategies, as well as effectively treating the affected ones can significantly reduce the disease prevalence if applied for more proportion of the male population. Furthermore, the implementation of vector management strategies also can considerably reduce the total prevalence of the disease. However, it is demonstrated that putting more effort in treating affected reservoir animals may not have any significant effect on the overall prevalence of the disease as compared to other possible mechanisms. The numerical simulation infers that a maximum of 60% of extra preventative measures targeted to only male population considerably reduces the total prevalence of VL by 80%. It is also possible to decrease the total prevalence of VL by 69.51% when up to 50% additional infected males receive treatment with full efficacy. Moreover, applying a maximum of 15% additional effort to reduce the number of vectors, decreases the total VL prevalence by 57.71%. Therefore, in order to reduce the disease burden of visceral leishmaniasis, public health officials and concerned stakeholders need to give more emphasis to the proportion of male humans in their intervention strategies.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

Zoonotic MERS-CoV transmission: modeling, backward bifurcation and optimal control analysis

Middle East Respiratory Syndrome Coronavirus (MERS-CoV) can cause mild to severe acute respiratory illness with a high mortality rate. As of January 2020, more than 2500 cases of MERS-CoV resulting in around 860 deaths were reported globally. In the absence of neither effective treatment nor a ready-to-use vaccine, control measures can be derived from mathematical models of disease epidemiology. In this manuscript, we propose and analyze a compartmental model of zoonotic MERS-CoV transmission with two co-circulating strains. The human population is considered with eight compartments while the zoonotic camel population consist of two compartments. The expression of basic reproduction numbers are obtained for both single strain and two strain version of the proposed model. We show that the disease-free equilibrium of the system with single stain is globally asymptotically stable under some parametric conditions. We also demonstrate that both models undergo backward bifurcation phenomenon, which in turn indicates that only keeping R 0 below unity may not ensure eradication. To the best of the authors knowledge, backward bifurcation was not shown in a MERS-CoV transmission model previously. Further, we perform normalized sensitivity analysis of important model parameters with respect to basic reproduction number of the proposed model. Furthermore, we perform optimal control analysis on different combination interventions with four components namely preventive measures such as use of masks, isolation of strain-1 infected people, strain-2 infected people and infected camels. Optimal control analysis suggests that combination of preventive measures and isolation of infected camels will eventually eradicate the disease from the community.

Modeling the effects of prosocial awareness on COVID-19 dynamics: Case studies on Colombia and India

The ongoing COVID-19 pandemic has affected most of the countries on Earth. It has become a pandemic outbreak with more than 50 million confirmed infections and above 1 million deaths worldwide. In this study, we consider a mathematical model on COVID-19 transmission with the prosocial awareness effect. The proposed model can have four equilibrium states based on different parametric conditions. The local and global stability conditions for awareness-free, disease-free equilibrium are studied. Using Lyapunov function theory and LaSalle invariance principle, the disease-free equilibrium is shown globally asymptotically stable under some parametric constraints. The existence of unique awareness-free, endemic equilibrium and unique endemic equilibrium is presented. We calibrate our proposed model parameters to fit daily cases and deaths from Colombia and India. Sensitivity analysis indicates that the transmission rate and the learning factor related to awareness of susceptibles are very crucial for reduction in disease-related deaths. Finally, we assess the impact of prosocial awareness during the outbreak and compare this strategy with popular control measures. Results indicate that prosocial awareness has competitive potential to flatten the COVID-19 prevalence curve.

Comments on "A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan"

Deterministic (ordinary differential equation) models for the transmission dynamics of vector-borne diseases that incorporate disease-induced death in the host(s) population(s) are generally known to exhibit the phenomenon of backward bifurcation (where a stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number of the model is less than unity). Further, it is well known that, in these models, the phenomenon of backward bifurcation does not occur when the disease-induced death rate is negligible (e.g., if the disease-induced death rate is set to zero). In a recent paper on the transmission dynamics of visceral leishmaniasis (a disease vectored by sandflies), titled "A Mathematical Study to Control Visceral Leishmaniasis: An Application to South Sudan," published in Bulletin of Mathematical Biology, Vol. 79, Pages 1110-1134, 2017, Ghosh et al. (2017) stated that their deterministic model undergoes a backward bifurcation even when the disease-induced mortality in the host population is set to zero. This result is contrary to the well-established theory on the dynamics of vector-borne diseases. In this short note, we illustrate some of the key errors in the Ghosh et al. (2017) study.