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5In this article, a mathematical model for the transmission of COVID-19 disease is formulated 6 and analysed. It is shown that the model exhibits a backward bifurcation at R 0 = 1 when recovered 7 individuals do not develop a permanent immunity for the disease. In the absence of reinfection, it is 8 proved that the model is without backward bifurcation and the disease free equilibrium is globally 9 asymptotically stable for R 0 < 1. By using available data, the model is validated and parameter 10 values are estimated. The sensitivity of the value of R 0 to changes in any of the parameter values 11 involved in its formula is analysed. Moreover, various mitigation strategies are investigated using 12 the proposed model and it is observed that the asymptomatic infectious group of individuals may 13 play the major role in the re-emergence of the disease in the future. Therefore, it is recommended 14 that in the absence of vaccination, countries need to develop capacities to detect and isolate at least 15 30% of the asymptomatic infectious group of individuals while treating in isolation at least 50% of 16 symptomatic patients to control the disease.

The novel coronavirus disease has ravaged many health systems around the world and has brought many economies to their knees. In the absence of an approved curing medicine or approved vaccine to date, the major control of the surge of infections is through use of Non-Pharmaceutical Interventions (NPIs) and imposing specific standard operating procedures (SOPs) in instances when the disease spread curbs are relaxed. It is thus essential to quantify the extent to which specific NPIs can be useful in containing the pandemic. To achieve this, we constructed a mathematical model that accounts for both person to person transmission as well as transmission through contact from pathogen-contaminated surfaces. The model assumes that there is change of behaviour resulting from the surge of the number of cases, hence a class of susceptible individuals who practise self-protection measures. Basic properties of the model including the conditions for existence and stability of steady states are explored. The model was fitted to new-cases data for South Africa and baseline parameter values were estimated. Sensitivity analysis of the model was performed to determine the most influential parameters on the disease threshold. Our results show that practising of selfprotection measures is vital in slowing the spread of the infection. In addition, it is evident from the results that minimizing contact through "physical distancing" as well as with contaminated surfaces can significantly help in containing the infection. The model was extended to account for testing and quarantining of both symptomatic and asymptomatic infected individuals. In addition, migration and potential use of a vaccine were explored. In the case of migration, the scenarios considered included aspects when there are both border control and illegal crossings as well as the case where the government is in full control with proper SOPs. Our results show that, although testing and isolating/quarantining of infected individuals is essential in

Communicated by: J. Banasiak MSC Classification: 34A34; 37N25; 92B05; 92D30 Syphilis, a major sexually transmitted disease, continues to pose major public health burden in both underdeveloped and developed nations of the world.This study presents a new 2-group sex-structured model for assessing the community-level impact of treatment and condom use on the transmission dynamics and control of syphilis. Rigorous analysis of the model shows that it undergoes the phenomenon of backward bifurcation. In the absence of this phenomenon (which is shown to arise because of the reinfection of recovered individuals), the disease-free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model can have multiple endemic equilibria when the reproduction threshold exceeds unity. Numerical simulations of the model, using data relevant to the transmission dynamics of the disease in Nigeria, show that, with the assumed 80% condom efficacy, the disease will continue to persist (ie, remain endemic) in the population regardless of the level of compliance in condom usage by men. Furthermore, detailed optimal control analysis (using Pontraygin's maximum principle) reveals that, for situations where the cost of implementing the controls (treatment and condom-use) considered in this study is low, channelling resources to a treatment-only strategy is more effective than channelling them to a condom-use only strategy. Furthermore, as expected, the combined condom-treatment strategy provides a higher population-level impact than the treatment-only strategy or the condom-use only strategy. When the cost of implementing the controls is high, the 3 strategies are essentially equally as ineffective.

Background. Leishmaniasis is a parasitic disease caused by obligate intracellular protozoans of the genus Leishmania. Objective. To assess the distribution of human leishmaniasis and assess community knowledge, attitude, and practice with regard to assumed risk factors and control options used by the society. Methods. Retrospective study from November 2013 to May 2014 was used. Six-year data from Metemma hospital record was reviewed and 89 people were interviewed. Results. The rates were 29% (n = 374/1270) and 26% (n = 328/1270) in 2005 E.C and 2003 E.C, respectively. 94% (1194/1270) of the affected individuals were in the age exceeding 15 years. At the same time, the rates in males and female were 97% (n = 1226/1270) and 3% (n = 44/1270), respectively. According to 88.8% (n = 79/89) of the respondents, transmission occurs through bite of sandflies, while 98.9% (n = 88/89) of the respondent's indicated that waste disposal in an open space was one of the risk factors for disease occurrence. Regarding the control measures, respondents replied that 73% (n = 65/89) of them use impregnated bed net and others use cleaning and proper waste disposal. Conclusion. The current finding indicated that the disease was common in the study area; as a result, proper use of impregnated bed net, early diagnosis and treatment, and reduction of different risk factors were essential.

A mathematical model presented in Berge T, Lubuma JM-S, Moremedi GM, Morris N Shava RK, A simple mathematical model for Ebola in Africa, J Biol Dyn 11(1): 42–74 (2016) for the transmission dynamics of Ebola virus is extended to incorporate vaccination and change of behavior for self-protection of susceptible individuals. In the new setting, it is shown that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number [Formula: see text] is less than or equal to unity and unstable when [Formula: see text]. In the latter case, the model system admits at least one endemic equilibrium point, which is locally asymptotically stable. Using the parameters relevant to the transmission dynamics of the Ebola virus disease, we give sensitivity analysis of the model. We show that the number of infectious individuals is much smaller than that obtained in the absence of any intervention. In the case of the mass action formulation with vaccination and education, we establish that the number of infectious individuals decreases as the intervention efforts increase. In the new formulation, apart from supporting the theory, numerical simulations of a nonstandard finite difference scheme that we have constructed suggests that the results on the decrease of the number of infectious individuals is valid.

Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R 0 , that is the average number of new infective individuals produced by one infective individual introduced into a completely susceptible population. If R 0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptotically stable endemic equilibrium when R 0 > 1. A nonstandard finite difference scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.

A deterministic model for the transmission dynamics of melioidosis disease in human population is designed and analyzed. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the basic reproduction number [Formula: see text] is less than one. It is further shown that the backward bifurcation dynamics is caused by the reinfection of individuals who recovered from the disease and relapse. The existence of backward bifurcation implies that bringing down [Formula: see text] to less than unity is not enough for disease eradication. In the absence of backward bifurcation, the global asymptotic stability of the disease-free equilibrium is shown whenever [Formula: see text]. For [Formula: see text], the existence of at least one locally asymptotically stable endemic equilibrium is shown. Sensitivity analysis of the model, using the parameters relevant to the transmission dynamics of the melioidosis disease, is discussed. Numerical experiments are presented to support the theoretical analysis of the model. In the numerical experimentations, it has been observed that screening and treating individuals in the exposed class has a significant impact on the disease dynamics.