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.

Mathematical Analysis of a Model for Assessing the Impact of Antiretroviral Therapy, Voluntary Testing and Condom Use in Curtailing the Spread of HIV.

This paper presents a deterministic model for evaluating the impact of anti-retroviral drugs (ARVs), voluntary testing (using standard antibody-based and a DNA-based testing methods) and condom use on the transmission dynamics of HIV in a community. Rigorous qualitative analysis of the model show that it has a globally-stable disease-free equilibrium whenever a certain epidemiological threshold, known as the effective reproduction number , is less than unity. The model has an endemic equilibrium whenever . The endemic equilibrium is shown to be locally-asymptotically stable for a special case. Numerical simulations of the model show that the use of the combined testing and treatment strategy is more effective than the use of the standard ELISA testing method with ARV treatment, even for the use of condoms as a singular strategy. Furthermore, the universal strategy (which involves the use of condoms, the two testing methods and ARV treatment) is always more effective than the combined use of the standard ELISA testing method and ARVs.

Modelling the transmission dynamics and control of the novel 2009 swine influenza (H1N1) pandemic.

The paper presents a deterministic compartmental model for the transmission dynamics of swine influenza (H1N1) pandemic in a population in the presence of an imperfect vaccine and use of drug therapy for confirmed cases. Rigorous analysis of the model, which stratifies the infected population in terms of their risk of developing severe illness, reveals that it exhibits a vaccine-induced backward bifurcation when the associated reproduction number is less than unity. The epidemiological consequence of this result is that the effective control of H1N1, when the reproduction number is less than unity, in the population would then be dependent on the initial sizes of the subpopulations of the model. For the case where the vaccine is perfect, it is shown that having the reproduction number less than unity is necessary and sufficient for effective control of H1N1 in the population (in such a case, the associated disease-free equilibrium is globally asymptotically stable). The model has a unique endemic equilibrium when the reproduction number exceeds unity. Numerical simulations of the model, using data relevant to the province of Manitoba, Canada, show that it reasonably mimics the observed H1N1 pandemic data for Manitoba during the first (Spring) wave of the pandemic. Further, it is shown that the timely implementation of a mass vaccination program together with the size of the Manitoban population that have preexisting infection-acquired immunity (from the first wave) are crucial to the magnitude of the expected burden of disease associated with the second wave of the H1N1 pandemic. With an estimated vaccine efficacy of approximately 80%, it is projected that at least 60% of Manitobans need to be vaccinated in order for the effective control or elimination of the H1N1 pandemic in the province to be feasible. Finally, it is shown that the burden of the second wave of H1N1 is expected to be at least three times that of the first wave, and that the second wave would last until the end of January or early February, 2010.

Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs.

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.

Role of incidence function in vaccine-induced backward bifurcation in some HIV models.

The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control. In such a scenario, the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. This paper addresses the role of the choice of incidence function in a vaccine-induced backward bifurcation in HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with their equivalents that employ mass action incidence. Furthermore, this result is independent of the type of vaccination program adopted. These results emphasize the need for further work on the incidence functions used in HIV models.

To cut or not to cut: a modeling approach for assessing the role of male circumcision in HIV control.

A recent randomized controlled trial shows a significant reduction in women-to-men transmission of HIV due to male circumcision. Such development calls for a rigorous mathematical study to ascertain the full impact of male circumcision in reducing HIV burden, especially in resource-poor nations where access to anti-retroviral drugs is limited. First of all, this paper presents a compartmental model for the transmission dynamics of HIV in a community where male circumcision is practiced. In addition to having a disease-free equilibrium, which is locally-asymptotically stable whenever a certain epidemiological threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium when the threshold is less than unity. The implication of this result is that HIV may persist in the population even when the reproduction threshold is less than unity. Using partial data from South Africa, the study shows that male circumcision at 60% efficacy level can prevent up to 220,000 cases and 8,200 deaths in the country within a year. Further, it is shown that male circumcision can significantly reduce, but not eliminate, HIV burden in a community. However, disease elimination is feasible if male circumcision is combined with other interventions such as ARVs and condom use. It is shown that the combined use of male circumcision and ARVs is more effective in reducing disease burden than the combined use of male circumcision and condoms for a moderate condom compliance rate.