Mathematical analysis of a model for zoonotic visceral leishmaniasis

Abstract

Zoonotic visceral leishmaniasis (ZVL), caused by the protozoan parasite Leishmania infantum and transmitted to humans and reservoir hosts by female sandflies, is endemic in many parts of the world (notably in Africa, Asia and the Mediterranean). This study presents a new mathematical model for assessing the transmission dynamics of ZVL in human and non-human animal reservoir populations. The model undergoes the usual phenomenon of backward bifurcation exhibited by similar vector-borne disease transmission models. In the absence of such phenomenon (which is shown to arise due to the disease-induced mortality in the host populations), the nontrivial disease-free equilibrium of the model is shown to be globally-asymptotically stable when the associated reproduction number of the model is less than unity. Using case and demographic data relevant to ZVL dynamics in Arac̣atuba municipality of Brazil, it is shown, for the default case when systemic insecticide-based drugs are not used to treat infected reservoir hosts, that the associated reproduction number of the model (normalℛ0) ranges from 0.3 to 1.4, with a mean of normalℛ0=0.85. Furthermore, when the effect of such drug treatment is explicitly incorporated in the model (i.e., accounting for the additional larval and sandfly mortality, following feeding on the treated reservoirs), the range of normalℛ0 decreases to normalℛ0∈[0.1,0.25em0.6], with a mean of normalℛ0=0.35 (this significantly increases the prospect of the effective control or elimination of the disease). Thus, ZVL transmission models (in communities where such treatment strategy is implemented) that do not explicitly incorporate the effect of such treatment may be over-estimating the disease burden (as measured in terms of normalℛ0) in the community. It is shown that normalℛ0 is more sensitive to increases in sandfly lifespan than that of the animal reservoir (so, a strategy that focuses on reducing sandflies, rather than the animal reservoir (e.g., via culling), may be more effective in reducing ZVL burden in the community). Further sensitivity analysis of the model ranks the sandfly removal rate (by natural death or by feeding from insecticide-treated reservoir hosts), the biting rate of sandflies on the reservoir hosts and the progression rate of exposed reservoirs to active ZVL as the three parameters with the most effect on the disease dynamics or burden (as measured in terms of the reproduction number normalℛ0). Hence, this study identifies the key parameters that play a key role on the disease dynamics, and thereby contributing in the design of effective control strategies (that target the identified parameters).

Mathematically modeling spillovers of an emerging infectious zoonosis with an intermediate host

“…In contrast, a zoonosis model must consider the disease dynamics in its nonhuman compartments, since these dynamics determine whether the pathogen reaches humans at all. Attempts have been made to model zoonotic spillovers [ 6 , 7 , 27 ], but without incorporating changes in the pathogen’s ecology over the course of an epidemic, these models are mathematically indistinguishable from those modeling a vector-borne disease with more hosts or a multispecies model. While a sizeable literature exists on mathematical models of vectorborne diseases, and this class of pathogen provides a useful comparison for the type of behavior modeled here, no model captures the unintentional opportunism of zoonoses or incorporates selective pressure on viruses [ 7 ].…”

Transmission dynamics of SARS-CoV-2: A modeling analysis with high-and-moderate risk populations

“…For mathematical convenience, the following equation represent the rate of change of the total population of humans, which is given by . Here, the prime denotes differentiation with respect to time, and thus, following [41] , we have Consider solutions of Eqn (1) , which is given by , and simplifying N it from Eqn. (2) , one can see that all solutions of the model starting in remain in for all .…”

Dynamical analysis of chikungunya and dengue co-infection model

“…We fitted each of the sub-model and the full model to the data using the Pearson's Chi-square and the least square methods (using the R statistical software) [21,35]. Firstly, each of the sub-models (6) and (9) were fitted to the cumulative number of human cases from 2010-2017 (see Table A1 for the number of CHIKV and DENV cases) using the data obtained from National Vector Borne Disease Control Programme (NVBDCP) India [31].…”

“…Thus, the region Ω is positive-invariant, and it is sufficient to consider solutions restricted in Ω. In this region, the usual existence, uniqueness and continuation results hold for the model Eqns (1)-(2) [21,43]. Table 2. 3.…”