Hybrid discrete-time-continuous-time models and a SARS CoV-2 mystery: Sub-Saharan Africa's low SARS CoV-2 disease burden.

Worldwide, the recent SARS-CoV-2 virus has infected more than 670 million people and killed nearly 67.0 million. In Africa, the number of confirmed COVID-19 cases was approximately 12.7 million as of January 11, 2023, that is about 2% of the infections around the world. Many theories and modeling techniques have been used to explain this lower-than-expected number of reported COVID-19 cases in Africa relative to the high disease burden in most developed countries. We noted that most epidemiological mathematical models are formulated in continuous-time interval, and taking Cameroon in Sub-Saharan Africa, and New York State in the USA as case studies, in this paper we developed parameterized hybrid discrete-time-continuous-time models of COVID-19 in Cameroon and New York State. We used these hybrid models to study the lower-than-expected COVID-19 infections in developing countries. We then used error analysis to show that a time scale for a data-driven mathematical model should match that of the actual data reporting.

Disease-Induced Hydra Effect with Overcompensatory Recruitment.

In ecological systems, the hydra effect is an increase in population size caused by an increase in mortality. This seemingly counterintuitive effect has been observed in several populations, including fish, blowflies, snails and plants, and has been modeled in both continuous and discrete time. A similar effect induced by disease has recently been observed empirically. Here we present theoretical and simulation results for an infectious disease-induced hydra effect, namely conditions under which the total population size, composed of those that are infectious as well as those that are susceptible, at an endemic equilibrium is greater than the population size at the disease-free equilibrium. (For an endemic k-cycle, this can be similarly defined using the average population.) We find this disease-induced hydra effect occurs when the intra-specific competition is strong and disease infection sufficiently inhibits the reproductive output of infected individuals. For our continuous time model, we give a necessary and sufficient condition for a disease-induced hydra effect. This condition requires overcompensatory recruitment. With a discrete time model, we show there is no disease-induced hydra effect without overcompensatory recruitment. We illustrate by simulations that a disease-induced hydra effect may occur with Ricker recruitment when the endemic system converges to either a fixed equilibrium or a 2-cycle.

Asymptotic behavior of a discrete-time density-dependent SI epidemic model with constant recruitment.

We use the epidemic threshold parameter, R 0 , and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables S n and I n represent the populations of susceptibles and infectives at time n = 0 , 1 , … , respectively. The model features constant survival "probabilities" of susceptible and infective individuals and the constant recruitment per the unit time interval [ n , n + 1 ] into the susceptible class. We compute the basic reproductive number, R 0 , and use it to prove that independent of positive initial population sizes, R 0 < 1 implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever R 0 > 1 and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.

Disease Extinction Versus Persistence in Discrete-Time Epidemic Models.

We focus on discrete-time infectious disease models in populations that are governed by constant, geometric, Beverton-Holt or Ricker demographic equations, and give a method for computing the basic reproduction number, [Formula: see text]. When [Formula: see text] and the demographic population dynamics are asymptotically constant or under geometric growth (non-oscillatory), we prove global asymptotic stability of the disease-free equilibrium of the disease models. Under the same demographic assumption, when [Formula: see text], we prove uniform persistence of the disease. We apply our theoretical results to specific discrete-time epidemic models that are formulated for SEIR infections, cholera in humans and anthrax in animals. Our simulations show that a unique endemic equilibrium of each of the three specific disease models is asymptotically stable whenever [Formula: see text].

Granuloma formation in leishmaniasis: A mathematical model.

Leishmaniasis is a disease caused by the Leishmania parasites. The two common forms of leishmaniasis are cutaneous leishmaniasis (CL) and visceral leishmaniasis (VL). VL is the more severe of the two and, if untreated, may become fatal. The hallmark of VL is the formation of granuloma in the liver or the spleen. In this paper, we develop a mathematical model of the evolution of granuloma in the liver. The model is represented by a system of partial differential equations and it includes migration of cells from the adaptive immune system into the granuloma; the rate of the influx is determined by the strength of the immune response of the infected individual. It is shown that parasite load decreases as the strength of the immune system increases. Furthermore, the efficacy of a commonly used drug, which increases T cells proliferation, increases in an individual with stronger immune response. The model also provides an explanation why, in contrast to humans, mice recover naturally from VL in the liver.

A Mathematical Model of Anthrax Transmission in Animal Populations.

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number [Formula: see text] is calculated, and existence of a unique endemic equilibrium is established for [Formula: see text] above the threshold value 1. Using data from the literature, elasticity indices for [Formula: see text] and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if [Formula: see text]. For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with [Formula: see text] and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.

Modelling Wolbachia infection in a sex-structured mosquito population carrying West Nile virus.

Wolbachia is possibly the most studied reproductive parasite of arthropod species. It appears to be a promising candidate for biocontrol of some mosquito borne diseases. We begin by developing a sex-structured model for a Wolbachia infected mosquito population. Our model incorporates the key effects of Wolbachia infection including cytoplasmic incompatibility and male killing. We also allow the possibility of reduced reproductive output, incomplete maternal transmission, and different mortality rates for uninfected/infected male/female individuals. We study the existence and local stability of equilibria, including the biologically relevant and interesting boundary equilibria. For some biologically relevant parameter regimes there may be multiple coexistence steady states including, very importantly, a coexistence steady state in which Wolbachia infected individuals dominate. We also extend the model to incorporate West Nile virus (WNv) dynamics, using an SEI modelling approach. Recent evidence suggests that a particular strain of Wolbachia infection significantly reduces WNv replication in Aedes aegypti. We model this via increased time spent in the WNv-exposed compartment for Wolbachia infected female mosquitoes. A basic reproduction number [Formula: see text] is computed for the WNv infection. Our results suggest that, if the mosquito population consists mainly of Wolbachia infected individuals, WNv eradication is likely if WNv replication in Wolbachia infected individuals is sufficiently reduced.

A bovine babesiosis model with dispersion.

Bovine Babesiosis (BB) is a tick borne parasitic disease with worldwide over 1.3 billion bovines at potential risk of being infected. The disease, also called tick fever, causes significant mortality from infection by the protozoa upon exposure to infected ticks. An important factor in the spread of the disease is the dispersion or migration of cattle as well as ticks. In this paper, we study the effect of this factor. We introduce a number, [Formula: see text], a "proliferation index," which plays the same role as the basic reproduction number [Formula: see text] with respect to the stability/instability of the disease-free equilibrium, and observe that [Formula: see text] decreases as the dispersion coefficients increase. We prove, mathematically, that if [Formula: see text] then the tick fever will remain endemic. We also consider the case where the birth rate of ticks undergoes seasonal oscillations. Based on data from Colombia, South Africa, and Brazil, we use the model to determine the effectiveness of several intervention schemes to control the progression of BB.

Mathematical modeling of contact tracing and stability analysis to inform its impact on disease outbreaks; an application to COVID-19.

We develop a mathematical model to investigate the effect of contact tracing on containing epidemic outbreaks and slowing down the spread of transmissible diseases. We propose a discrete-time epidemic model structured by disease-age which includes general features of contact tracing. The model is fitted to data reported for the early spread of COVID-19 in South Korea, Brazil, and Venezuela. The calibrated values for the contact tracing parameters reflect the order pattern observed in its performance intensity within the three countries. Using the fitted values, we estimate the effective reproduction number Re and investigate its responses to varied control scenarios of contact tracing. Alongside the positivity of solutions, and a stability analysis of the disease-free equilibrium are provided.

Unequal effects of SARS-CoV-2 infections: model of SARS-CoV-2 dynamics in Cameroon (Sub-Saharan Africa) versus New York State (United States).

Worldwide, the recent SARS-CoV-2 virus disease outbreak has infected more than 691,000,000 people and killed more than 6,900,000. Surprisingly, Sub-Saharan Africa has suffered the least from the SARS-CoV-2 pandemic. Factors that are inherent to developing countries and that contrast with their counterparts in developed countries have been associated with these disease burden differences. In this paper, we developed data-driven COVID-19 mathematical models of two 'extreme': Cameroon, a developing country, and New York State (NYS) located in a developed country. We then identified critical parameters that could be used to explain the lower-than-expected COVID-19 disease burden in Cameroon versus NYS and to help mitigate future major disease outbreaks. Through the introduction of a 'disease burden' function, we found that COVID-19 could have been much more severe in Cameroon than in NYS if the vaccination rate had remained very low in Cameroon and the pandemic had not ended.

A discrete-time risk-structured model of cholera infections in Cameroon.

In a recent paper, Che et al. [5] used a continuous-time Ordinary Differential Equation (ODE) model with risk structure to study cholera infections in Cameroon. However, the population and the reported cholera cases in Cameroon are censored at discrete-time annual intervals. In this paper, unlike in [5], we introduce a discrete-time risk-structured cholera model with no spatial structure. We use our discrete-time demographic equation to 'fit' the annual population of Cameroon. Furthermore, we use our fitted discrete-time model to capture the annually reported cholera cases from 1987 to 2004 and to study the impact of vaccination, treatment and improved sanitation on the number of cholera infections from 2004 to 2019. Our discrete-time cholera model confirms the results of the ODE model in [5]. However, our discrete-time model predicts a decrease in the number of cholera cases in a shorter period of cholera intervention (2004-2019) as compared to the ODE model's period of intervention (2004-2022).

Mathematical modeling of the influence of cultural practices on cholera infections in Cameroon.

The Far North Region of Cameroon, a high risk cholera endemic region, has been experiencing serious and recurrent cholera outbreaks in recent years. Cholera outbreaks in this region are associated with cultural practices (traditional and religious beliefs). In this paper, we introduce a mathematical model of the influence of cultural practices on the dynamics of cholera in the Far North Region. Our model is an SEIR type model with a pathogen class and multiple susceptible classes based on traditional and religious beliefs. Using daily reported cholera cases from three health districts (Kaélé, Kar Hay and Moutourwa) in the Far North Region from June 25, 2019 to August 16, 2019, we estimate parameter values of our model and use Akaike information criterion (AIC) to demonstrate that our model gives a good fit for our data on cholera cases. We use sensitivity analysis to study the impact of each model parameter on the threshold parameter (control reproduction number), Rc, and the number of model predicted cholera cases. Finally, we investigate the effect of cultural practices on the number of cholera cases in the region.

Mathematical model for optimal use of sulfadoxine-pyrimethamine as a temporary malaria vaccine.

In this paper, we introduce a deterministic malaria model for determining the drug administration protocol that leads to the smallest first malaria episodes during the wet season. To explore the effects of administering the malaria drug on different days during the wet season while minimizing the potential harmful effects of drug overdose, we define 40 drug administration protocols. Our results fit well with the clinical studies of Coulibaly et al. at a site in Mali. In addition, we provide protocols that lead to smaller number of first malaria episodes during the wet season than the protocol of Coulibaly et al.

Age structured discrete-time disease models with demographic population cycles.

We use juvenile-adult discrete-time infectious disease models with intrinsically generated demographic population cycles to study the effects of age structure on the persistence or extinction of disease and the basic reproduction number, [Formula: see text]. Our juvenile-adult Susceptible-Infectious-Recovered (SIR) and Infectious-Salmon Anemia-Virus (ISA[Formula: see text] models share a common disease-free system that exhibits equilibrium dynamics for the Beverton-Holt recruitment function. However, when the recruitment function is the Ricker model, a juvenile-adult disease-free system exhibits a range of dynamic behaviours from stable equilibria to deterministic period k population cycles to Neimark-Sacker bifurcations and deterministic chaos. For these two models, we use an extension of the next generation matrix approach for calculating [Formula: see text] to account for populations with locally asymptotically stable period k cycles in the juvenile-adult disease-free system. When [Formula: see text] and the juvenile-adult demographic system (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove that the juvenile-adult disease goes extinct whenever [Formula: see text]. Under the same period k juvenile-adult demographic assumption but with [Formula: see text], we prove that the juvenile-adult disease-free period k population cycle is unstable and the disease persists. When [Formula: see text], our simulations show that the juvenile-adult disease-free period k cycle dynamics drives the juvenile-adult SIR disease dynamics, but not the juvenile-adult ISAv disease dynamics.

Risk structured model of cholera infections in Cameroon.

Since 1991, Cameroon, a cholera endemic African country, has been experiencing large cholera outbreaks and cholera related deaths. In this paper, we use a "fitted" demographic equation (disease-free equation) to capture the total population of Cameroon, and then use a fitted low-high risk structured cholera differential equation model to study reported cholera cases in Cameroon from 1987 to 2004. For simplicity, our model has no spatial structure. The basic reproduction number of our fitted cholera model, R0, is bigger than 1 and our model predicted cholera endemicity in Cameroon. In addition, the fitted risk structured model predicted a decreasing trend from 1987 to 1994 and an increasing trend from 1995 to 2004 in the pre-intervention reported number of cholera cases in Cameroon from 1987 to 2004. Using the fitted risk structured cholera model, we study the impact of vaccination, treatment and improved sanitation on the number of cholera infections in Cameroon from 2004 to 2022. The dual strategies of either vaccination and treatment or vaccination and improved sanitation or the combined strategy of vaccination, treatment and improved sanitation reduce the basic reproduction number of Cameroon from 1.1803 to 0.9982, 1.1803 to 0.9987 and 1.1803 to 0.9952, respectively, and the number of cholera cases by 99.6735%, 98.7498% and 99.7280%, respectively. Thus, each of these three strategies is capable of eliminating cholera in Cameroon with the combined strategy having the lowest value for the effective reproduction number, RE, and the highest percentage decrease in the number of cholera cases. Finally, using sensitivity analysis, we study the impact of our model parameters on the demographic threshold, basic reproduction number, effective reproduction number and on the total number of our model's predicted cholera cases.

Demographic population cycles and ℛ0 in discrete-time epidemic models.

We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, [Formula: see text], to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where [Formula: see text]. When [Formula: see text] and the demographic equation (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove the local asymptotic stability of the disease-free period k cycle. That is, the disease goes extinct whenever [Formula: see text]. Under the same period k demographic assumption but with [Formula: see text], we prove that the disease-free period k population cycle is unstable and the disease persists. Using the Ricker recruitment function, we apply our results to discrete-time infectious disease models that are formulated for Susceptible-Infectious-Recovered (SIR) infections with and without vaccination, and Infectious Salmon Anemia Virus (ISA[Formula: see text]) infections in a salmon population. When [Formula: see text], our simulations show that the disease-free period k cycle dynamics drives the SIR disease dynamics, but not the ISAv disease dynamics.

Controlling imported malaria cases in the United States of America.

We extend the mathematical malaria epidemic model framework of Dembele et al. and use it to ``capture" the 2013 Centers for Disease Control and Prevention (CDC) reported data on the 2011 number of imported malaria cases in the USA. Furthermore, we use our ``fitted" malaria models for the top 20 countries of malaria acquisition by USA residents to study the impact of protecting USA residents from malaria infection when they travel to malaria endemic areas, the impact of protecting residents of malaria endemic regions from mosquito bites and the impact of killing mosquitoes in those endemic areas on the CDC number of imported malaria cases in USA. To significantly reduce the number of imported malaria cases in USA, for each top 20 country of malaria acquisition by USA travelers, we compute the optimal proportion of USA international travelers that must be protected against malaria infection and the optimal proportion of mosquitoes that must be killed.

Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models.

We use a periodically forced density-dependent compensatory Leslie model to study the combined effects of environmental fluctuations and age-structure on pioneer populations. In constant environments, the models have globally attracting positive fixed points. However, with the advent of periodic forcing, the models have globally attracting cycles. We derive conditions under which the cycle is attenuant, resonant, and neither attenuant nor resonant. These results show that the response of age-structured populations to environmental fluctuations is a complex function of the compensatory mechanisms at different life-history stages, the fertile age classes and the period of the environment.

Spatially discrete metapopulation models with directional dispersal.

We use an age-structured discrete-time metapopulation model linking two sub-populations through larval transport and directed movements of adults to study the implications of linkages among subpopulations for the stability and resilience of exploited species. Our two-habitat model, a generalization of Fogarty's inshore-offshore lobster population model, includes isolated habitats under compensatory (monotone) or overcompensatory (oscillatory) dynamics [M.J. Fogarty, Implications of migration and larval interchange in American lobster (Homarus americanus) stocks: spatial structure and resilience, in: G.S. Jamieson, A. Campbell (Eds.), Proc. of North Pacific Symposium on Invertebrate Stock Assessment and Management, Can. Spec. Publ. Fish. Aquat. Sci. 125 (1998) 273]. Pre-migration local dynamics are selected from general classes of functions that capture the effects of competition for resources via contest (compensatory) and scramble (overcompensatory) intraspecific competitions. We explore the implications of these mechanisms on the long-term survival of exploited species. In particular, we use threshold parameters R(d)1 for Habitat 1 and R(d)2 for Habitat 2 together with precise mathematical definitions to prove that species persistence is possible at high levels of fishing in one habitat and low to moderate levels of fishing in the other. Our results support Fogarty's conclusion that conservative management of larval source populations could contribute to the resilience of exploited species.