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The geographic areas in the United States most affected by the coronavirus disease 2019 (COVID-19) pandemic have changed over time. On May 7, 2020, CDC, with other federal agencies, began identifying counties with increasing COVID-19 incidence (hotspots) to better understand transmission dynamics and offer targeted support to health departments in affected communities. Data for January 22-July 15, 2020, were analyzed retrospectively (January 22-May 6) and prospectively (May 7-July 15) to detect hotspot counties. No counties met hotspot criteria during January 22-March 7, 2020. During March 8-July 15, 2020, 818 counties met hotspot criteria for ≥1 day; these counties included 80% of the U.S. population. The daily number of counties meeting hotspot criteria peaked in early April, decreased and stabilized during mid-April-early June, then increased again during late June-early July. The percentage of counties in the South and West Census regions* meeting hotspot criteria increased from 10% and 13%, respectively, during March-April to 28% and 22%, respectively, during June-July. Identification of community transmission as a contributing factor increased over time, whereas identification of outbreaks in long-term care facilities, food processing facilities, correctional facilities, or other workplaces as contributing factors decreased. Identification of hotspot counties and understanding how they change over time can help prioritize and target implementation of U.S. public health response activities.

Sub-Saharan Africa harbours the majority of the burden of Lassa fever. Clinical diseases, as well as high seroprevalence, have been documented in Nigeria, Sierra Leone, Liberia, Guinea, Ivory Coast, Ghana, Senegal, Upper Volta, Gambia, and Mali. Deaths from Lassa fever occur all year round but naturally peak during the dry season. Annually, the number of people infected is estimated at 100,000 to 300,000, with approximately 5,000 deaths. There have been some work done on the dynamics of Lassa fever disease transmission, but to the best of our knowledge, none has been able to capture the seasonal variation of Mastomys rodent population and its impact on the transmission dynamics. In this work, a periodically forced seasonal nonautonomous system of a nonlinear ordinary differential equation is developed that captures the dynamics of Lassa fever transmission and seasonal variation in the birth of Mastomys rodents where time was measured in days to capture seasonality. It was shown that the model is epidemiologically meaningful and mathematically well posed by using the results from the qualitative properties of the solution of the model. A time-dependent basic reproduction number RLt is obtained such that its yearly average is written as R˜L<1, when the disease does not invade the population (means that the number of infected humans always decreases in the seasons of transmission), and R˜L>1, when the disease remains constantly and is invading the population, and it was detected that R˜L≠RL. We also performed some evaluation of the Lassa fever disease intervention strategies using the elasticity of the equilibrial prevalence in order to predict the optimal intervention strategies that can be useful in guiding the local national control program on Lassa fever disease to make a proper decision on the intervention packages. Numerical simulations were carried out to illustrate the analytical results, and we found that the numerical simulations of the model showed that possible combined intervention strategies would reduce the spread of the disease. It was established that, to eliminate Lassa fever disease, treatments with ribavirin must be provided early to reduce mortality and other preventive measures like an educational campaign, community hygiene, isolation of infected humans, and culling/destruction of rodents must be applied to also reduce the morbidity of the disease. Finally, the obtained results gave a primary framework for planning and designing cost-effective strategies for good interventions in eliminating Lassa fever.

As insect populations decline, due to climate change and other environmental disruptions, there has been an increased interest in understanding extinction probabilities. Generally, the life cycle of insects occurs in well-defined stages: when counting insects, questions naturally arise about which life stage to count. Using tsetse flies (vectors of trypanosomiasis) as a case study, we develop a model that works when different life stages are counted. Previous branching process models for tsetse populations only explicitly represent newly emerged adult female tsetse and use that subpopulation to keep track of population growth/decline. Here, we directly model other life stages. We analyse reproduction numbers and extinction probabilities and show that several previous models used for estimating extinction probabilities for tsetse populations are special cases of the current model. We confirm that the reproduction number is the same regardless of which life stage is counted, and show how the extinction probability depends on which life stage we start from. We demonstrate, and provide a biological explanation for, a simple relationship between extinction probabilities for the different life stages, based on the probability of recruitment between stages. These results offer insights into insect population dynamics and provide tools that will help with more detailed models of tsetse populations. Population dynamics studies of insects should be clear about life stages and counting points.

BackgroundA relatively simple life history allows us to derive an expression for the extinction probability of populations of tsetse, vectors of African sleeping sickness. We present the uncertainty and sensitivity analysis of extinction probability for tsetse population, to offer key insights into parameters in the control/eradication of tsetse populations.MethodsWe represent tsetse population growth as a branching process, and derive closed form estimates of population extinction from that model. Statistical and mathematical techniques are used to analyse the uncertainties in estimating extinction probability, and the sensitivity of the extinction probability to changes in input parameters representing the natural life history and vital dynamics of tsetse populations.ResultsFor fixed values of input parameters, the sensitivity of extinction probability depends on the baseline parameter values. For example, extinction probability is more sensitive to the probability that a female is inseminated by a fertile male when daily pupal mortality is low, whereas the extinction probability is more sensitive to daily mortality rate for adult females when daily pupal mortality, and extinction probabilities, are high. Global uncertainty and sensitivity analysis showed that daily mortality rate for adult females has the highest impact on the extinction probability.ConclusionsThe strong correlation between extinction probability and daily female adult mortality gives a strong argument that control techniques to increase daily female adult mortality may be the single most effective means of ensuring eradication of tsetse population.Author summaryTsetse flies (Glossina spp) are vectors of Trypanosomiasis, a deadly disease commonly called sleeping sickness in humans and nagana in livestock. The relatively simple life history of tsetse enabled us to model its population growth as a stochastic branching process. We derived a closed-form expression for the probability that a population of tsetse goes extinct, as a function of death, birth, development and insemination rates in female tsetse. We analyzed the sensitivity of the extinction probability to the different input parameters, in a bid to identify parameters with the highest impact on extinction probability. This information can, potentially, inform policy direction for tsetse control/elimination. In all the scenarios we considered, the daily mortality rate for adult females has the greatest impact on the magnitude of extinction probability. Our findings suggest that the mortality rate in the adult females is the weakest link in tsetse life history, and this fact should be exploited in achieving tsetse population control, or even elimination.

Background A relatively simple life history allows us to derive an expression for the extinction probability of populations of tsetse, vectors of African sleeping sickness. We present the uncertainty and sensitivity analysis of the extinction probability, to offer key insights into factors affecting the control or eradication of tsetse populations. Methods We represent tsetse population growth as a branching process, and derive closed form estimates of population extinction from that model. Statistical and mathematical techniques are used to analyse the uncertainties in estimating extinction probability, and the sensitivity of the extinction probability to changes in input parameters representing the natural life history and vital dynamics of tsetse populations. Results For fixed values of input parameters, the sensitivity of extinction probability depends on the baseline parameter values. Extinction probability is most sensitive to the probability that a female is inseminated by a fertile male when daily pupal mortality is low, whereas the extinction probability is most sensitive to daily mortality rate for adult females when daily pupal mortality, and extinction probabilities, are high. Global uncertainty and sensitivity analysis show that daily mortality rate for adult females has the highest impact on the extinction probability. Conclusions The high correlation between extinction probability and daily female adult mortality gives a strong argument that control techniques which increase daily female adult mortality may be the single most effective means of ensuring eradication of tsetse population., Author summary Tsetse flies (Glossina spp.) are vectors of African trypanosomiasis, a deadly disease commonly called sleeping sickness in humans and nagana in livestock. The relatively simple life history of tsetse enabled us to model its population growth as a stochastic branching process. We derived a closed-form expression for the probability that a population of tsetse goes extinct, as a function of death, birth, development and insemination rates in female tsetse. We analyzed the sensitivity of the extinction probability to the different input parameters, in a bid to identify parameters with the highest impact on extinction probability. This information can, potentially, inform policy direction for tsetse control/elimination. In all the scenarios we considered for the global sensitivity analysis, the daily mortality rate for adult females had the greatest impact on the magnitude of extinction probability. Our findings suggest that the mortality rate in the adult females is the weakest link in tsetse life history, and this fact should be exploited in achieving tsetse population control, or even eradication.

Increases in temperature over recent decades have led to a significant reduction in the populations of tsetse flies (Glossinaspp) in parts of the Zambezi Valley of Zimbabwe. If this is true for other parts of Africa, populations of tsetse may actually be going extinct in some parts of the continent. Extinction probabilities for tsetse populations have not so far been estimated as a function of temperature. We develop a time-homogeneous branching process model for situations where tsetse flies live at different levels of fixed temperatures. We derive a probability distributionpk(T) for the number of female offspring an adult female tsetse is expected to produce in her lifetime, as a function of the fixed temperature at which she is living. We show thatpk(T) can be expressed as a geometric series: its generating function is therefore a fractional linear type. We obtain expressions for the extinction probability, expected number of female offspring per female tsetse, and time to extinction. No tsetse population can escape extinction if subjected, for extended periods, to temperatures outside the range 16 °C - 32°C. Extinction probability increases more rapidly as temperatures approach and exceed the upper and lower limits. If the number of females is large enough, the population can still survive even at high temperatures (28°C - 31°C). Small decreases or increases in constant temperature in the neighbourhoods of 16°C and 31°C, respectively, can drive tsetse populations to extinction. Further study is needed to estimate extinction probabilities for tsetse populations in field situations where temperatures vary continuously.Author summaryTsetse flies (Glossinaspp) are the vectors of the African sleeping sickness. We derived an expression for the extinction probability, and mean time to extinction, of closed populations of the flies experiencing different levels of fixed temperatures. Temperatures play a key role in tsetse population dynamics: no tsetse populations can escape extinction at constant temperatures < 16°C > 32°C. The effect of temperature is more severe if tsetse populations are already depleted. Increasingly high temperatures due to climate change may alter the distribution of tsetse populations in Africa. The continent may witness local extinctions of tsetse populations in some places, and appearances in places hitherto too cold for them.

Significant reductions in populations of tsetse (Glossina spp) in parts of Zimbabwe have been attributed to increases in temperature over recent decades. Sustained increases in temperature might lead to local extinctions of tsetse populations. Extinction probabilities for tsetse populations have not so far been estimated as a function of temperature. We develop a time-homogeneous branching process model for situations where tsetse live at different levels of fixed temperature. We derive a probability distribution pk(T) for the number of female offspring an adult female tsetse is expected to produce in her lifetime, as a function of the fixed temperature at which she is living. We show that pk(T) can be expressed as a geometric series: its generating function is therefore a fractional linear type. We obtain expressions for the extinction probability, reproduction number, time to extinction and growth rates. The results are valid for all tsetse, but detailed effects of temperature will vary between species. No G. m. morsitans population can escape extinction if subjected, for extended periods, to temperatures outside the range 16°C–32°C. Extinction probability increases more rapidly as temperatures approach and exceed the upper and lower limits. If the number of females is large enough, the population can still survive even at high temperatures (28°C–31°C). Small decreases or increases in constant temperature in the neighbourhoods of 16°C and 31°C, respectively, can drive tsetse populations to extinction. Further study is needed to estimate extinction probabilities for tsetse populations in field situations where temperatures vary continuously., Author summary Tsetse flies (Glossina spp) are the vectors of the African sleeping sickness. We derived expressions for the extinction probability, and mean time to extinction, of closed populations of G. m. morsitans experiencing different levels of fixed temperature. Temperature plays a key role in tsetse population dynamics: no population of G. m. morsitans can escape extinction at constant temperatures < 16°C or > 32°C. The effect of temperature is more severe if tsetse populations are already depleted. Increasingly high temperatures due to climate change may alter the distribution of tsetse populations in Africa. The continent may witness local extinctions of tsetse populations in some places, and appearances in places hitherto too cold for tsetse.

A published study used a stochastic branching process to derive equations for the mean and variance of the probability of, and time to, extinction in population of tsetse flies (Glossina spp) as a function of adult and pupal mortality, and the probabilities that a female is inseminated by a fertile male. The original derivation was partially heuristic and provided no proofs for inductive results. We provide these proofs, together with a more compact way of reaching the same results. We also show that, while the published equations hold good for the case where tsetse produce male and female offspring in equal proportion, a different solution is required for the more general case where the probability (β) that an offspring is female lies anywhere in the interval (0, 1). We confirm previous results obtained for the special case where β = 0.5 and show that extinction probability is at a minimum for β > 0.5 by an amount that increases with increasing adult female mortality. Sensitivity analysis showed that the extinction probability was affected most by changes in adult female mortality, followed by the rate of production of pupae. Because females only produce a single offspring approximately every 10 days, imposing a death rate of greater then about 3.5% per day will ensure the eradication of any tsetse population. These mortality levels can be achieved for some species using insecticide-treated targets or cattle – providing thereby a simple, effective and cost-effective method of controlling and eradicating tsetse, and also human and animal trypanosomiasis. Our results are of further interest in the modern situation where increases in temperature are seeing the real possibility that tsetse will go extinct in some areas, without the need for intervention, but have an increased chance of surviving in other areas where they were previously unsustainable due to low temperatures.Author summaryWe derive equations for the mean and variance of the probability of, and time to, extinction in population of tsetse flies (Glossina spp), the vectors of trypanosomiasis in sub-Saharan Africa. In so doing we provide the complete proofs for all results, which were not provided in a previously published study. We also generalise the derivation to allow the probability that an offspring is female to lie anywhere in the interval (0, 1). The probability of extinction was most sensitive to changes in adult female mortality. The unusual tsetse life cycle, with very low reproductive rates means that populations can be eradicated as long as adult female mortality is raised to levels greater than about 3.5% per day. Simple bait methods of tsetse control, such as insecticide-treated targets and cattle, can therefore provide simple, affordable and effective means of eradicating tsetse populations. The results are of further interest in the modern situation where increases in temperature are seeing the real possibility that tsetse will go extinct in some areas, but have an increased chance of surviving in others where they were previously unsustainable due to low temperatures.