Mathematical model of rabies vaccination in the United States.

Rabies is one of the oldest viral diseases and it has been present on every continent except Antarctica. Within the U.S. human rabies cases are quite rare. In the eastern USA, raccoons are the main reservoir hosts and pet vaccination serves as an important barrier against human rabies exposure. In this paper, we develop a compartmental model for rabies transmission amongst raccoons and domestic pets. We find the disease-free equilibria, reproduction numbers for the raccoons and domestic pets. We also determine the vaccination coverage/rates, both for raccoons and pets, needed to achieve the elimination of rabies.

A dynamic game of lymphatic filariasis prevention by voluntary use of insecticide treated nets.

Lymphatic filariasis (LF) has been targeted for elimination as a public health concern by 2030 with a goal to keep the prevalence of LF infections under the 1% threshold. While mass drug administration (MDA) is a primary strategy recommended by WHO, the use of insecticide treated nets (ITN) plays a crucial role as an alternative strategy when MDA cannot be used. In this paper, we use imitation dynamics to incorporate human behavior and voluntary use of ITN into the compartmental epidemiological model of LF transmission. We find the equilibrium states of the dynamics and the ITN usage as it depends on epidemiological parameters and the cost of ITNs. We investigate the conditions under which the voluntary use of ITNs can keep the LF prevalence under the 1% threshold. We found that when the cost of using the ITNs is about 105 smaller than the perceived cost of LF, then the voluntary use of ITNs will eliminate LF as a public health concern. Furthermore, when the ITNs are given away for free, our model predicts that over 80% of the population will use them which would eliminate LF completely in regions where Anopheles are the primary vectors.

A mathematical model of invasion and control of coconut rhinoceros beetle Oryctes rhinoceros (L.) in Guam.

The coconut rhinoceros beetle (CRB), is one of the most damaging pests to coconut palms causing severe economic harm. Its expansion from Asia to the Pacific in the early 20th century has been stopped by virus control. However, a new haplotype CRB-Guam has recently escaped this control and invaded Guam, other Pacific islands, and has even established itself in the Western Hemisphere. In this paper, we present a compartmental ODE model of CRB population and control. We carefully consider CRB life stages and its interplay with coconut palms as well as "the green waste", the organic matters used by CRB for breeding sites. We calibrate and validate the model based on data count of CRBs trapped in Guam between 2008 and 2014. We derive the basic reproduction number determining the CRB population growth without any control measures. We also identify control levels required to eliminate CRBs. We show that, in the absence of viable virus control, the sanitation, i.e., the removal of the green waste is the most efficient way to control the population. Our model predicts that the sanitation efforts need to roughly double from the current levels to eliminate CRB from Guam. Furthermore, we demonstrate that a rare event like Typhoon Dolphin that hit Guam in 2015 can lead to a quick rise of the CRB population.

Quasi-neutral evolution in populations under small demographic fluctuations.

We study an eco-evolutionary dynamics in finite populations of two haploid asexually reproducing allelic types. We focus on the quasi-neutral case when individual types differ only in their intrinsic birth and death rates but have the same expected lifetime reproductive output. We assume that the population size can fluctuate stochastically. We solve the Kolmogorov forward equation in the population whose size fluctuates only minimally and show that the fixation probability is decreasing with the increasing turnover rate. We also show that when the mutant's turnover is small enough, selection favors the mutant replacing residents. Similarly, when the turnover is high enough, selection opposes the replacement. This basic result has previously been demonstrated numerically for the contact process and shown analytically for the Moran process; the current paper extends this analysis to provide an analytical proof for the contact process. We also demonstrate numerically that our results extend for general fluctuating populations and beyond the quasi-neutral case.

Estimation of standard deviations and inverse-variance weights from an observed range.

A variety of methods have been proposed to estimate a standard deviation, when only a sample range has been observed or reported. This problem occurs in the interpretation of individual clinical studies that are incompletely reported, and also in their incorporation into meta-analyses. The methods differ with respect to their focus being either on the standard deviation in the underlying population or on the particular sample in hand, a distinction that has not been widely recognized. In this article, we contrast and compare various estimators of these two quantities with respect to bias and mean squared error, for normally distributed data. We show that unbiased estimators are available for either quantity, and recommend our preferred methods. We also propose a Taylor series method to obtain inverse-variance weights, for samples where only the sample range is available; this method yields very little bias, even for quite small samples. In contrast, the naïve approach of simply taking the inverse of an estimated variance is shown to be substantially biased, and can place unduly large weight on small samples, such as small clinical trials in a meta-analysis. Accordingly, this naïve (but commonly used) method is not recommended.

A game-theoretic model of rabies in domestic dogs with multiple voluntary preventive measures.

Game theory is now routinely applied to quantitatively model the decision making of individuals presented with various voluntary actions that can prevent a given disease. Most models consider only a single preventive strategy and the case where multiple preventative actions are available is severely understudied. In our paper, we consider a very simple SI compartmental model of rabies in the domestic dog population. We study two choices of the dog owners: to vaccinate their dogs or to restrict the movements of unvaccinated dogs. We analyze the relatively rich patterns of Nash equilibria (NE). We show that there is always at least one NE at which the owners utilize only one form of prevention. However, there can be up to three different NEs at the same time: two NEs at which the owners use exclusively only the vaccination or movement restriction, and the third NE when the owners use both forms of prevention simultaneously. However, we also show that, unlike the first two types of NEs, the third kind of NE is not convergent stable.

A mathematical model of Guinea worm disease in Chad with fish as intermediate transport hosts.

Guinea-worm disease (GWD) was thought to be almost eliminated in Chad when it reemerged in 2010. The disease now shows a peculiar pattern of spreading along Chari River and its tributaries, rather than clustering around a particular drinking water source. We create a mathematical model of GWD that includes the population dynamics of the parasite as well as the dynamics of its hosts (copepods, fish, humans, and domestic dogs). We calibrate our model based on data from the literature and validate it on the recent GWD annual incidence data from Chad. The effective reproduction number predicted by our model agrees well with the empirical value of roughly 1.25 derived directly from the data. Our model thus supports the hypothesis that the parasite now uses fish as intermediate transport hosts. We predict that GWD transmission can be most easily interrupted by avoiding eating uncooked fish and by burying the fish entrails to prevent transmission through dogs. Increasing the mortality of copepods and even partially containing infected dogs to limit their access to water sources is another important factor for GWD eradication.

Kleptoparasitic interactions modeling varying owner and intruder hunger awareness.

We consider a game theoretical model of kleptoparasitic interaction between two individuals, the Owner and the Intruder. The Owner is in possession of a resource and must decide whether to defend the resource against the Intruder or flee. If the Owner defends, the Intruder must decide whether to fight with the Owner or flee. The outcome of the fight depends on the hunger of the individuals, the hungrier the individual is, the more likely they are to win the fight. We consider three scenarios: (a) both individuals know their own and their opponent's hunger, (b) individuals only know their own hunger but not that of their opponent, and (c) individuals do not know their own nor the opponent's hunger levels. We determine Nash equilibrium strategies in each scenario. We conclude that Owner is generally willing to defend more often than the Intruder is willing to attack. Also, the Intruder's payoff is largest in the full information case; but the Owner may benefit in the no information or partial information cases when the cost of the fight is neither too large nor too small.

A game-theoretical analysis of poliomyelitis vaccination.

Poliomyelitis is a worldwide disease that has nearly been eradicated thanks to the Global Polio Eradication Initiative. Nevertheless, the disease is currently still endemic in three countries. In this paper, we incorporate the vaccination in a two age-class model of polio dynamics. Our main objective is to see whether mandatory vaccination policy is needed or if polio could be almost eradicated by a voluntary vaccination. We perform game theoretical analysis and compare the herd immunity vaccination levels with the Nash equilibrium vaccination levels. We show that the gap between two vaccination levels is too large. We conclude that the mandatory vaccination policy is therefore needed to achieve a complete eradication.

Estimating the sample variance from the sample size and range.

For meta-analysis studies and systematic reviews, it is important to pool the data from a set of similar clinical trials. To pool the data, one needs to know their SD. Many trial reports, however, contain only the median, the minimum and maximum values, and the sample size. It is therefore important to be able to estimate the SD S from the sample size n and range r. For small n ≤ 100, we improve existing estimators of r/S, the "divisor," denoted by ξ(n) . This in turn yields improved estimators of the SD in the form S^=r/ξ^(n) on simulated as well as real datasets. We provide numerical values of the proposed estimator as well as approximation by a simple formula 3ln(n)-1.4025 . Furthermore, for large n, we provide estimators ξ^(n) of the divisor ξ(n) for the normal, exponential, and other bounded and unbounded distributions.

Game-Theoretical Model of Retroactive Hepatitis B Vaccination in China.

Hepatitis B (HepB) is one of the most common infectious diseases affecting over two billion people worldwide. About one third of all HepB cases are in China. In recent years, China made significant efforts to implement a nationwide HepB vaccination program and reduced the number of unvaccinated infants from 30 to 10%. However, many individuals still remain unprotected, particularly those born before 2003. Consequently, a catch-up retroactive vaccination is an important and potentially cost-effective way to reduce HepB prevalence. In this paper, we analyze a game theoretical model of HepB dynamics that incorporates government-provided vaccination at birth coupled with voluntary retroactive vaccinations. Given the uncertainty about the long-term efficacy of the HepB vaccinations, we study several scenarios. When the waning rate is relatively high, we show that this retroactive vaccination should be a necessary component of any HepB eradication effort. When the vaccine offers long-lasting protection, the voluntary retroactive vaccination brings the disease incidence to sufficiently low levels. Also, we find that the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in an individual's self-interest to vaccinate (and potentially re-vaccinate) at a rate just slightly above the vaccine waning rate.

A compartmental model for Schistosoma japonicum transmission dynamics in the Philippines.

Schistosomiasis is a chronic and debilitating neglected tropical disease (NTD), second only to malaria as one of the most devastating parasitic diseases. Caused by a parasitic flatworm of the genus Schistosoma, infection occurs when skin comes in contact with contaminated freshwater that contains schistosome-hosting snails. The disease continues to be endemic in many regions of the Philippines, where it poses a significant public health challenge due to a lack of healthcare resources. In the Philippines, additional mammalian reservoirs for the S. japonicum parasite, especially bovines such as carabaos, also facilitate the spread of schistosomiasis. We extend existing compartment models to include human, snail, bovine, and free-living Schistosoma for a comprehensive look at the transmission dynamics of the disease. Sensitivity analysis of model parameters shows that the carabaos themselves can sustain the endemicity of schistosomiasis. Thus, we consider the control method of farming mechanization to avoid contaminated freshwater sources. We find that a reduction of contaminated water contacts by at least 77% will break the transmission cycle and eliminate the disease. However, reducing the contact by about 70% will still result in decrease of human schistosomiasis prevalence to under 1% in 15 years or less. Achieving such high reduction of contact rates could be a daunting task, especially in rural areas. Still, the potential to eliminate or at least reduce the schistosomiasis prevalence should be considered an additional benefit of mechanization efforts in the Philippines.

Imperfect vaccine can yield multiple Nash equilibria in vaccination games.

As infectious diseases continue to threaten communities across the globe, people are faced with a choice to vaccinate, or not. Many factors influence this decision, such as the cost of the disease, the chance of contracting the disease, the population vaccination coverage, and the efficacy of the vaccine. While the vaccination games in which individuals decide whether to vaccinate or not based on their own interests are gaining in popularity in recent years, the vaccine imperfection has been an overlooked aspect so far. In this paper we investigate the effects of an imperfect vaccine on the outcomes of a vaccination game. We use a simple SIR compartmental model for the underlying model of disease transmission. We model the vaccine imperfection by adding vaccination at birth and maintain a possibility for the vaccinated individual to become infected. We derive explicit conditions for the existence of different Nash equilibria, the solutions of the vaccination game. The outcomes of the game depend on the complex interplay between disease transmission dynamics (the basic reproduction number), the relative cost of the infection, and the vaccine efficacy. We show that for diseases with relatively low basic reproduction numbers (smaller than about 2.62), there is a little difference between outcomes for perfect or imperfect vaccines and thus the simpler models assuming perfect vaccines are good enough. However, when the basic reproduction number is above 2.62, then, unlike in the case of a perfect vaccine, there can be multiple equilibria. Moreover, unless there is a mandatory vaccination policy in place that would push the vaccination coverage above the value of unstable Nash equilibrium, the population could eventually slip to the "do not vaccinate" state. Thus, for diseases that have relatively high basic reproduction numbers, the potential for the vaccine not being perfect should be explicitly considered in the models.

Modeling the "F" in "SAFE": The dynamic game of facial cleanliness in trachoma prevention.

Trachoma, a neglected tropical disease (NTDs) caused by bacterium Chlamydia trachomatis, is a leading cause of infectious blindness. Efforts are underway to eliminate trachoma as a public health problem by using the "SAFE" strategy. While mathematical models are now standard tools used to support elimination efforts and there are a variety of models studying different aspects of trachoma transmission dynamics, the "F" component of the strategy corresponding to facial cleanliness has received very little attention so far. In this paper, we incorporate human behavior into a standard epidemiological model and develop a dynamical game during which individuals practice facial cleanliness based on their epidemiological status and perceived benefits and costs. We found that the number of infectious individuals generally increases with the difficulty to access a water source. However, this increase happens only during three transition periods and the prevalence stays constant otherwise. Consequently, improving access to water can help eliminate trachoma, but the improvement needs to be significant enough to cross at least one of the three transition thresholds; otherwise the improved access will have no noticeable effect.

A game-theoretic model of lymphatic filariasis prevention.

Lymphatic filariasis (LF) is a mosquito-borne parasitic neglected tropical disease. In 2000, WHO launched the Global Programme to Eliminate Lymphatic Filariasis (GPELF) as a public health problem. In 2020, new goals for 2030 were set which includes a reduction to 0 of the total population requiring Mass Drug Administrations (MDA), a primary tool of GPELF. We develop a mathematical model to study what can happen at the end of MDA. We use a game-theoretic approach to assess the voluntary use of insect repellents in the prevention of the spread of LF through vector bites. Our results show that when individuals use what they perceive as optimal levels of protection, the LF incidence rates will become high. This is in striking difference to other vector-borne NTDs such as Chagas or zika. We conclude that the voluntary use of the protection alone will not be enough to keep LF eliminated as a public health problem and a more coordinated effort will be needed at the end of MDA.

Unified approach to optimal estimation of mean and standard deviation from sample summaries.

Recently, various methods have been developed to estimate the sample mean and standard deviation when only the sample size, and other selected sample summaries are reported. In this paper, we provide a unified approach to optimal estimation that can be easily adopted when only some summary statistics are reported. We show that the proposed estimators have the lowest variance among linear unbiased estimators. We also show that in the most commonly reported cases, that is, when only a three-number or five-number summary is reported, the newly proposed estimators match the previously developed estimators. Finally, we demonstrate the performance of the estimators numerically.

An ODE model of yaws elimination in Lihir Island, Papua New Guinea.

Yaws is a chronic infection that affects mainly the skin, bone and cartilage and spreads mostly between children. The new approval of a medication as treatment in 2012 has revived eradication efforts and now only few known localized foci of infection remain. The World Health Organization strategy mandates an initial round of total community treatment (TCT) with single-dose azithromycin followed either by further TCT or by total targeted treatment (TTT), an active case-finding and treatment of cases and their contacts. We develop the compartmental ODE model of yaws transmission and treatment for these scenarios. We solve for disease-free and endemic equilibria and also perform the stability analysis. We calibrate the model and validate its predictions on the data from Lihir Island in Papua New Guinea. We demonstrate that TTT strategy is efficient in preventing outbreaks but, due to the presence of asymptomatic latent cases, TTT will not eliminate yaws within a reasonable time frame. To achieve the 2030 eradication target, TCT should be applied instead.

Game-Theoretical Model of the Voluntary Use of Insect Repellents to Prevent Zika Fever.

Zika fever is an emerging mosquito-borne disease. While it often causes no or only mild symptoms that are similar to dengue fever, Zika virus can spread from a pregnant woman to her baby and cause severe birth defects. There is no specific treatment or vaccine, but the disease can be mitigated by using several control strategies, generally focusing on the reduction in mosquitoes or mosquito bites. In this paper, we model Zika virus transmission and incorporate a game-theoretical approach to study a repeated population game of DEET usage to prevent insect bites. We show that the optimal use effectively leads to disease elimination. This result is robust and not significantly dependent on the cost of the insect repellents.

Voluntary vaccination may not stop monkeypox outbreak: A game-theoretic model.

Monkeypox (MPX) is a viral zoonotic disease that was endemic to Central and West Africa. However, during the first half of 2022, MPX spread to almost 60 countries all over the world. Smallpox vaccines are about 85% effective in preventing MPX infections. Our objective is to determine whether the vaccines should be mandated or whether voluntary use of the vaccine could be enough to stop the MPX outbreak. We incorporate a standard SVEIR compartmental model of MPX transmission into a game-theoretical framework. We study a vaccination game in which individuals decide whether or not to vaccinate by assessing their benefits and costs. We solve the game for Nash equilibria, i.e., the vaccination rates the individuals would likely adopt without any outside intervention. We show that, without vaccination, MPX can become endemic in previously non-endemic regions, including the United States. We also show that to "not vaccinate" is often an optimal solution from the individual's perspective. Moreover, we demonstrate that, for some parameter values, there are multiple equilibria of the vaccination game, and they exhibit a backward bifurcation. Thus, without centrally mandated minimal vaccination rates, the population could easily revert to no vaccination scenario.

Mathematical modelling Treponema infection in free-ranging Olive baboons (Papio anubis) in Tanzania.

Yaws is a chronic infection caused by the bacterium Treponema pallidum susp. pertenue (TPE) that was thought to be an exclusive human pathogen but was recently found and confirmed in nonhuman primates. In this paper, we develop the first compartmental ODE model for TPE infection with treatment of wild olive baboons. We solve for disease-free and endemic equilibria and give conditions on local and global stability of the disease-free equilibrium. We calibrate the model based on the data from Lake Manyara National Park in Tanzania. We use the model to help the park managers devise an effective strategy for treatment. We show that an increasing treatment rate yields a decrease in disease prevalence. This indicates that TPE can be eliminated through intense management in closed population. Specifically, we show that if the whole population is treated at least once every 5-6 years, a disease-free equilibrium can be reached. Furthermore, we demonstrate that to see a substantial decrease of TPE infection to near-elimination levels within 15 years, the whole population needs to be treated every 2-3 years.