Modeling the role of information and limited optimal treatment on disease prevalence

Highlights

• A model for individuals’ behavioural response to disease prevalence is proposed.

• Lyapunov stability is performed and then an optimal control problem is proposed.

• Cost for information is accounted in cost functional using order four nonlinearity.

• Use of information and treatment is found to be better than when they are applied alone.

• Information plays important role in reduction of disease burden and economic load, with or without treatment.

Abstract

Disease outbreaks induce behavioural changes in healthy individuals to avoid contracting infection. We first propose a compartmental model which accounts for the effect of individual's behavioural response due to information of the disease prevalence. It is assumed that the information is growing as a function of infective population density that saturates at higher density of infective population and depends on active educational and social programmes. Model analysis has been performed and the global stability of equilibrium points is established. Further, choosing the treatment (a pharmaceutical intervention) and the effect of information (a non-pharmaceutical intervention) as controls, an optimal control problem is formulated to minimize the cost and disease fatality. In the cost functional, the nonlinear effect of controls is accounted. Analytical characterization of optimal control paths is done with the help of Pontryagin's Maximum Principle. Numerical findings suggest that if only control via information is used, it is effective and economical for early phase of disease spread whereas treatment works well for long term control except for initial phase. Furthermore, we observe that the effect of information induced behavioural response plays a crucial role in the absence of pharmaceutical control. Moreover, comprehensive use of both the control interventions is more effective than any single applied control policy and it reduces the number of infective individuals and minimizes the economic cost generated from disease burden and applied controls. Thus, the combined effect of both the control policies is found more economical during the entire epidemic period whereas the implementation of a single policy is not found economically viable.

Introduction

Infectious diseases have potentially serious consequences on population or community and also may drastically affect their overall social well-being, economy, health and other related developments. In addition, due to high morbidity and mortality, the prevention and control of disease transmission is of utmost importance. In recent years, infectious diseases such as influenza, flu and SARS have posed major challenges across the world. They not only increased the disease burden but also added a huge economic burden on society and thus have been affecting development plans of many countries (CBC News, 2003, Global Status Report on Communicable Diseases or Infectious Diseases,, The Economic and Social Impact of Emerging Infectious Disease,, Gupta et al., 2005, Russell, 2004). For example, approximately 1.4 million deaths are caused by various types of hepatitides (Hepatitis A–E) every year (http://www.who.int/campaigns/hepatitis_day/2014/en/). In case of SARS outbreak in 2003, the total estimated economic loss due to productivity loss, health care expenditures, employment and travel, etc. was approximately 50 billion dollars across the world (CBC News, 2003, The Economic and Social Impact of Emerging Infectious Disease,).

Due to reemerging and complex nature of diseases, detection and prevention are challenging tasks for policy makers worldwide even though medical facilities have significantly improved nowadays. There are various tools and techniques available which predict the dynamics of the disease transmission and also suggest suitable control interventions. Among them, mathematical modeling has been found to be a very successful tool and has been extensively used (Brauer and Castillo-Chávez, 2012, Gaff and Schaefer, 2009, Hethcote, 2000, Lee et al., 2010, Lenhart and Workman, 2007). Some of the pharmaceutical and non-pharmaceutical control interventions, viz. isolation, vaccination, quarantine and treatment, etc. have been studied in Gaff and Schaefer (2009), Joshi et al. (2006), Jung et al. (2009), Lee et al. (2010), Lenhart and Workman (2007), and Zaman et al. (2008).

In addition, the individual's behavioural change due to the information of disease has also been observed during the disease outbreak. This information induced behavioural change is found to be useful as it reduces the force of infection (Ahituv et al., 1996, Castilho, 2006, Joshi et al., 2008, Liu and Cui, 2008, Misra et al., 2011a, Philipson, 1996). The information can be spread via educational campaigns, media or by population itself through social activities and hence it can be used as an intervention. Indeed, people's response to the threat of disease often depends on their perception of risk, which could be influenced by the public and the private information disseminated widely by the media (Tchuenche et al., 2011). Therefore, there is growing interest among researchers to study the impact of these behaviour influencing factors on the spread of infectious diseases (Castilho, 2006, Collinson and Heffernan, 2014, Cui et al., 2008a, Funk et al., 2010, Funk et al., 2009, Joshi et al., 2008, Kiss et al., 2010, Liu et al., 2007, Liu and Cui, 2008, Margevicius and Joshi, 2013, Misra et al., 2011a, Wang et al., 2013, Xiao et al., 2013). To incorporate the behavioural change in a model due to awareness of disease, mainly two approaches have been used in literature. In one approach, a correction in incidence rate is considered to account for reduction in the contact rate due to awareness of the disease (Cui et al., 2008a, Cui et al., 2008b, d'Onofrio and Manfredi, 2009, Kumar and Srivastava, 2014, Collinson and Heffernan, 2014, Li et al., 2008, Liu et al., 2007, Liu and Cui, 2008, Tchuenche and Bauch, 2012, Tchuenche et al., 2011, Wang and Xiao, 2014) and in the other approach a subclass of individuals with awareness was considered (Funk et al., 2009, Funk et al., 2010, Joshi et al., 2008, Kassa and Ouhinou, 2015, Kiss et al., 2010, Margevicius and Joshi, 2013, Misra et al., 2011a, Misra et al., 2011b, Samanta et al., 2013). Apart from this, some studies have incorporated the impact of behavioural response on the control interventions such as vaccination (see Bauch et al., 2003, Manfredi and D'Onofrio, 2013, Reluga et al., 2006, Wells and Bauch, 2012 and references there in). The effect of this kind of human behavioural response that also affects the disease progression has been explained and discussed in the book of Manfredi and D'Onofrio (2013). Part II of this book has been devoted to study the modeling of the human behavioural response on the control the epidemic threats. Recently, Greenhalgh et al. (2015) presented a brief and nice commentary on literature related to awareness and their effects on the dynamics of diseases. A brief development of some of these studies relevant to our work is discussed in the following.

A wide range of tools, including mass media, have been used in an effort to control and eliminate epidemic diseases (Tchuenche and Bauch, 2012). The effect of media coverage on disease transmission is accounted by considering a correction in the incidence rate with a saturated function of infective individuals in Cui et al. (2008b) and Liu and Cui (2008). Authors found that only media effect cannot eradicate disease completely, since there is no effect of awareness on the basic reproduction number, but it can reduce the disease prevalence. Some authors used a decaying exponential function of infection spreading populations in the force of infection (Cui et al., 2008a, Liu et al., 2007, Xiao et al., 2015). Xiao et al. (2015) found that the media impact delays epidemic peak and reduces the severity of outbreak. Their findings also suggest that the media impact is not always effective to reduce disease transmission rather a switching pattern for media was found in early stage of the outbreaks.

Joshi et al. (2008) divided the susceptible population into three subclasses based on the information level generated from social and educational campaigns in an SIR model for the HIV epidemic of Uganda. They compared their findings with the data of HIV/AIDS epidemic of Uganda and found that the information reduces the infection level. Misra et al., 2011a, Misra et al., 2011b proposed models in which susceptible individuals become aware due to awareness spread via media by assuming awareness as a compartment. In Misra et al. (2011b), the interaction of media with susceptible individuals is taken as a saturated function of awareness. These individuals then move to aware class. They observed that media or awareness helps in controlling the disease burden but is not able to eradicate disease. Buonomo et al., 2008, Buonomo et al., 2010, Buonomo et al., 2012, d'Onofrio and Manfredi (2009), and d'Onofrio et al., 2007a, d'Onofrio et al., 2007b, d'Onofrio et al., 2008 considered the effect of information-related change in the contact pattern as well as on vaccination process. d'Onofrio et al. (2007b) proposed a model that accounts for the impact of information on vaccination process for the new born. They observed that delayed information and information based vaccination coverage cause the oscillations when endemic equilibrium changes its stability. They numerically validated the obtained results considering various forms of the information based vaccination coverage. In further related works (Buonomo et al., 2008, Buonomo et al., 2012, d'Onofrio et al., 2007a), the authors established global stability and bifurcation for SIR type models with information and vaccination. Buonomo et al. (2010) and d'Onofrio et al. (2008) showed how the effect of information recovers the rational exemption to vaccination for fatal and non-fatal diseases globally. Buonomo et al. (2013) considered the effect of information based vaccination at the time of new birth in an SEIR model. They established that this information induced vaccination may trigger oscillations which is different from the SIR case where these oscillations occur via information delay.

Some other modeling approaches such as game theory, network modeling etc. have also been used to address the impact of information and human behavioural response on disease. Bauch et al. (2003) proposed and studied an epidemic model for the dynamics of smallpox along with vaccination policies with help of game theoretic approach. Their study accentuated that group level interest for vaccination is more applicable than individual's one. Reluga et al. (2006) found that individual level interest for vaccination may destabilize the system and lead to the oscillatory behaviour whereas vaccination uptake at community or group level may help in eradication of disease. Wells and Bauch (2012) studied the dynamics of seasonal influenza coupled with individual vaccinating decision using the network modeling (They found that individual's long term memory for past events (like perceived vaccine failures and individual's personal experience, etc.) may have a stable impact on vaccination coverage.

From above, we note that most of the articles have incorporated the effect of information either in the disease transmission term or have considered a new class of aware individuals. We quantify the impact of the information of disease in terms of ‘behavioural response of susceptible individuals’. As the susceptible individuals get information of disease they tend to adopt protective measures such as using face masks, sanitizer, vaccination, isolation, etc. Hence these susceptible individuals, by changing their behaviour and protecting themselves from infection, no longer participate in disease transmission dynamics and move to removed class. As time passes, these individuals lose their protectiveness and again move to susceptible class and this cycle goes on.

Diseases pose huge economic burden which primarily includes opportunity loss, health care related expenditures, losses due to mortality and dependent's care, and loss of employment, etc. Also, the costs for implementing control interventions such as treatment, vaccination, etc. are involved. Thus for policy makers, it is not only important to control the spread of disease but also to minimize the overall cost incurred during a specified time period. A suitable choice for a control policy to use a single control intervention or multiple control interventions depends on availability of resources. For multiple control interventions, it may be important to understand that in what ratio and for how long such controls should be applied. To address these issues, optimal control theory is used. In last two decades, many optimal control problems in epidemiology have been studied for various control interventions considering both pharmaceutical and non-pharmaceutical interventions (Behncke, 2000, Castilho, 2006, Gaff and Schaefer, 2009, Goldman and Lightwood, 2002, Joshi et al., 2015, Kassa and Ouhinou, 2015, Lenhart and Workman, 2007, Zeiler et al., 2010, Zaman et al., 2008).

Behncke (2000) studied the effect of pharmaceutical and health-promotional campaigns as control measure with the importance of financial support for SIR models. Qualitative analysis showed that the control interventions suppress the disease level, whereas financial support promotes promotional campaigns and these campaigns help in lowering the disease transmission during the course of epidemic. Goldman and Lightwood (2002) assessed the impact of economy on medical treatment with the help of SIS model. They found that treatment plays an important role in reducing of disease prevalence. Also, they noted that treatment can eradicate disease completely under certain conditions. Castilho (2006) focused on control of epidemic with the help of educational campaigns. Keeping in mind the limitation of budget and time, they proposed two strategies: first – reduction in the disease transmission through educational campaigns and second – increase in the removal rate of infective population via removal campaigns. In order to achieve a predefined goal, authors observed that both the control policies have to be executed simultaneously for the entire period of epidemic which not only suppress the disease burden but also minimize total incurred cost.

Gaff et al. (2011) studied different combinations of control interventions – vaccination and treatment for a set of SIR, SIRS and SEIR models. They observed that if the vaccination is available then it is a suitable intervention but with high expense, whereas combination of both vaccination and treatment will be more effective with relatively cheaper cost. Lee et al. (2010) investigated the role of limited antiviral treatment and isolation for pandemic influenza using them as optimal controls. Here, authors proposed five control strategies using various combinations of treatment and isolation during the course of epidemic. Their numerical findings suggested that the combined impact of antiviral treatment and isolation is much more than the single one and in this case the number of clinically and hospitalized infective individuals remains relatively low.

Effect of information or education has been used as control in the dynamics of HIV in Joshi et al. (2015) and Kassa and Ouhinou (2015) also. Kassa and Ouhinou (2015) studied a control problem for HIV in Botswana taking effect of education, treatment and other protective measures as controls. Numerically they established that the combination of these controls minimizes the economic load and also reduces the HIV burden. Recently Joshi et al. (2015) studied an optimal control model and considered level of information as only control intervention. Their numerical results suggested that in the presence of optimal information the disease transmission is lower and the number of infective individuals is minimized.

As there is no study of combination of both the treatment and the effect of information as control interventions, we further modify our model as optimal control problem by considering the treatment and the effect of information as controls. We also assume that the availability of treatment is limited. Overview of the rest of the paper is as follows: in the following section, we formulate a mathematical model which accounts for the information induced behavioural response. In Section 3, we perform equilibrium analysis and establish global stability of the equilibrium points. Further, we formulate corresponding optimal control problem in Section 4. In the subsequent sections we perform numerical experimentations for model system and discuss the outcomes.