Infectious disease transmission has been widely discussed among nations over the last two decades. All countries across the globe are worried about this issue since the communicable diseases are borderless transmitted and there is a significant increase of the mortality rate caused by the escalating number of people who are suffered from particular transmissible diseases like HIV-AIDS and Tuberculosis. Epidemiologists have struggled for describing the transmission pattern of the infectious diseases and how to forecast the possible approaches that can be implemented in the future. To tackle this problematic situation, mathematical models have been developed as a viable remedy for describing the phenomena of how the infectious diseases are spread (Hethcote, 2000) and feasible predictions of further development of the transmissible disease phenomena (Koopman, 2002). According to Schichl (2010) mathematical modeling is a range of process to determine a mathematical formula which can describe the real world problems for providing knowledge, solutions, and suggestions that can be applied in further applicative fields. The models mathematically represent abstractions of natural phenomena that are difficult to be explained and understood (Tedeschi, 2006), like the communicable disease transmission. However, mathematical models have been criticized regarding their functionality in terms of providing abstractions of the spread of transmissible diseases, the accuracy of projections and conceptual frameworks for strategic-decision makers. This essay will discuss these three main issues to illustrate that mathematical models are incredibly useful tools for helping communicable disease prevention.

**The abstraction of reality**

In terms of infectious disease phenomena abstractions, Ferguson, Keeling, Edmunds, Gani, Grenfell, Anderson, and Leach (2003) believe that it is still questionable whether a complex model might provide correct assumptions of the real disease problems. These scholars think that this is because most complex models input lots of parameters to explain certain phenomena. It has also been claimed that the outputs of mathematical models never provide any exact estimations of reality (Keeling, 2005). Nevertheless, Hollingsworth (2009) asserts that mathematical models for epidemiology present several cognitive outcomes like the numbers of thresholds, primary reproduction, and substitution. He highlights that the more complicated an epidemiological model the more it can provide a better abstraction of the reality. As Hethcote (2000) and de Castro Medeiros, Castilho, Braga, de Souza, Regis, and Monteiro (2011) state, this is because the epidemiological models allow us to make a combination of complicated data. As a consequence, these can be used to make identification of how the communicable diseases can be transmitted (Hethcote, 2000). It has also been shown that mathematical models in epidemiology offer certain basic abstractions such as the period of generations, the levels of epidemic growth and the principal and effective numbers of prolificacy which take an essential part in describing the characteristics of communicable diseases (Hollingsworth, 2009). Therefore, it seems clear that complex mathematical models are beneficial tools in providing acceptable assumptions which reflect how the communicable diseases are transmitted.

Another issue is been raised by Klein, Laxminarayan, Smith, and Gilligan (2007), who argue that there are only fewer numbers of mathematical models that have explained clearly about the logic characters of individual cases; therefore, unfortunately, it has a negative impact on describing the clustering cases. Nonetheless, Punyacharoensin, Edmunds, De Angelis, and White (2011) state that there has been a significant increase in the numbers of mathematical models that are used in explaining the transmission of infectious diseases recently. Take mathematical models of the HIV transmission, for example, there is a considerable increase in the numbers of the models that have been used to explain the HIV transmission and its surveillances over the last 25-year period (Punyacharoensin et al., 2011). This is because those models link the reality with observations which can be beneficial in enriching the reliability of the real word problems (Byl, 2003; Hollingsworth, 2009; de Castro Medeiros et al., 2011). Research which has been carried out by de Castro Medeiros et al. (2011), for example, has contributed in providing a better explanation of how dengue fever is transmitted. Moreover, the communicable disease models can show an abstraction of the real phenomena which serve information about the hosts’ position and the possible combination which can be created between the history of endemic diseases and numeric illustrations about the transmission stages (Riley, 2007). Thus, despite what Klein et al. (2007) claim, it can clearly be seen that modeling mathematics for epidemiology has been used popularly in describing the transmission of infectious diseases recently.

**Prediction accuracy**

It has been argued that the precision of such predictions, which is provided by a mathematical model for explaining the transmission of viruses, is doubtful (Colizza, Barrat, Barthelemy, Valleron, & Vespignani, 2007). These scholars think that this is because the models include the numbers of irrelevant assumptions like the character of viruses. They also believe that there is no any consideration about how the viruses spread with varieties frequency in remote areas. However, Kitching, Hutber, and Thrusfield (2005) criticize this idea by asserting that in general, mathematical models in epidemiology can predict how long an epidemic can last, the range of its geographic and the prevalence numbers. They highlight this as the most ultimate reason why mathematical models of epidemiology are built and used. They also prove that in San Francisco, for instance, the resistant incidence and its prevalence have been predicted by epidemiological models. Hethcote (2000) emphasizes what Kitching et al. (2005) have indicated by stating that as beneficial tools during the experiment, mathematical models have power to test and build theories, solve distinctive problems, give quantitative assessments on hypothesis, determine sensitive alteration of values of estimators and conjecture important data estimators. Moreover, it has also been demonstrated that mathematical models for epidemiology offer probability to deepen analysis about the disease problems (de Castro Medeiros et al., 2011). Therefore, due to these reasons, it is totally true that modeling mathematics for epidemiology offers accurate predictions about the phenomena of infectious diseases.

Furthermore, Colizza, Barrat, Barthélemy, and Vespignani (2006) assume that how models can predict effectively is still a key problem related to epidemiological modeling of global phenomena. One probably reason is because mathematical models offer probabilistic calculations which mean the accuracy of such predictions are being decayed as the projections are made (Keeling, 2005). Nevertheless, Logan (2005) censures these scholars for this claim. He demonstrates that as what physical models can predict the mechanical structure, biological models also can forecast how living structures run. Epidemiological models can be used to forecast the general phenomena in the future and provide estimations of the projection uncertainty (Hethcote, 2000; Blower, Schwartz, & Mills, 2003). As what Chubb and Jacobsen (2010) explain, mathematical models help epidemiological research in making evaluations of the variables’ significance, predicting the possible outputs of a project and providing extra data clarification. Additionally, Ferguson et al. (2003) demonstrate that the power of forecasting the possible process that occur individually and interaction within the population is one of the undeniable advantages of the use of mathematical models. There is also an absolute possibility to predict the geographical area of disease infection by using a probabilistic model that has been created for describing how the diseases trans-missed globally (Hufnagel, Brockmann & Geisel, 2004). Although the effectiveness of the prediction provided by epidemiological models is still problematic but it seems incredibly true that without using the models we could not forecast the future possible trends of communicable disease transmission.

**Conceptual frameworks for the policy makers**

Concerning to public health policies, it has been claimed that even though epidemiological models have contributed to solve global issues related to infectious diseases, they still lack benefits for the formulation and surveillance programs (Woolhouse, 2003). Klein et al. (2007) believe that one reason is because the models generalize the infection pattern by providing a similar assumption of how diseases are infected todays with the previously. Nonetheless, Blower et al. (2003) condemn these scholars for claiming as so. They assert that modeling mathematics offer strong recommendations for the policy makers to formulate public health strategies. This is due to the models warning them about the effectiveness of insulations that might reduce the infection burdens and the efficiency of vaccines that should be delivered (Bozzette, Boer, Bhatnagar, Brower, Keeler, Morton, & Stoto, 2003). Moreover, Keeling (2005) has demonstrated that the epidemiological models provide a help to inform the policy makers about the best way to maximize the limitation of resources. The models also suggest what essential data which are needed in other surveys in the future (Hethcote, 2000). Additionally, it has been demonstrated that modeling mathematics is incredibly capable to provide a brief evaluation of the prevention policies for controlling the chronic diseases like cancer and diabetes (Dasbach, Elbasha, & Insinga, 2006). Thus, it is clear that in relation to the public health policies, modeling mathematics plays an important part in supplying information which is unexpectedly needed by the decision makers.

Additionally, Klein et al. (2007) say that there has been a failure in constructing policies to overcome the issue of infectious diseases transmission using mathematical models’ analysis. These scholars assume that there was no consideration in the analysis about the possible cost that might be needed in the future. They claim what happen in 1990 as an indication that mathematical models has failed to provide guidance for the policy makers to construct strategies for combating the HIV spread. However, it has been demonstrated that epidemiological models contribute as a useful tool for designing and planning further observations related to epidemiology (Keeling & Rohani, 2008; Bozzette et al., 2003; Punyacharoensin et al., 2011). As Hethcote (2000) states, epidemiological models are a useful tool for making comparisons, evaluation and optimization of detection diversities, surveillance and prevention programs, and recovery actions. For instance, Donaldson and Alexandersen (2002) have demonstrated that a mathematical model plays an essential part in constructing the emergency strategies that should be taken once foot-mouth-disease (FMD) is transmitted. Moreover, Dasbach et al. (2006) assert that the use of the epidemiological models in the cervical cancer cases, for example, has been applied in the development and evaluation of the screening strategies for the cancer issues. Therefore, there are no reasons to question about the success of mathematical models in supporting the formulation strategies related to preventive actions of the infectious diseases.

**Conclusion**

In conclusion, this essay has outlined the importance of mathematical models for overcoming the infectious disease transmission issues through covering three main focus areas, namely, abstraction of the reality, prediction accuracy, and conceptual frameworks for policy makers. Even though there is same critic related to modeling mathematics due to the complexity and the clarity of the assumptions which are covered by the models, the uncertainty of their accuracy in predicting the future trends, and their contributions in giving direction for the policy makers to formulate the preventive actions and strategies, it has been shown that there is a meteoric increase in the numbers of epidemiological models which are used to explain how the communicable diseases are spread. One reason is because the models are able to provide acceptable assumptions in reflecting the transmission of the infectious diseases. Additionally, they also offer accurate projections by drawing assessments quantitatively on hypothesis and determining sensitive alteration of values of estimators. Furthermore, the models enable the decision makers to confidently design preventive strategies by supplying crucial information like the effectiveness of insulations and the efficiency of vaccines that should be delivered. Therefore, by using mathematical models we would be able to have better illustrations about how the diseases are transmitted and anticipate the further actions for preventing the communicable disease transmission. Without modeling mathematics for epidemiology, it may be impossible to combat the infectious diseases and numbers of people who are suffered from the infectious diseases will continue to increase.

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