Cumulative viral load, an indicator of virulence, is controlled by the host's immune response.


 A viral strain may infect a host, proliferate rapidly, become controlled by immune reactions, and eventually be eliminated from the body. The virulence, or the magnitude of harm to the host due to infection, depends on the abundance and duration of the viral strain in the body, and the importance of the damaged tissue of the host. In this study, we investigated how the cumulative viral load (time-integral of the number of infected cells) depends on various factors, such as the viral growth rate, the effectiveness of immune cells to kill infected cells, speed of immune activation, formation of memory cells, and longevity of immune cells. In addition, viruses may produce a mutant with different antigen types, escape the immune reaction targeting the original type, and inflate virulence. We derived four simple formulas for the cumulative viral load that holds in different parameter regions. We analyzed the sensitivity of the cumulative viral load to the parameters in the model. Additionally, we discussed the reported correlation between virulence and molecular evolution rate. We conclude that viral virulence can be mitigated by enhancing the speed and effectiveness of immune reactions and by reducing the viral growth rate.

in the future. We consider the time integral of the viral abundance in the body until the 62 virus is cleared from the body and refer to it as the "cumulative viral load" [1,2,3]. 63 Virulence, or harm to host health, would increase with the cumulative viral load. 64 If the virus mutates to a type that has a different antigen specificity, the 65 immune reactions targeting the original type cannot suppress the mutant strain. Hence 66 the mutant strain escapes immune surveillance. To suppress the mutant viral strain, the 67 host immune system needs to develop cytotoxic T cells that are reactive to the new 68 antigen type. The host would suffer additional harm until the second strain is 69 eliminated. If one or more additional mutants of different antigen types are created, they 70 may cause a further increase in the cumulative viral load. We can evaluate the total 71 number of cells infected by the viral strain(s), including the initial type and all of its 72

descendants. 73
In the present study, we consider memory T cells that have a very long life. For 121 simplicity of analysis, we neglect the mortality of memory T cells, as assumed in [7]. 122 The dynamics are given as: 123 (1c) 126 Eq. (1a) indicates the dynamics of the number of infected cells in a host. We 127 assume that in the absence of an immune reaction, they proliferate exponentially with a 128 growth rate . Their increase is checked by the immune activity of cytotoxic T cells .  Eq. (1c) indicates the dynamics of memory T cells that are reactive to the viral 141 antigens. We assume that a small fraction of these cells become memory T cells with a 142 long life. Hence, their number increases at a rate proportional to the viral abundance , 143 with proportionality coefficient . We here neglect the mortality of memory T cells, as 144 assumed in [7]. 145 We consider a situation in which the initial dose of the viral infection is very 146 small, indicated by a small and positive number . Hence, the initial condition of Eq. 147 (1) is 148 (0) = , (0) = 0, and (0) = 0.
(2) 149 The trajectory and the final state depend on the value of . In this study, we are 150 interested in the limit when is very small ( → 0). .
(3a) 164 In Fig. 1(c), this is illustrated as the area under curve ( ). The amount of harm due to 165 viral infection is proportional to with the coefficient of proportionality ℎ " . 166 [harm due to the pathogen infection]＝ℎ " (3b) 167 ℎ " indicates the importance of the tissue attacked by the virus. If a mutation produces a 168 viral strain that escapes immune control, the host immune system must activate immune 169 cells with the mutant's antigen specificity, making the virulence even larger than Eq. 170 (3b). We will discuss this effect in a later section. 171 172

Four regions of simple parameter dependence 173
We first attempt to estimate , given by Eq. (3a). (5a) 190 Now, we focus on the time integral on the right-hand side of Eq. (5a). It is 191 where the last equality is derived from Eq. (4c) and Q (0) = 0. Because the dynamics 193 in Eq. (4) include only and , Eq. (5b) is a function of these two quantities. 194 When is large and is not small, ( , ) = , (6a) 204 When is small and is large, ( , ) = 2 , (6b) 205 When is very small and is large, When both and are small, The last value was numerically calculated for = = 0. Please see Appendix B for 208 the derivation of these limiting behaviors and the conditions under which they are valid. 209

Cumulative viral load in four parameter regions 220
Eq. (5) indicates that the cumulative viral load is the product of ( , ) 221 and the factor 3 ⁄ . By combining this with the four approximate formulas in Eq. 222 (6), we have formulas for the cumulative viral load , in these four different regions: 223 When is large and is not small, = When is small and is large, When is very small and is large, = 3.
(7c) 226 When both and are small, = 1.306 We can see that the cumulative viral load follows formulas different between 228 parameter regions, and hence its parameter dependence varies with and . 229  Table 1 shows these three cases, in which different 243 approximate formulas of cumulative viral load hold. In all three regions, the cumulative 244 viral load increases with , decreases with , but is independent of . For other 245 parameters ( , , and ), the dependence varies between regions, depending on = 246 ⁄ .
indicates the relative importance of direct enhancement of cytotoxic T cells 247

( ) and indirect enhancement via memory T cells ( and ). 248
In contrast, if the virus grows very slowly compared to the turnover time for

Sensitivity to intermediate parameter values 257
The parameter sensitivity of the cumulative viral load is represented in terms of 258 elasticity; the elasticity of for parameter This quantity is useful because it is independent of the choice of "unit" of each quantity, 260 and has been widely used to represent parameter sensitivity in economics and ecology 261 [10,11]. In this study, we adopted elasticity to represent the parameter sensitivity of the 262 cumulative viral load. = " + .   3). The expected number of first-generation strains is . These three strains produced 325 two second-generation strains ( 3 = 2). The expected number of second-generation 326 strains is 3 . Each of these two-step mutants may produce three-step mutants. The 327 expected total number of these three-step mutants is 8 . If we continue these 328 calculations, the expected number of total mutant strains starting from a single initial 329 strain is 1 + + 3 + 8 + 9 +. . . = . 330 The total cumulative viral load in the presence of mutations generating novel 331 antigen types is the one without mutations multiplied by this factor: 332 [total cumulative viral load] =

Four parameter regions 353
The time-integral of the viral load, , has mathematical properties worthy of 354 detailed examination. After rescaling variables, it becomes a quantity depending on 355 two combinations of variables: = ⁄ and = ⁄ . As shown in Fig. 2, the four 356 parameter regions exhibit simple parameter dependence. We derived four explicit 357 formulas that were quite accurate, as shown in Fig. 3. In this section, we discusse the effect of a large cumulative viral load, an 403 important factor of virulence, on the rate of molecular evolution, which is a question 404 that has not been discussed intensively. In contrast, the evolution of virulence levels for 405 pathogens has been the focus of many theoretical studies for more than two decades [12, 406 13, 14, 15, 16, 17, 18]. A traditional argument was that the virus that recently became to 407 attacked a host species may be very virulent, but they tend to evolve more benign, The parameter dependence of the cumulative viral load would be useful for 417 understanding the variation in virulence among strains and for designing drug therapy. 418 The analyses of parameter sensitivity (or elasticity) suggest effective options for 419 reducing the virulence of viral pathogens as follows: 420 When the longevity of the cytotoxic T cells is longer than the doubling time of 421 viral proliferation, we have the following three options: 422 (1) Reduce the growth rate of the viral strain (decrease ). 423 (2) Enhance the effectiveness of the immune system in killing the virus (increase ). 424 shown by sensitivity analyses (Table 1 and Fig. 4), which could be useful in designing 446 immunotherapy for viral pathogens. 447 Table 1 Directions  [I] [III] [IV] [II]